INTRODUCTION TO CALCULUS AND ANALYSIS Volume
One
Other Books by the Same Authors Richard Courant 1937; Vol. II, first edition, 1936
Differential and Integral Calculus, Interscience Publishers, Vol. I, second edition,
1950
Dirichlet's Principles, Conformal Mapping and Minimal Surfaces, Interscience Publishers,
I, 1953; Vol. II, 1962; Vol. III, in press.
Methods of Mathematical Physics (and D. Hilbert), Interscience Publishers, Vol.
Supersonic Flow and Shock Waves (and K. 0. Friedrichs), Interscience Publishers,
1948
Fritz John 1964
Partial Differential Equations (and L. Bers and M. Schechter), Interscience Publishers,
Equations, Interscience Publishers, 1955
Plane Waves and Spherical Means Applied to Partial Differential
Introduction to CALCULUS AND ANALYSIS Volume One
Richard Courant and Fritz John Courant I nstitute of Mathematical Sciences New York University
lnterscience Publishers
A Division of John New York
·
Wiley
London
and Sons, Inc. · Sydney
Copyright © 1965 by Richard Courant
All Rights Reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher. Library of Congress Catalog Card Number: 65-/6403 Printed in the United States of America
Preface
During the latter part of the seventeenth century the new mathe matical analysis emerged as the dominating force in mathematics. It is characterized by the amazingly successful operation with infinite processes or limits. Two of these processes, differentiation and inte gration, became the core of the systematic Differential and Integral Calculus, often simply called "Calculus," basic for all of analysis. The importance of the new discoveries and methods was immediately felt and caused profound intellectual excitement. Yet, to gain mastery of the powerful art appeared at first a formidable task, for the avail able publications were scanty, unsystematic, and often lacking in clarity. Thus, it was fortunate indeed for mathematics and science in general that leaders in the new movement soon recognized the vital need for writing textbooks aimed at making the subj ect ac cessible to a public much larger than the very small intellectual elite of the early days.
One of the greatest mathematicians of modern times,
Leonard Euler, established in introductory books a firm tradition and these books of the eighteenth century have remained sources of inspira tion until today, even though much progress has been made in the clarification and simplification of the material. After Euler, one author after the other adhered to the separation of differential calculus from integral calculus, thereby obscuring a key point, the reciprocity between differentiation and integration.
Only in
R . Courant's German Vorlesungen iiber Differential und Integralrechnung, appeared in the Springer-Verlag 1 927
when the first edition of
was this separation eliminated and the calculus presented as a unified subj ect.
From that German book and its subsequent editions the present work originated. With the cooperation of James and Virginia McShaue a greatly expanded and modified English edition of the "Calculus" w�s prepared and published by Blackie and Sons in Glasgow since v
1 934,
and
vi Preface
distributed in the United States in numerous reprintings by Inter science-Wiley. During the years it became apparent that the need of college and uni versity instruction in the United States made a rewriting of this work desirable. Yet, it seemed unwise to tamper with the original versions which have remained and still are viable. Instead of trying to remodel the existing work it seemed preferable to supplement it by an essentially new book in many ways related to the European originals but more specifically directed at the needs of the present and future students in the United States. Such a plan became feasible when Fritz John, who had already greatly helped in the prepara tion of the first English edition , agreed to write the new book together with R. Courant. While it differs markedly in form and content from the original, it is animated by the same i ntention : To lead the student directly to the heart of the subject and to prepare him for active application of his knowledge. It avoids the dogmatic style which conceals the motivation and the roots of the calculus in intuitive reality. To exhibit the interac tion between mathematical analysis and its various applications and to emphasize the role of intuition remains an i mportant aim of this new book. Somewhat strengthened precision does not, as we hope, inter fere with this aim. Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is unsuitable for students who seek intellectual independence rather than indoctrina tion ; disregard for applications and intuition leads to isolation and atrophy of mathematics. It seems extremely i mportant that students and instructors should be protected from smug purism. The book is addressed to students on various levels, to mathema ticians, scientists, engineers . It does not pretend to make the subj ect easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult ; most of them supplement the material in the text. In an additional pamphlet more
Preface vii
problems and exercises of a routine character will be collected, and moreover, answers or hints for the solutions will be given. Many colleagues and friends have been helpful. Albert A . Blank not only greatly contributed incisive and constructive criticism, but he also played a maj or role in ordering, augmenting, and sifting of the problems and exercises, and moreover he assumed the main responsi bility for the pamphlet. Alan Solomon helped most unselfishly and effectively in all phases of the preparation of the book. Thanks is also due to Charlotte John, Anneli Lax, R. Richtmyer, and other friends, including James and Virginia McShane. The fi rst volume i s concerned primarily with functions of a single variable, whereas the second volume will discuss the more ramified theories of calculus for functions of several variables. A final remark should be addressed to the student reader. It might prove frustrating to attempt mastery of the subj ect by studying such a book page by page following an even path. Only by selecting shortcuts first and returning time and again to the same q uestions and d ifficulties can one gradually attain a better understanding from a more elevated point. An attempt was made to assist users of the book by marking with an asterisk some passages which might impede the reader at his fi rst at tempt. Also some of the more difficult problems are marked by an asterisk. We hope that the work in the present new form will be useful to the young generation of scientists. We are aware of many imperfections and we sincerely invite critical comment which might be helpful for later improvements.
Richard Courant Fritz John J une 1965
Contents
Chapter
1
1
Introduction 1.1
1.2
1.3
1.4 1.5
The Continuum of Numbers a. The System of Natural Numbers and Its Extension. Counting and Measuring, 1 b. Real Numbers and Nested Intervals, 7 c. Decimal Fractions. Bases Other Than Ten, 9 d. Definition of Neighborhood, 1 2 e . Inequalities, 1 2
1
The Concept of Function b . Definition o f the a. Mapping-Graph, 1 8 Concept of Functions of a Continuous Variable. Domain and Range of a Function, 2 1 c . Graphical Representation. Monotonic Functions, 24 d. Continuity, 31 e . The Intermediate Value Theorem. Inverse Functions, 44
17
The Elementary Functions b. Algebraic a. Rational Functions, 47 Functions, 49 c. Trigonometric Functions, 49 d. The Exponential Function and the Logarithm, 5 1 e . Compound Functions, Symbolic Products, Inverse Functions, 52 Sequences
47
55
Mathematical Induction
57
ix
x
Contents
1.6
The Limit of a Sequence
I
8. Un
=
C. Un
=
-,
n
61
_!!___! ,
b.
azm
= -;
I
m
d. an
63
I
Uzm-1
=
zyp, 64
60
62 Zm'
e. an = a", 65 f. Geometrical Illustration of the Limits of
h.
a"
1.7
1.8
and
Un
=
n+
zyp, 65 � 69
=
g. The Geometric Series, 67 i.
an
=
�
-
y;;, 69
Further Discussion of the Concept of Limit
Definition of Convergence and Divergence, 70 b. Rational Operations with Limits, 71 c. Intrinsic Convergence Tests. Monotone Sequences, 73 d. Infinite Series and the Summation Symbol, 75 e. The Number e, 77 f. The Number 1r as a Limit, 80 a.
70
The Concept of Limit for Functions of a Continuous Variable
82
Some Remarks about the Elementary Functions, 86
a.
Supplements
87
S.1
89
Limits and the Number Concept
The Rational Numbers, 89 b. Real Numbers Determined by Nested Sequences of Rational Intervals, 90 c. Order, Limits, and Arithmetic Operations for Real Numbers, 92 d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria, 94 e. Least Upper Bound and Greatest Lower Bound, 97 f. Denumerability of the Rational Numbers, 98 a.
S.2
Theorems on Continuous Functions
S.3
Polar Coordinates
S.4
Remarks on Complex Numbers
PROBLEMS
99 101
103 106
Contents xi
Chapter
2
The Fundamental Ideas of the Integral and Differential Calculus
119
2.1
The Integral a. Introduction, 1 20 b. The Integral as an Area, 1 2 1 c. Analytic Definition of the Integral. Notations, 1 22
2.2
128 Elementary Examples of Integration a. Integration of Linear Function, 1 28 b. Integration of x2, 1 30 c. Integration of x"' for Integers a � - I , 131 d. Integration of x"' for Rational a Other Than -I, 1 34 e. Integration of sin x and cos x, 1 35
2.3
Fundamental Rules of Integration a. Additivity, 1 36 b. Integral of a Sum of a Product with a Constant, 1 37 c. Estimating Integrals, 1 38, d. The Mean Val ue Theorem for Integrals, 1 39
2.4
The Integral as a Function of the Upper Limit (Indefinite Integral) 143
2.5
Logarithm Defined by an Integral a. Definition of the Logarithm Function, 145 b. The Addition Theorem for Logarithms, 147
145
2.6
Exponential Function and Powers a. The Logarithm of the Number e, 1 49 b. The Inverse Function of the Logarithm. The Exponential Function, 1 50 c. The Exponential Function as Limit of Powers, 1 52 d. Definition of Arbitrary Powers of Positive Numbers, 1 52 e. Logarithms to Any Base, 1 53
149
2.7
The Integral of an Arbitrary Power of x
154
2.8
The Derivative a. The Derivative and the Tangent, 1 56 b. The Derivative as a Velocity, 162
155
120
136
xii
Contents c. Examples of Differentiation, 163 d. Some Fundamental Rules for Differentiation, 165 e. Differentiability and Continuity of Functions, 166 f. Higher Derivatives and Their Significance, 169 g. Derivative and Difference Quotient. Leibnitz's Notation, 171 h. The Mean Value Theorem of Differential Calculus, 173 i. Proof of the Theorem, 175 j. The Approximation of Functions by Linear Functions. Definition of Differentials, 179 k. Remarks on Applications to the Natural Sciences, 183
2.9
The Integral, the Primitive Function, and the Fundamental Theorems of the Calculus
184 The Derivative of the Integral, 184 b. The Primitive Function and Its Relation to the Integral, 186 c. The Use of the Primitive Function for Evaluation of Definite Integrals, 189 d. Examples, 191
a.
Supplement
The Existence of the Definite Integral
PROBLEMS
192
The Techniques of Calculus
201
Part A Differentiation and Integration of the Elementary Functions
201
of a Continuous Function
Chapter
3
3.1
196
The Simplest Rules for Differentiation and Their Applications
Rules for Differentiation, 201 b. Differentiation of the Rational Functions, 204 c. Differentiation of the Trigonometric Functions, 205
201
a.
3.2
The Derivative of the Inverse Function General Formula, 206 b. The Inverse of
a.
the nth Power; Lie nth Root, 210 Inverse Trigonometric Functions-
c.
The
206
Contents d. The Corresponding
Multivaluedness, 210 Integral Formulas, 215 3.3
Derivative and
Differentiation of Composite Functions c.
3.4
e.
Integral of the Exponential Function, 216
a. Definitions, 217
xiii
b. The Chain Rule, 218
217
The Generalized Mean Value Theorem of the
Differential Calculus, 222
Some Applications of the Exponential Function a. Definition of the Exponential Function by
223
Means of a Differential Equation, 223 b. Interest Compounded Continuously.
Radioactive Disintegration, 224 c. Cooling or Heating of a Body by a Surrounding Medium, 225
d. Variation of the
Atmospheric Pressure with the Height above the Surface of the Earth, 226 e. Progress of a Chemical Reaction, 227
3.5
f. Switching an
Electric Circuit on or off, 228 The Hyperbolic Functions a. Analytical Definition, 228
b. Addition
228
Theorems and Formulas for Differentiation 231 c.
3.6
The Inverse Hyperbolic Functions, 232
d. Further Analogies, 234
Maxima and Minima a.
Convexity and Concavity of Curves, 236
236
b. Maxima and Minima-Relative Extrema.
3. 7
Stationary Points, 238 The Order of Magnitude of Functions a. The Concept of Order of Magnitude. The
Simplest Cases, 248
b. The Order of
Magnitude of the Exponential Function and of the Logarithm, 249 c. General Remarks, 251
d. The Order of Magnitude of a Function in the Neighborhood of an Arbitrary Point, 252
Function Tending to Zero, 252 f. The "0" and "o" Notation for Orders of Magnitude, 253 e.
The Order of Magnitude (or Smallness) of a
248
xiv Contents APPENDIX
A.1
The Function y = eiix2, 255 b. The Function y = eiix, 256 c. The Function y = tanh 1/x, 257 d. The Function y = x tanh 1/x, 258 e. The Function y = x sin 1/x, y(O) = 0, 259 Some Special Functions
255
255
a.
A.2
Remarks on the Differentiability of Functions 259
Part B Techniques of Integration 3.8 3.9
261
Table of Elementary Integrals
263
The Method of Substitution
263
The Substitution Formula. Integral of a Composite Function, 263 b. A Second Derivation of the Substitution Formula, 268 c. Examples. Integration Formulas, 270 a.
3.10 Further Examples of the Substitution Method 271 274 3.1 1 Integration by Parts a. General Formula, 274 b. Further Examples of Integration by Parts, 276 c. Integral Formula for (b) + f(a), 278 d. Recursive Formulas, 278 •e. Wallis's Infinite Product for 1r, 280 3.12 Integration of Rational Functions 282 b. Integration a. The Fundamental Types, 283 of the Fundamental Types, 284 c. Partial Fractions, 286 d. Examples of Resolution into Partial Fractions. Method of Undetermined Coefficients, 288 3.13 Integration of Some Other Classes of Functions
Preliminary Rennarks on the Rational Representation of the Circle and the Hyperbola, 290 b. Integration of R(cos x, sin x), 193 c. Integration of a.
290
Contents xv R(cosh x, sinh x), 294 R( x, ·v}-=.--:;2), 294
d. Integration of
vg-=-1), 295 f. Integration of g. Integration of R( x, �). 295 R( x, v ax2 + 2bx +c), 295 h. Further R( x,
e.
Integration of
Examples of Reduction to Integrals of Rational Functions, 296
i. Remarks on the Examples,
297
Part C Further Steps in the Theory of Integral Calculus 3.14
Integrals of Elementary Functions a.
298 298
Definition of Functions by Integrals. Elliptic
Integrals and Functions, 298
b. On
Differentiation and Integration, 300
3.15
Extension of the Concept of Integral a.
Integrals, 301
b. Functions with Infinite
Discontinuities, 303 Areas, 304 e.
301
Introduction. Definition of "Improper" c.
Interpretation as g. The Dirichlet
Infinite Interval of Integration, 306
Gamma Function, 308 Integral, 309
f. The
d. Tests for Convergence, 305
h. Substitution. Fresnel
Integrals, 310
3.16
The Differential Equations of the
312
Trigonometric Functions a.
Introductory Remarks on Differential
Equations, 312 b. Sin x and cos x defined by a Differential Equation and Initial Conditions,
312
Chapter
4
PROBLEMS
314
Applications in Physics and Geometry
324
4.1
324
Theory of Plane Curves a.
Parametric Representation, 324
b. Change
of Parameters, 326 c. Motion along a Curve. Time as the Parameter. Example of the
xvi Contents Cycloid, 328 d. Classifications of Curves. Orientation, 333 e. Derivatives. Tangent and Normal, in Parametric Representation, 343 f. The Length of a Curve, 348 g. The Arc Length as a Parameter, 352 h. Curvature, 354 i. Change of Coordinate Axes. I nvariance, 360 j. Uniform Motion in the Special Theory of Relativity, 363 k. Integrals Expressing Area within Closed Curves, 365 I. Center of Mass and Moment of a Curve, 373 m. Area and Volume of a Surface of Revolution, 374 n. Moment of Inertia, 375 4.2
4.3
Examples a. The Common Catenary, 378 Lemniscate, 378
Cycloid,
c.
376
b.
The
376
The Ellipse and the
Vectors in Two Dimensions 379 Definition of Vectors by Translation. Notations, 380 b. Addition and Multiplication of Vectors, 384 c. Variable Vectors, Their Derivatives, and Integrals, 392 d. Application to Plane Curves. Direction, Speed, and Acceleration, 394 a.
4.4
Motion of a Particle under Given Forces 397 Newton's Law of Motion, 397 b. Motion of Falling Bodies, 398 c. Motion of a Particle Constrained to a Given Curve, 400
a.
4.5
Free Fall of a Body Resisted by Air
402
4.6
The Simplest Type of Elastic Vibration
404
Motion on a Given Curve The Differential Equation and Its Solution, 405 b. Particle Sliding down a Curve, 407 c. Discussion of the Motion, 409 d. The Ordinary Pendulum, 410 e. T he Cycloidal Pendulum, 411
405
4. 7
a.
4.8
4.9
Contents
xvii
Motion in a Gravitational Field 413 a. Newton's Universal Law of Gravitation, 413 b. Circular Motion about the Centt"r of Attraction , 415 c. Radial Motion-Escape Velocity, 416
Work and Energy 418 Work Done by Forces during a Motion, 418 b. Work and Kinetic Energy. Conservation of Energy, 420 c. The Mutual Attraction of Two Masses, 421 d. The Stretching of a Spring, 423 e. The Charging of a Condenser, a.
423
APPENDIX
A.1 Properties of the Evolute A.2 Areas Bounded by Closed Curves. Indices PROBLEMS
Chapter
5
Taylor's Expansion
424
424
430 435
440
440
5.1
Introduction: Power Series
5.2
Expansion of the Logarithm and the Inverse 442 Tangent b. The Inverse a. The Logarithm, 442 Tangent, 444
5.3
5.4
5.5
Taylor's Theorem 445 a. Taylor's Representation of Polynomials, 445 b. Taylor's Formula for Nonpolynomial Functions, 446 Expression and Estimates for the Remainder 447 a. Cauchy's and Lagrange's Expressions, 447 b. An Alternative Derivation of Taylor's Formula, 450 Expansions of the Elementary Functions a. The Exponential Function, 453
453
xviii Contents
b. Expansion of sin x, cos x, sinh The Binomial Series, 456
c.
x,
cosh
x,
454
Geometrical Applications Contact of Curves, 458 b. On the Theory of Relative Maxima and Minima, 461
5.6
457
a.
APPENDIX I
462
A.I.l
Example of a Function Which Cannot Be Expanded in a Taylor Series 462
A.I.2
Zeros and Infinites of Functions 463 a. Zeros of Order n, 463 b. Infinity of Order v,
463
A.I.3
Indeterminate Expressions
A.I.4
The Convergence of the Taylor Series of a Function with Nonnegative Derivatives of all Orders 467
APPENDIX
II INTERPOLATION
464
470
A.II.l The Problem of Interpolation. Uniqueness 470
Chapter
6
A.II.2 Construction of the Solution. Newton's Interpolation Formula
471
A.II.3 The Estimate of the Remainder
474
A.II.4 The Lagrange Interpolation Formula
476
PROBLEMS
477
Numerical Methods
481
6.1
482
Computation of mtegrals Approximation by Rectangles, 482 b. Refined Approximations-Simpson's Rule, 483 a.
Contents xix
6.2
Other Examples of Numerical Methods
a. The "Calculus of Errors", 490 b. Calculation of 1r, 492 c. Calculation of Logarithms, 493 6.3
Numerical Solution of Equations
a. Newton's Method, 495 b. The Rule of False Position, 497 c. The Method of Iteration, 499 d. Iterations and Newton's Procedure, 502
APPENDIX
A.l
Stirling's Formula
PROBLEMS
Chapter
7
Infinite Sums and Products 7.1
7.2
7.3 7.4
490
494
504
504 507
510
The Concepts of Convergence and Divergence 511 Basic Concepts, 511 b. Absolute Convergence and Conditional Convergence, 513 c. Rearrangement of Terms, 517 d. Operations with Infinite Series, 520 a.
Tests for Absolute Convergence and 520 Divergence a. The Comparison Test. Majorants, 520 b. Convergence Tested by Comparison with the Geometric Series, 521 c. Comparison with an Integral, 524 Sequences of Functions Limiting Processes with Functions and Curves, 527
a.
526
529 Uniform and Nonuniform Convergence a. General Remarks and Definitions, 529 b. A Test for Uniform Convergence, 534 c. Continuity of the Sum of a Uniformly Convergent Series of Continuous Functions, 535 d. Integration of Uniformly Convergent Series, 536 e. Differentiation of Infinite Series, 538
xx Contents 7.5
7.6
7.7
Power Series a. Convergence Properties of Power Series Interval of Convergence, S40 b. Integration and Differentiation of Power Series, 542 c. Operations with Power Series, 543 d. Uniqueness of Expansion, 544 e. Analytic Functions, 545
540
Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples 546 a. The Exponential Function, 546 b. The Binomial Series, 546 c. The Series for arc sin x, 549 d. The Series for ar sinh X = Jog [X + VCJ + x2)], 549 e. Example of Multiplication of Series, 550 f. Example of Term-by-Term Integration (Elliptic Integral), 550 Power Series with Complex Terms 551 a. Introduction of Complex Terms into Power Series. Complex Representations of the Trigonometric Function, 551 b. A Glance at the General Theory of Functions of a Complex Variable, 553
APPENDIX
A. I Multiplication and Division of Series a. Multiplication of Absolutely Convergent Series, 555 b. Multiplication and Division of Power Series, 556
555
555
A.2 Infinite Series and Improper Integrals
557
A.3 Infinite Products
559
A.4 Series Involving Bernoulli Numbers
562
PROBLEMS
564
Contents
Chapter
8
8.2
Periodic Functions a.
General Remarks. Periodic Extension of a Function, 572 b. I ntegrals Over a Period, 573 c. Harmonic Vibrations, 574 Superposition of Harmonic Vibrations
a.
b.
8.3
8.4
Harmonics. Trigonometric Polynomials, Beats, 577
576
Complex Notation a.
General Remarks, 582 b. Application to Alternating Currents, 583 c. Complex Notation for Trigonometrical Polynomials, 585 d. A Trigonometric Formula, 586 Fourier Series a.
Fourier Coefficients,
588
j·oo
sin
z
i
571
"Trigonometric Series 8.1
xx
587
dz =
b.
Basic Lemma,
572
576
582
587
1!"
, 589 2 d. Fourier Expansion for the Function (x) = x, 591 e. The Main Theorem on Fourier Expansion, 593 c.
8.5
8.6
Proof of
0
��
z
-
Examples of Fourier Series
Preliminary Remarks, 598 b. Expansion of the Function (x) = xz, 598 c. Expansion of x cos x, 598 d. The Functionf(x) = lxl, 600 e. A Piecewise Constant Function, 600 f. The Function sin lxl, 601 g. Expansion of cos Jl.X. Resolution of the Cotangent into Partial Fractions. The Infinite Product for the Sine, 602 h. Further Examples, 603
a.
Results, 604
Bessel's Inequality, 604
Further Discussion of Convergence a.
b.
598
604
xxii
Contents 605
c. Proof of Corollaries (a), (b), and (c),
d. Order of Magnitude of the Fourier
Coefficients Differentiation of Fourier Series, 607
8. 7
Approximation by Trigonometric and Rational Polynomials 608 a. General Remark on Representations of Functions, 608 Theorem, 608
b. Weierstrass Approximation c. Fejers Trigonometric
Approximation of Fourier Polynomials by Arithmetical Means, 610
d. Approximation
in the Mean and Parseval's Relation, 612 APPENDIX
I
614
A.I.l
Stretching of the Period Interval. Fourier's Integral Theorem 614
A.I.2
Gibb's Phenomenon Discontinuity
A.I.3
at Points
of 616
Integration of Fourier Series
APPENDIX II
618 619
A.II.l Bernoulli Polynomials and Their Applications a. Definition and Fourier Expansion, 619 b. Generating Functions and the Taylor Series
619
of the Trigonometric and Hyperbolic Cotangent, 621
c. The Euler-Maclaurin
Summation Formula, 624
Asymptotic Expressions, 626
d. Applications. e. Sums of
f. Euler's Constant and
Power Recursion Formula for Bernoulli Numbers, 628
Stirling's Series, 629 PROBLEMS
631
Contents xxiii
Chapter
9
Differential Equations for the Simplest Types of Vibration 9.1
Vibration Problems of Mechanics and Physics 634
a.
b.
9.2
The Simplest Mechanical Vibrations, Electrical Oscillations, 635
634
Solution of the Homogeneous Equation. Free Oscillations
The Fornal Solution, 636 b. Physical Interpretation of the Solution, 638 c. Fulfilment of Given Initial Conditions. Uniqueness of the Solution, 639
a.
9.3
The
Nonhomogeneous
Equation.
Forced
Oscillations
General Remarks. Superposition, 640 b. Solution of the Nonhomogeneous Equation, 642 c. The Resonance Curve, 643 d. Further Discussion of the Oscillation, 646 e. Remarks on the Construction of Recording Instruments, 647 a.
List of Biographical Dates Index
633
636
640
650 653
1 Introduction
Since antiq uity the intuitive notions of contin uous change, growth, and motion , have challenged scientific minds. Yet, the way to the understanding of continuous variatio n was opened only in the seven teenth century when modern science emerged and rapid ly developed in close conjunction with integral and d i fferential calculus, briefly called calculus, and mathematical analysis. The basic notions of Calculus are derivative and integral : the derivative is a measure for the rate of change, the integral a measure for the total effect of a process of conti n uous change. A precise under standing of these concepts and their overwhelming fruitfulness rests upon the concepts of limit and of function which in turn depend upon an understanding of the continuum of numbers. Only grad ually, by penetrating more and more i nto the substance of Calculus, can one appreciate its power and beauty. In this introductory chapter we shal l explain the basic concepts of number, function, and limit, at first simply and intuitively, and then with carefu l argument. 1.1
The Continuum of Numbers
The positive integers or natural numbers I , 2, 3, . . . are abstract symbols for in dicati ng "how many" objects there are in a collection or set of d iscrete elements. These symbols are stripped of all reference to the concrete qualities of the objects counted, whether they are persons, atoms, houses, or any objects whatever. The natural numbers are the adequate instrument for counting elements of a collection or "set." However, they do not suffice for another equally important objective : to measure quantities such as the length of a curve and the volume or weight of a body. The question, 1
2 Introduction
Ch.
1
"how much ?", cannot be answered immediately in terms of the natural numbers. The profound need for expressing measures of quantities in terms of what we would like to call numbers forces us to extend the number concept so that we may describe a continuous gradation of measures. This extension is called the number continuum or the system of "real n umbers" (a nondescriptive but generally accepted name). The extension of the number concept to that of the continuum is so convincingly natural that it was used by all the great mathematicians and scientists of earlier times without probing questions. Not until the nineteenth century did mathematicians feel compelled to seek a firmer logical foundation for the real number system. The ensuing precise formulation of the concepts, in turn, led to further progress in mathe matics. We shall begin with an unencumbered intuitive approach, and later on we shall give a deeper analysis of the system of real numbers.' a. The System of Natural Numbers and Its Extension. Counting and Measuring
The Natural and the Rational Numbers. The sequence of "natural" numbers I, 2, 3, . . . is considered as given to us. We need not discuss how these abstract entities, the numbers, may be categorized from a philosophical point of view. For the mathematician, and for anybody working with numbers, it is important merely to know the rules or laws by which they may be combined to yield other natural numbers. These laws form the basis of the familiar rules for adding and multiplying n umbers in the decimal system ; they i nclude the commutative laws a + b = b + a and ab = ba, the associative laws a + (b + c) = (a + b) + c and a(bc) = (ab)c, the distributive law a(b + c) = ab + ac, the cancellation law that a + c = b + c implies a = b, etc. The inverse operations, subtraction and division, are not always possible within the set of natural numbers ; we cannot subtract 2 from I or divide I by 2 and stay within that set. To make these operations possible without restriction we are forced to extend the concept of number by inventing the number 0, the "negative" integers, and the fractions. The totality of all these numbers is called the class or set of rational numbers; they are all obtained from unity by using the "rational operations" of calculation, namely, addition, s ubtraction, multiplication, and division. 2 A rational number can always be written i n the form pfq, where p 1 A more complete exposition is given in What Is Mathematics? by Courant and Robbins, Oxford University Press, 1962. The word "rational" here does not mean reasonable or logical but is derived from the word "ratio" meaning the relative proportion of two magnitudes. 2
Sec. 1 . 1
The Continuum of Numbers 3
and q are integers and q ¥- 0. We can make thi s representation u nique by requiring tha't q i s positive and that p and q have no common factor larger than I. Within t h e domain o f rational num bers a l l t h e rational operations, addition, m u ltiplication, subtraction , and divi sion (except division by zero), can be performed and prod uce aga i n rational n umbers. As we know from elementary arithmetic, operations with rational n umbers obey the same laws as operations with natural num bers: thus the rational numbers extend the system of positive i ntegers in a com pletely straightforward way.
Graphical Representation of Rational Numbers. Rational numbers are usually represented graph ically by points on a straight l i ne L, the number axis. Tak i n g an arbitrary poi n t of Las the origi n or poi nt 0 ------+--+-- L p 0 I--
-x
--------�--�-+- L 0 p -----1
Figure 1.1
The number axis.
and another arbitrary poi n t as the poi nt I, we use the d istance between these two poi nts to serve as a scale or unit of meas u rement and define the direction from 0 to I as "posi ti ve." The l i n e with a d i rection thus imposed is called a directed line. It is customary to depict L so that the point I i s to the right of the poi n t 0 ( Fig. !.I). The location of any point P on L is completely determi ned by two pieces of i nformation: the di stance of P from the origin 0 and the d i rection from 0 to P (to the right or left of 0). The point P on L representing a positive rational number l ies at d istance x units to the right of 0. A negative rational number x i s represented by the point -x u n i ts to the left ofO. In ei ther case the di stance from 0 to the poi nt which represents is called the absol ute val ue of x, wri tten lxl, and we have
lxl
=
{
x, - x,
if x is positive or zero,
if x i s negative.
x
We note that lxl i s never negative and equals zero only when x = 0.
4
Ch. 1
Introduction
From elementary geometry we recall that with ruler and compass it is possible to construct a subdivision of the unit length into any number of equal parts. It follows that any rational length can be constructed and hence that the point representing a rational number x can be found by purely geometrical methods. In this way we obtain a geometrical representation of rational numbers by points on L, the rational points. Consistent with our notation for the points 0 and I, we take the liberty of denoting both the rational number and the corresponding point on L by the same symbol x. The relation x < y for two rational numbers means geometrically that the point x lies to the left of the point y. Tn that case the distance between the points is y x units. If x > y, the distance is x y units. In either case the distance between two rational points x, y of L i s I Y xl units and is again a rational number. -
-
-
--�+--r----�+-�-- L P P +I 0 ! 1 p
q
q
q-q-
Figure 1.2 A segment on L with end points a, b where a < b will be called an interval. The particular segment with end points 0, I is called the unit interval. If the end points are i ncl uded in the i nterval, we say the interval
is closed; if the end poi nts are excluded, the i nterval is called open. The open i nterval, denoted by (a, b), consists of those points x for which a < x < b, that is, of those points that lie "between" a and b. The closed interval, denoted by [a, b ], consists of the points x for which a s x s b.1 In either case the length of the interval is b a. The points corresponding to the i ntegers 0, ±I, ±2, . . subdivide the number axis into intervals of unit length. Every point on L is either an end point or interior point of one of the intervals of the subdivision. If we further subdivide every interval i nto q equal parts, we obtain a subdivision of L into intervals of length 1/q by rational points of the form p/q. Every point P of Lis then either a rational point of the form p/q or lies between two successive rational points p/q and (p + I)/q (see Fig. 1.2). Since successive points of subdivision are 1 /q units apart, it follows that we can find a rational point p/q whose distance from P does not exceed 1/q units. The number 1/q can be made as small as we please by choosing q as a sufficiently large positive integer. For example, choosing q = 1 0n (where n is any natural number) we can -
.
1
The relation a � x (read "a less than .:>r equal to x") is interpreted as "either x, or a = x . " We interpret the double signs � and ± in similar fashion .
a <
Sec. 1.1
The Continuum of Numbers 5
find a "decimal fraction" x = p/10" whose distance from P is less than 1 / 1 0" . Although we do not assert that every point of L is a rational point we see at least that rational points can be found arbitrarily close to any point P of L.
Density The arbitrary closeness of rational points to a given point P of L is expressed b y saying : The rational points are dense on the number axis. It is clear that even smaller sets of rational numbers are dense, for example, the points x = p/10", for all natural numbers n and integers p. Density i mplies that between any two distinct rational points a and b there are infinitely many other rational points. In particular, the point halfway between a and b, c = Ha + b), corresponding to the arithmetic mean of the numbers a and h, is again rational. Taking the midpoints of a and c, of b and c, and continuing in this manner, we can obtain any number of rational points between a and b. An arbitrary point P on L can be located to any degree of precision by using rational points. At first glance it might then seem that the task of locating P by a number has been achieved by introducing the rational numbers. After al l , in physical reality q uantities are never given or known with absolute precision but always only with a degree of uncertainty and therefore might just as well be considered as measured by rational numbers.
Incommensurable Quantities . Dense as the rational numbers are, they do not suffice as a theoretical basis of measurement by numbers. Two quantities whose ratio is a rational number are called commen surable because they can be expressed as integral multiples of a common unit. As early as in the fifth or sixth century B.C. G reek mathematicians and philosophers made the surprising and profoundly exciting dis covery that there exist quantities which are not commensurable with a given u nit. In particular, line segments exist which are not rational multiples of a given u nit segment. It is easy to give an example of a length incommensurable with the unit length : the diagonal / of a square with the sides of unit length . For, by the theorem of Pythagoras, the square of this length I must be equal to 2. Therefore, if I were a rational number and consequently equal to pfq, where p and q are positive integers, we should have p2 = 2q2 We can assume that p and q have no common factors, for such common factors could be canceled out to begin with . According to the above equation, p2 is an even number ; hence p itself must be even, say p = 2p'. Substituting 2p' for p gives us 4p' 2 = 2q 2 or q 2 = 2p'2; consequently, q2 •
,
6
Ch. I
Introduction
is even and so q is also even. This proves that p and q both have the factor 2. However, this contradicts our hypothesis that p and q have no common factor. Since the assumption that the diagonal can be represented by a fraction p/q leads to a contradiction, it i s false. This reasoning, a characteristic example of indirect proof, shows that the symbol .J2 cannot correspond to any rational number. Another example is 1r , the ratio of the circumference of a circle to its diameter. The proof that 1r is not rational is much more complicated and was obtained only in modern times (Lambert, 1 761 ). It is easy to find many incommensurable quantities (see Problem I, p. 1 06); in fact, i ncom mensurable quantities are in a sense far more common than the commensurable ones (see p. 99). Irrational Numbers
Because the system of rational numbers is not sufficient for geom etry, it is necessary to invent new numbers as measures of incommen surable q uantities : these new numbers are called "irrational . " The ancient Greeks did not emphasize the abstract number concept, but considered geometric entities, such as line segments, as the basic elements. In a purely geometrical way, they developed a logical system for dealing and operating with i ncommensurable quantities as well as commensurable (rational) ones. This important achieve ment, i nitiated by the Pythagoreans, was greatly advanced by Eudoxus and is expressed at length in Euclid's famous Elements. In modern times mathematics was recreated and vastly expanded on a foundation of number concepts rather than geometrical ones. With the introduction of analytic geometry a reversal of emphasis developed i n the ancient relationship between numbers and geometrical quantities and the classical theory of incommensurables was all but forgotten or disre garded . It was assumed as a matter of course that to every point on the number axis there corresponds a rational or irrational number and that this totality of "real" numbers obeys the same arithmetical laws as the rational numbers do. Only later, in the nineteenth century, was the need for justifying such an assumption felt and was eventually completely satisfied in a remarkable booklet by Dedekind which makes fascinating reading even today.1
1 R. Dedekind, "Nature and Meaning of Number" i n Essays on Number, London and Chicago, 1901. (The first of these essays, "Conti n uity and Irrational Numbers," supplies a detailed account of the definition and laws of operation with real num bers.) Reprinted under title Essays on the Theory of Numbers, Dover, New York, 1964. The original of these translations appeared i n 1887 under the title W as sind und wass sollen d i e Zahlen ?" "
Sec. 1.1
The Continuum of Numbers 7
In effect, Dedekind showed that the "naive" approach practiced by all the great mathematicians from Fermat and Newton to Gauss and Riemann was on the right track : That the system of real n umbers (as symbols for the lengths of segments, or otherwise defined) is a consistent and complete instrument for scientific measurement, and that in this system the rules of computation of the rational n umber system remain valid . Without harm, one could leave i t at that and turn directly to the substance of calculus. However, for a deeper understanding of the concept of real n umber, which is necessary for our later work, the following account as well as the Supplement to this chapter should be studied. h. Real Numbers and Nested Intervals
For the moment let us think of the points on a line L as the basic elements of the continuum . We postulate that to each point on L there corresponds a " real number" x, its coordinate, and that for these n umbers x, y the relationships j ust described for the rational numbers retain their meaning. I n particular, the relationship x < y indicates order on L and the expression IY x l means the d istance between the point x and the point y. The basic problem is to relate these numbers (or measurements on the geometrically given continuum of points) to the rational n umbers considered originally and hence ultimately to the integers. In addition, we have to explain how to operate with the elements of this "num ber-contin uum" in the same way as with the rational numbers. Eventually, we shall formulate the concept of the continuum of numbers independently of the intuitive geometric con cepts, but for the present we postpone some of the more abstract discussion to the Supplement. How can we describe an irrational real number? For some numbers such as .J2 or rr, we can give a simple geometric characterization, but that i s not always feasible. A method flexible enough to yield every real point consists in describing the value x by a sequence of rational approximations of greater and greater precision. Specifically, we shall approximate x simultaneously from the right and from the left with successively increasing accuracy and in such a way that the margin of error approaches zero. In other words, we u se a "sequence" of rational intervals containing x, with each interval of the sequence containing the next one, such that the length of the i nterval, and with i t the error of the approximation, can be made s maller than any specified positive number by taking intervals sufficiently far along in the sequence. -
Ch. I
8 Introduction To begin, let
x
be confined to a closed i nterval /1
[a1, b1], that is,
=
a1 .::;; x .::;; b1,
where a1 and b1 are rational (see Fig. 1 .3). Within /1 we consider a "subinterval" I2 = [a2, b2] contai ning x, that is, a, .::;; a 2 .::;; x .::;; b2 .::;; b" where a2 and b2 are rational. For example, we may choose for I2 one of the halves of /1, for x must lie in one or both of the half-intervals. Within I2 we consider a subinterval /3 = [a3, b3] which also contains x : a1 .::;; a2 .::;; a3 .::;; x .::;; b 3 .::;; b2 .::;; b1,
where a9 and b3 are rational, etc. We require that the length of the interval In tends to zero with increasing n ; that is, that the length of In is less than any preassigned positive number for all sufficiently large n. A set of closed i ntervals I1, I2, I3 , each containing the •
•
.
--��-----+-1--+---+--�- L a1 a2 b n + l bn an Un + l bz b1 X
Figure 1 .3
A nes te d sequence of intervals.
next one and such that the lengths tend to zero will be called a "nested sequence of intervals." The point x is u niquely determined by the nested sequence ; that is, no other point y can lie in all I since the distance between x and y would exceed the length of In once n is suffi ciently large. Since h ere we always choose rational points for the end points of the In and since every i nterval w ith rational end points i s described b y two rational numbers, w e see that every point x of L, that is, every real number, can be precisely described with the help of infinitely many rational numbers. The converse statement is not so obvious ; we shall accept it as a basic axiom . n•
POSTULATE OF NESTED I N TERVALS . If I1 , 12, 13, form a nested sequence of intervals with rational end points, there is a point x contained in all I.-' •
.
•
As we shall see, this is an axiom of continuity : it guarantees that no gaps exist on the real axis. We shall use the axiom to characterize the real continuum and to j ustify all operations with limits which are 1 It is important to emphasize for a nested <equence that the inter v al s I,. are closed. If, for example, I,. denotes the open inter v al 0 < x < 1/n, th en each /,. contains the following one and the lengths of the intervals tend to zero ; but there is no x contained in all I,. .
The Continuum of Numbers 9
Sec. 1.1
basic for calculus and analysis. (There also are many other ways of formulating this 1\Xiom as we sha l l see later.) c. Decimal Fractions. Bases Other than Ten
Infinite Decimal Fractions. One of the many ways of defining real n umbers is the familiar description in terms of infinite decimals. It i s entirely possible t o take the infinite decimals a s the basic objects rather than the point s of the number axis, but we would rather proceed in a more suggestive geometrical way by defining the infinite decimal repre sentation of real numbers in ter ms of nested sequences of intervals. Let the number axis be subdivided i nto unit intervals by the points corresponding to i n tegers. A poi n t x eit he r lies between two successive points of subdivision or is i tsel f one of the dividing poin ts . In either case there is at least one integer c0 such that c0 � x � c0
+ I,
so that x belongs to the closed i nterval I0 = (c0, c0 + 1 ]. We d ivi d e , c0 + 190 • /0 i n to ten equal parts by po i n ts c0 + /0 , c0 + 120 , The poi nt x must then belong to at least one of the closed subintervals of /0 (possi bly to two adj acen t o ne s if x is one of the poi nts of s ub d i vi s i on ). In other words, there is a digit c1 ( that is, one of the i ntegers 0, I , 2, . . . , 9) s u ch that x belongs to the closed interval /1 given by •
•
•
Dividing / 1 in turn into ten equal parts, we find a digit c2 such that
lies in the interval /2 given by
We repeat this process. A fter given by
n steps x
x
is confined to an interval In
1 I 1 1 1 c0 + - c 1 + · · · + - c < x< c0 + - c + · · · + - c n + - , 1 0" 1 0" 10 1 1 0n 10 n
w here c1, c2 , are all digits. The i n terval /" has l e ngth 1 / 1 0", which tends to zero for i n crea sin g n . It i s clear that the In form a nested set of •
•
•
i ntervals, and hence that x is determined uniquely by the ln. Since the /" are known, once the numbers c0, c1 , c2 , • • • are given we find that an arbitrary real number can be described com p letely by an infi nite sequence of integers c0, c1, c 2 , , where all except the first are digits, .
•
•
10 Introduction
Ch. 1
having values from zero to nine only. In ordinary decimal notation the connection between x and c0, c1, c2, is indicated by writing x =
c0
+ O.c1 C2C3 • • • • .
•
•
( Usually, the integer c0 itself is also written i n decimal notation if c0 is positive.) Conversely, by the axiom of continuity, every such expression denoting an i nfinite decimal fraction represents a real number. It is possible that there are two different decimal representations of the same number ; for example,
· · · = 1 .00000 · · · .
I = 0. 99999
In our construction the integer c0 is determined uniquely by x unless x itself is an i ntege r . In that case we could choose either c0 = x or c0 = x - I . Once a choice has been made the d i gi t c1 is unique unless x is one of the new points subdividing !0 into ten equal parts. Con tinuing we find that c0 an d all ck are determined u n i que ly by unless occurs as a p o i n t of subdivision at some stage. I f this should happen for the fi r st ti me at the nth stage, then
x
x = c0
x
I c , + I c +"·+" 10 1
-
10"
wh e re c1 o c2 , c ,. are d i gi ts and w here c, > 0, s ince otherwise x would have bee n a point of subdivision at an earlier stage. lt fol lows that /,+ 1 is either th e i n terval [x, x + 1 / I O" P ] o r the interval [x - 1 / 1 0"+1 , x]. In t h e first case x will be the left-hand end point of all later i nte r va l s !, + 2 , !,_, 3 , • • • and i n the second case, the right-hand end point. We are then led either to the decimal representation •
•
•
,
,
x = c0 + O.c1c2 • • • c,OOO · · ·
or the representation x = c0
+ O.c1c2 •
• • (c, - 1 )99999
···.
Hence the only case in which an ambiguity can arise is for a rational n u m be r x which can be written as a fraction having a power of ten for its d e n ominator. We can el iminate even this a mbi g ui ty by excluding decimal representations in which all d i gits from a certain point on are nines. In the decimal representation of real n u m bers the special role played by the number ten is purely i ncidental. The only evident reason for the widespread use of the decimal system is the ease of c ou nti n g by tens on our fingers (digits). Any integer p greater than one can serve equally well . We could use p equa� subdivisions at each stage. A real
The Continuum of Numbers I I
Sec. 1.1
number x would then be represented i n the form
where c0 is an i nteger, and now c1, c 2 , have one of the values 0, 1, 2, . . . , p 1 . This representation again characterizes x by a nested set of i ntervals, namely •
•
•
-
1 1 1 1 1 Co + - C1 + " " " + ---;; Cn � X � Co + C1 + " · + ---;; C n + ---;; . p p p p p -
"
1f x is positive or zero, the integer c0 is also positive or zero and itself has a finite expansion of the form Co
=
do + pd1 + p 2 d2 + · · · + p kdk,
where d0, d1, dk take one of the values 0, I, . . . , p complete representation of x "to the base p" takes the form X = dkdk-1 . . . dl do.C1C 2 C3 . • • • •
•
•
c0
,
-
1.
The
If x is negative, we m ay use this kind of representation for - x. 101.01 '----"""'
--�--�-----r--,_--�--_,-r+--+- L 0
Figure 1 . 4 The fraction
101
100
11
10
�.'
lol.i' l l l
i n the binary sy stem.
Bases other than 10 have actually been used extensively. Following the lead of the ancient Babylonians, astronomers for many centuries consistently represented numbers as "sexagesimal" fractions with p = 60 as the base. Binary Representation. The "binary" system with the base p = 2 has special theoretical interest and is useful i n the logical design of computing machines. In the binary system the digits h ave only two possible values, zero and one. The number �; , for example, would be written 1 0 1 .0 1 corresponding to the formula 21
- =
4
1 2 2 . 1 + 2 0 + 1 •
.
1 +
1 . l . 1 0+ 22 2
-
-
(see Fig. 1 .4) .
Calculating with Real Numbers. Although the definition of real numbers and their infinite decimal or binary representations, etc., are straightforward, it may not seem obvious that one can operate with the
12
Introduction
Ch.
I
number continuum exactly as with rational numbers, performing the rational operations and retaining the laws of arithmetic, such as the associative, the commutative, and the d istributive laws. The proof is simple, although somewhat tedious. Instead of i mpeding the way to the live substance of analysis by taking up the question here, we shall accept temporarily the possibility of ordinary arithmetic calculation with the real numbers. A deeper understanding of the logical structure underlying the number concept will come when we discover the idea of limit and its implications . (See the Supplement to this chapter, p. 89.) d. Definition of Neighborhood
Not only the rational operations but also order relations or in equalities for real numbers obey the same rules as for the rational numbers. Pairs of real numbers a and b with a < b again give rise to closed intervals [a, b] (given by a :::; x :::; b) and open intervals (a, b) (given by a < x < b). Frequently we shall be led to associate with a point x0 the various open i ntervals that contain that point or specifically have it as center, which we shall call neighborhoods of the point. More precisely, for any positive E the £-neighborhood of the point x0 consists of the values x for which x0 - E < x < x0 + £, th a t is, it i s the interval (x0 - £, x0 + £) . Any open interval (a, b) containing a point x0 always also contain s a whole neighborhood of x0. H aving defined intervals with real end points we can now form nest ed sequences of i ntervals using the same definition as in the case of rational end points. It i s most i mportant for the logical consistency of calculus that for any nested sequence of i ntervals with real end points there i s a real n umber contained in all of them . (See Supplement, p. 95.) e. Inequalities
Basic Rules
Inequalities play a far larger role in higher mathematics than i n elementary mathematics. Often the precise value o f a quantity x is difficult to determine, whereas it may be easy to make an estimate of x , that is, to show that x i s greater than some known quantity a and less than some other quantity b. For many purposes, only the information contained in such an estimate of x is significant . We shall therefore briefly recall some of the elementary rul e s about i nequalities . The basic fact is that the sum and product of two positive real numbers are again positive ; that is, if a > 0 and b > 0, then a + b > 0
Sec. 1. 1
The Continuum of Numbers
13
and ab > 0. Moreover, we rely on the fact that the inequality a > b is equivalent to a - b > 0. Consequently, two i nequalities a > b and c > d can be added to yield the inequality a + c > b + d since
(a + c) - (b + d) = (a - b) + ( c - d) is positive as the sum of two positive numbers. (Subtracting the inequalities to obtain a - c > b - d is not legitimate. Why?) An inequality can be multiplied by a positiz•e number ; that is, i f a > b and c > 0, then ac > be. For the proof, we observe that
ac - be = (a - b)c is positive since it is the product of positive numbers. If c is negative, we can conclude from a > b that ac < be. More generally, it follows from a > b > 0 and c > d > 0 that ac > bd. I t is geometrically obvious that ineq uality is transitive ; that is, i f a > b and b > c , then a > c. Transitivity 1 also follows immediately from the positivity of the sum
(a - b) + (b - c) = a -
c.
The preceding rules also hold i f we replace the sign > by � everywhere. Let a and b be positive numbers and observe that
a2 - b2 = (a + b)(a - b). Since a + b is positive, we conclude that a2 > b2 fol lows from a > b. Thus an ineq uality between positive n u mbers can be "squared . " Similarly, a 2 � b 2 whenever a � b � 0. From t h e eq uation l a - b = -- ( a 2 - b 2), a + b
valid for all positive a and b, it follows that the converse is also true ; that is, for positive a and b, a2 > b 2 i mplies a > h. Applying this result to the n u m bers a = .,J;, b = .)");, for arbitrary positive real numbers x, y, we find 2 that .)-;, > .)"); when x > y. More generally, .j; � .)"y whenever x � y � 0. H ence it is legi timate to take the
1 Transitivity justifies the use of the compound formula "a < b < c . . . " to express "a < b and b < c, etc." Avoid non transitive arrangements like x < y > z; these are confusing and mislead ing. 2
Here and hereafter the symbol
v;
for
z
:?: 0 denotes that nonnegative number
whose square is z . With this convention lei = v;:; for any real c since l e i l ei ' = c' . From this we obtain the important identity JxyJ = Jxl · JyJ since
JxyJ '
=
(xy) ' = x'y ' = ( J xJ J y J )'. ·
� 0 and
Ch. 1
14 Introduction
square root of both sides of an inequality between nonnegative real numbers. S uppose that a and b are positive and n is a positive integer. In the factorization a " - b " = (a - b) (a "-1 + a " -2b +
·
·
·
+ b "-1 )
the second factor is positive. Thus a " - b" has the same sign as a - b ; i f a " > b", then a > b and i f a " < b", then a < b. Most i nequalities we shall encounter occur in the form of estimates for the absolute value of a number. We recall that lx l is defined to be x for x � 0 and -x for x < 0. We may also say that lxl is the larger of the two numbers x and -x when x is not zero and is equal to both of them when x is zero. The i nequality lxl � a then states that neither x nor - x exceeds a, that is, that x � a and - x � a. Since - x � a i s equivalent t o x � -a, we see that the i nequality l xl � a means that x -------r---r--,_-- L xo + a Yn - a xo Figure 1 .5 The interval l x - x,l � a .
lies in the closed interval - a � x � a with center 0 a n d length 2a. The inequality lx - x0 1 � a then states that - a � x - x0 � a or that x0 - a � x � x0 + a, thus, that x lies i n the closed interval with center x0 and length 2a (see Fig. 1 . 5). S i milarly the £-neighborhood (x0 - £, x0 + £) of a po i nt x0, that is, the open interval x0 £ < x < x0 + £, can be described by the inequ a l ity l x - x0 1 < £ . ,
-
Triangle Inequality
One of the most important inequalities involving absolute val ues i s the so-called triangle inequality Ia + b l � l al + lbl
for any real a, b. The name "triangle inequality" i s more appropriate for the equivalent statement Ia - ti l � Ia - Y l
+ l y - ti l
for wh i ch we have set a = a - y, b = y tl. The geometrical inter pretation of this statement is that the direct distance from a to t1 i s less than o r equal t o the sum o f the distances via a third point y ; (this also corresponds to the fact that in any triangle the sum of the two sides exceeds the third side). A formal proof of the triangle inequality is easily given. We dis tinguish the cases a + b � 0 and a + b < 0. In the first case the -
The Continuum of Numbers
Sec. 1.1
!S
i nequality states that a + b :::;; lal + l b l : but this follows trivially by addition of the inequalities a :::;; lal and b :::;; l b l . In the second case the triangle i nequality reduces t o - (a + b) :::;; lal + l b l , which again follows by addition from - a :::;; l a l , - b :::;; l b l . W e immediately derive a n analogous inequality for three quantities : I a + b + c l :S: Ia I + lb l + l e i ;
for, by applyi ng t he tri a ngle i nequali ty twice,
Ia + b + c l = l (a + b) + c l :S: I a + b l + l ei :S: l a l + l b l + l e i .
I n t h e same way, the m ore general inequality
i s derived. Occasi onally we need estim ates for I a + b l from below. We observe that l a l = l (a + b) + ( - bl l :S: Ia + b l + 1 - b l = Ia + bl + l b l a n d hence that t h e i n equality ho lds.
I a + b l � lal - l b l
The Cauchy-Schwarz Inequality
Some of the most i m portant i ne qualities exploit the obv i o u s fact that the square of a real n u m ber is never negative and that conse quently a sum of squares also cannot be negative. One of the most frequently used results obtai ned i n this way i s the Cauchy-Schwarz i ne q uality ( a 1 b 1 + a b + · · · + a , b ,)2 2 2 :::;; (a / + a 2 + . . . + a ,2 )(hi2 + h 2 + . . . + hn2 ). 2 2 P u tti n g A = a12 + a 2 + · · · + a ,2, 2 B = a 1 b1 + a b + · · + a nbn, 2 2 c = b12 + b22 + . . . + b ,2, ·
the i n equality becomes A C � B 2 • To prove it we observe that for any
real t
since the right-hand side i s a sum of squares. Expanding each square
16
Ch. 1
,
Introduction
and arranging according to powers of t we find that
0 � A + 28t + Ct2
for all t, where A, 8, C have the same meaning as before. Here C z. 0. We may assume that C > 0, since certainly 82 = A C = 0 when C = 0. S u bstitutin g then for t the special value t = - B/C [corre spondin g to the minimum of the quadratic e x pres s ion
( + �r ( �) J,
A + 28t + Ct2 = C t
we find
+
A -
and hence A C - 82 z. 0.
Figure 1 . 6 I n the
Geometric a n d arith metic means of x and y.
special case n = 2 we can choose
where x and y are positive numbers. Th e i n eq uality then takes the form (2)7cy)2 � (x + y)2 or
--
;x+ y . < --.j x y 2
This inequal ity states that the geometric mean .J 7cy of two positive numbers x, y never exceeds thei r arithmetic mean (x + y)/2. The geometric mean of two numbers x , y can be interpreted as the length of the altitude of a right triangle dividing the hypotenuse into seg ments of length x and y respectively. The ineq uality then states that
Sec. 1 . 2
The Concept of Function 1 7
i n a right triangle the altitude does not exceed half the hypotenuse (see Fig. 1 .6).1
1.2 The Concept of Function
From the beginning of m odern mathematics in the I 7th century the concept of function has been at the very center of mathematical thought. (Leibnitz appears to have been the first to use the word " function".) Although the idea of functional relationships is significant far beyond the mathematical domain, we shall naturally focus our attention o n functions i n t h e mathematical sense, that i s , on the con nectio n o f mathematical quantities b y mathematical relations or prescriptions o r "ope rati ons." A very large part of mathematics a n d t h e natural sciences is dominated by functional relationships, for they occur everywhere i n analysis, geometry, mechanics, and other fields. F o r example, the pressure i n a n ideal gas i s a functio n of density and temperat ure ; the position of a moving molecule i s a function of the time ; the volume and surface of a cylinder are functions of its radius and height. When ever the values of certa i n quantities a, h, c , . are determined by those of certa i n others x, y, z , . . , we say that a, h, c, . depend o n x, y, z , . . . o r are functions of x , y, z , . . . Examples of fu nctional relations are given by formal expressions such as the following. .
.
.
.
.
.
(a) The formula A = a2 defines A as a fu nction of a. For can i n terpret A as the area of a s q uare o f side a. (b) The formula
a > 0 we
y = ) J - x2 of x for all x for
defines y as a fun ction which - I � x � I . For this fu nction expresses the side y of a right triangle with hypot enuse I in terms of the other side x . (c) The equations 2
x>0
X =
I,
11
= -(
assign val ues of x and y to each 1 and thus define x and y as fu nctions of t . I f we i nterpret x and y a s t h e rectangular coordinates of a p o i n t P i n the plane and t as the time, then our e q uatio n s describe the location o f P at the time t; in other words, they describe the motion of the point P. (d) The equations
a= 2 x
+ X
y2
---
,
1 The interested re1der w i l l find more material in An Introduction to Inequalities, by E. F. Beckenbach and R. Bel l man, Random House, 1 96 1 , and Geometric Inequalities, by N. Kazarinotf, Random H ouse, 1 96 1 .
1 8 Introduction
Ch. 1
define a and b as functions of x and y for x2 + y2 'J'f 0. Interpreting the pairs of values x, y and a, b as rectangular coordinates of two points, we see that the equations assign to each point (x, y) [with the exception of the origin (0, 0)] an "image" (a, b). The reader can verify easily that the image (a, b) always lies on the same ray from the origin as the original or "antecedent" (x, y) and has the reciprocal distance from the origin . We speak of "mapping (x , y) onto (a, b) by means of the equations expressing a, b in terms of x, y.
"
"
"
In the preceding examples the functional law is expressed by simple formulas which determine certain quantities in terms of certai n others. 1 The quantities appearing on the left-hand sides, the "dependent variables," are expressed in terms of the "independent variables" o n t h e right T h e mathematical law assigning unique values of the dependent variables to given values of the independent variables is called a function. It is unaffected by the names x, y, etc., for these variables. In Example c we have an independent variable t and two dependent variables x, y, whereas in Example d there are two i ndepend ent variables x, y and two dependent variables a, b. The dependence of y on x by a functional relation is frequently indicated by the brief expression "y is a function of x. " 2 .
a. Mapping-Graph
Domain and Range of a Function We usually interpret the independent variables geometrically as coordinates of a poiht i n one or more dimensions. In Example b this would be a point on the x-axis, in Example d a p oin t in the x,y-plane. Sometimes the independent variables are free to take all values, as i n examples a and c . Often, however, there is some restriction, inherent or imposed, and our functions are not defined for all values. The set of values or the points for which a function is defined form the domain" of the function. In E x ample a the domain is the whole a-axis, in b the interval - I ::::;; x ::::;; I , in c the whole t-axis, and in d the points of the x,y-plane different from the origin. To each point P in the domain our functions assign definite values "
1 Later we shall gradually realize the need for considering functions not capable of such representation by simple formulas. (See, for example, p . 2 5 . ) ' This locution is used freely in the sciences, but some of the more pedantic texts avoid it. There is no point in hampering <>urselves by an undue concern for hair splitting "precision" when it has no relation to the substance.
Sec. 1.2
The Concept of Function 1 9
for the dependent variables . These values also can b e interpreted as coordinates of a point Q, the image of P. We say that P is "mapped" by our functions onto the poi nt Q. Thus in Example d the point p = ( !) ) of the x,y-plane is mapped onto the point Q = n. n of the a,b-plane. The image points Q form the range of the function. 1 Each Q in the range is the image of one (or more) points in the domain of the function. I n Example c points of the t-axis have as their images points i n the x,y-plane. The t-axis is m apped into t h e x,y-plane. B u t n o t every point of the x,y-plane occurs as i mage, only those for which y = x 2 Thus the range of the mapping is the parabola y = - x2• We say , the t-axis is mapped onto th e parabola y = - x2 , i n the sense that the image points fill this parabola. In Example d the range consists of the points (a, b) i n the a,b-plane w hose coordinates can be written i n the form a = xj(x 2 + y2), b = y j (x2 + y 2) with suitable x, y for which x2 + y 2 ,e. 0. I n other words, the range consists of those points (a, b) for which the preceding equations have a solution (x, y). As seen i m mediately the range consists of the points (a, b) for which a and b do not both vanish ; each such point (a, b) is i m age of the point : r = aj(a2 + b2), y = bj(a2 + b2) . Every geometrical figure in the x,y-plane is then mapped onto a corresponding figure i n the a,b-plane wh ich consists of the images of the points of the fi rst figure. For example, a circle x 2 + y2 = r2 about the origin i s m apped onto t h e circle a 2 + b 2 = l /r 2 in t h e a,b-plane. I n this and the following chapters we shall deal al most exclusively with a single independent variable, say x, and a si ngle dependent vari able, say y, as indicated in Example b. 2 Ordinarily we represent such a function in the standard way by its graph in the x,y-plane, that is, by the curve consisting of those points (x, y) whose ordi nate is i n the specified functional relationship to the abscissa x (see Fig. 1 . 7). For Example b the graph is the upper half of a circle of radius one about the origi n. The interpretation of the fu nction as a mapping of a domain on the x-axis o nto a range on the y-axis leads to a different visualization of functions. We interpret x and y not as coordinates of the same point i n the x,y-plane, but as points on two different, independent number -
•
1 It is often convenient to talk of the point Q as "a function" of P, although in the analytic representation several functions expressing the different coordinates of Q appear. 2 However, it should be emphasized from the beginning that functions of several variables occur just as naturally in many instances. They will be discussed systemat ical ly in Volume I I .
20
Ch. l
Introduction y
y
--------
� I I I I I I
--� 0�--------------�x�--------� x Figure 1 .7
Graph of function.
axes. Then the function maps a point x on the x-axis i nto a point y on the y-axis. Such mappings arise frequently in geometry, such as the "affine" mapping which originates by projecting a point x on the x-axis onto a point y on a parallel y-axis from a center 0 located in the plane of the two axes (see Fig. 1 . 8). This mappi ng can be expressed analyt ically, as easily ascertained, by the li near function y = ax + b with 0
y
Y=
ax ex
+b +d
y Figure 1 .8
Mappings.
The Concept of Function 2 1
Sec. 1.2
constants a and b. Obviously, it is a "one-to-one" mapping i n which inversely to the image y, there corresponds a unique original x . Another, more general, example is the "perspective mapping" defined by the same sort of projection, only with the two axes not necessarily parallel . Here the analytical expression is given by a rational linear function of the form y = (ax + b)/(cx + d), with constants a, b, c, d. Any projection of a surface S in space i nto another surface S' from some center N can be viewed as a mapping whose domain is S and whose range lies on S ' . For example, we can map a sphere onto an equatorial plane by projecting each point P of the sphere onto a point P' of the plane by rays from the North Pole (see Fig. 1 .9). This mapping N
s·
Figure 1.9
p·
Stereographic projection .
is the "stereographic projection" used frequently for maps of the earth. The i nterpretation of functions as " maps" i s suggested by examples of this type. When more independent or dependent variables are involved, the definition of functions by mapping provides a more flexible and suitable interpretation than that by graphs. This fact will become fully apparent i n the second volume. h. Definition of the Concept of Functions of a Continuous Variable. Domain and Range of a Function
A function of a si ngle i ndependent variable x assigns values y to values x. The domain of the function is the totality of values x for which the function is defined . I n the cases that concern us most t he domain of the function consists of one or several i ntervals (see Fig. 1 . 1 0). We say then that y is a function of a continuous variable (in contrast to other cases where, for example, the function might only be defined for rational or for integral values of x) . Here the "intervals"
Ch . I
22 Introduction
forming the domain may or may not contain their end points and may also extend to infinity in one or both directions.1 Thus the function y = .J I - x2 is defined in the closed interval - I � x � + I , the function y = 1 /x in the two semi-infinite open intervals x < 0 and x > 0, the function y = x2 in the infinite "interval" - CfJ < x < + CfJ y
- - - - --
I
------ - - -,
11 Y = f(x) I I I I - - - - - - -+ - - - - - - 1 I I I I I
I I I
-cot------t====�==�--� x Figure 1.10
Domain
Domain and range of a function in graphical representation.
co nsi sti n g of al l x, the functio n y = .J(x2 - I )(4 - x2) i n the two separate intervals 1 � x � 2 and -2 � x � - 1 . Functions are denoted b y symbols such as f, F, g, etc. The corre s p o n ding relations between x and the associated y-values are written i n the form y = f (x) o r y = F(x) or y = g(x), e tc . , o r also so metim e s y = y (x) to indicate2 that y depends on x. If, for example, f(x) is defined b y the expressio n x2 + 1 we havef(3) = 3 2 + I = I O,f( - I ) =
( - 1 )2 + 1 = 2 .
1 Ordinarily we will reserve the word "interval" for "bounded," that is, "finite" intervals, that have definite finite end points ; then one m ight indicate the more comprehensive concept as used in the text, by the word "convex sets," meaning sets which when containing two points must contain all intermediate ones. 2 In this notation we try to emphasize the variables and do not explicitly indicate the functional operation by a symbol such as f The notation .f:
x- y
for the function f mapping x into y is also sometimes encountered.
The Concept of Function 2 3
Sec. 1.2
Nature of Functional Relation In the general definition of a function j(x) nothing is said about the nature of the relation by which the dependent variable is found when the independent variable i s given. As said before, often the function is given i n "closed form" by a simple expression like f(x) = x 2 + I or f(x) = .J I + sin 2 x, and i n the early days of the calculus such explicit expressions were mostly what mathematicians meant by functions. Often mechanical devices generate geometric curves or y
Figure 1 . 1 1
graphs which then define functions. A striking example is t h e cycloid, a curve described by a point fixed on a circle which rol ls along the x-axis (see Fig. 1 . 1 1 ) . Its functional analytical expression by formulas will be given later (see p. 328). Logically, we are not restricted to such geometrically or mechanically generated functions. Any rule by which a value of y i s assigned to values of x constitutes a function. I n some theoretical i nvestigations the wide generality or vagueness of the function concept is, in fact, an advantage. However, for applications, particularly in the calculus, the general concept of function i s u nnecessarily wide. To make meaningful mathematical developments possible, the "arbitrary" laws of correspondence by which a value of y is assigned to x must be subjected to radical restrictions. During the past century and a half mathematicians have recognized and formulated in precise terms the essential restrictions that have to be imposed on the overly general concept in order to obtain functions that i ndeed have the u seful properties one would expect intuitively.
24 Introduction •
Ch. 1
Extended or Restricted Domains of Functions
Even for functions given by explicit formulas, it is important to realize that any complete description of a function must include a definition of the domain of the function . For us the "function" f described by "j(x) = :r2 for 0 < x < 2" is not strictly the same function as the function g given by "g (x) = x 2 in the larger domain - 2 < x < 2," although j(x) and g(x) have the same values in the interval 0 < x < 2 where both are defined . Generally, we call a function fa "restriction" of a function g (or g an "extension" off), if, wherever f is defined, g is also defined and assumes the same values. Of course, the same function f can arise by restriction from many different functions. In our example above f is also a restriction of the function h defined by h(x) = x 2 for 0 < x < 2, h(x) = -x2 for -2 < x � 0. As a matter of fact this example illustrates the process inverse to that of forming restrictions of a function which might be called "piecing together'' ; we can generate new functions by simply defining them by different explicit expres sions in different portions of the domain.
c. Graphical Representation. Monotonic Functions
The fundamental idea of analytical geometry is to give an analytical representation to a curve originally defined by some geometrical property. This is done usually by regarding one of the rectangular coordi nates, say y, as a function y = f(x) of the other coordi nate x ; for example, a parabola is represented b y the function y = x2, the circle with radius I about the origin by the two functions y = J I - x2 and y = - J I - x2 • In the first example we may think of the function as defined in the in fi nite interval - oo < x < oo ; in the second we must restrict ourselves to the interval - I � x � I , since outside this interval the function has no meaning.1 Conversely, if i nstead of starting with a curve defined geometrically we consider a function y = f(x) given analytically, we can represent the functional dependence of y on x graphically, using a rectangular coordinate system in the usual way (cf. Fig. 1 . 7). If for each abscissa x we take the corresponding ordinate y = f(x), we obtain the geo metrical representation of the function . The restrictions to be i mposed on the function concept should secure for its geometrical representation the shape of a "reasonable" geometrical curve. This, it is true, expresses an intuitive feeling rather than a strict mathematical condition. How ever, we shall soon formulate conditions, such as continuity, differenti ability, etc., which insure that the graph of a function is a curve capable 1 We do not ordinarily consider imaginar) or complex val ues of x and y.
Sec. 1.2
The Concept of Function 25
of being visualized geometrically. This would not be the case if we admitted " pathological" functions such as the following : For every rational value of x, the function y has the value I ; for every irrational value of x, the value of y is 0. This functional prescription assigns a definite value of y to each x ; but in every i nterval of x, no matter how small, the value of y j umps from 0 to I and back an i nfinite number of times. This example demonstrates that the general unrestricted func tion concept may lead to graphs which we would not consider as curves. Multiz•alued Functions
We consider only functions y = f(x) assigning a unique value of y to each value of x in the domain, as, for example, y = x2 or y = sin x . Yet, for a curve described geometrically, it may happen, as for the circle x2 + y2 = I , that the whole course of the curve i s not given by just one (single-valued) function, but requires several functions-in the case of the circle, the two functions y = _;I x2 and y = - _; I x2 • The same is true for the hyperbola y2 - x2 = I , which is represented by the two functions y = _; I + x2 and !I = _; I + x2• Such curves therefore do not determine unambiguously the corresponding functions. It i s sometimes said that the curve is represented by a multivalued function ; the separate functions representing it are then called the si ngle-valued branches of the multivalued function belonging to the curve. For the sake of clarity we shall always use the word "function" to mean a single-valued function. For example, the symbol _;; (for x 2 0) will always denote the nonnegative number whose square is x. If a curve is the graph of one fu nction, it is intersected by any parallel to the y-axis in at most one point, since to each point x in the i nterval of definition there corresponds j ust one value of !f . The unit circle represented by the two functions -
-
-
y = _; J - x2
and
y = - J I - x2,
is intersected by such parallels to the y-axis in more than one point. The portions of a curve corresponding to different single-valued branches are sometimes connected with each other so that the complete curve is a single figure which can be drawn with one stroke of the pen, for example, the circle (cf. Fig. 1 . 1 2) ; on the other hand, these portions may be completely separated, as for the hyperbola (cf. Fig. 1 . 1 3) . Examples. Let us consider some further examples of the graphical representation of functions.
26 Introduction
Ch.
1
y
y
Figure 1 . 1 2
Figure 1.13
(a) y is proportional to x, y = ax. The graph (see Fig. 1 . 1 4) is a straight line through the origin of the coordinate system. (b) y is a "linear function" of x,
y = ax + b. y
Figure 1 .14
Linear functions.
Sec. 1.2
The Concept of Function 27
The graph is a straight line through the point x = 0, y = b, which, if a ,r. 0, also p a sses through the point x = bfa, y = 0, and if a = 0 is horizontal. (c) y is inversely proportional to x, -
a y=-. X
In particular, for a = I
1
y=-, X
so that y = 1
for x = 1 ,
y= 2
for x = ! .
y= !
for x = 2 .
The graph ( cf. Fig. 1 . 1 5) is a rectangular hyperbola, a curve symmetrical with respect to the bisectors of the angles between t he coordi nate axes. This function is obviously not defined for the value x = 0 since division by zero has no meaning. I n the neighborhood of the exceptional point x = 0, the function has arbitrarily large values, both positive and negative ; this is the simplest example of an infinite discontinuity, a concept which we shall discuss l ater (see p. 35). y
X
Figure 1 . 1 5
Infinite discontinuity.
28 Introduction
Ch. I
y
(d) y is the square of x,
Figure 1.16
Parabola.
2 y = x .
As is well known, this function is represented by a parabola (see Fig. 1 . 1 6). Similarly, the function y = x3 is represented by the so-called cubical parabola (see Fig. 1 . 1 7). y
Figure 1.17 Cubical parabola.
Sec.
1.2
The Concept of Function 29
Monotone Functions
A functio n which for all values of x in an interval has the same value y = a is called a constant ; it is represented graphically by a horizontal straight line. A function y = f(x) for which an i ncrease in the value of x always results i n an i ncrease in the value of y that is, for which j(x) < f(x') w henever x < x') is called a monotonic increasing function ; if, on the other hand, an i ncrease in the value of x always implies a decrease i n the value of y, the function is called a monotonic decreasing function . Such functions are represented graphically by curves which always rise or always fall as x traverses the i nterval of definition toward y
y
Figure 1 . 1 8
Monotone functions.
increasing values (see Fig. 1 . 1 8) . A monotone function always maps different values of x in to different y ; that i s, the mapping is one-to-one.
Even and Odd Functions
If the curve represented by y = f(x) is symmetrical with respect to the y-axis, that is, if x = -a and x = a yield the same function value
f( -x) = f(x)
we call the function an even function. For example, the function y = x2 is even (see Fig. 1 . 1 6) . If, on the other hand, the curve is symmetrical with respect to the origi n ; that is, if /( -x) = -J(x),
we say the function is an odd function ; thus the functions y y = x3 (see Fig. 1 . 1 7) and y = 1 /x (see Fig. 1 . 1 5) are odd.
= x,
Ch. 1
30 Introduction y
Figure 1 . 19
Graph of y > x'.
It is frequently helpful to consider the geometrical representation of an inequality. For example, the inequality y > x2 is represented by the domain above the parabola y = x 2 (Fig. 1 . 1 9) . The interior of the unit circle centered at the origin (Fig. 1 .20) is described by the inequality x2 + y2 < 1 . Often several inequal ities describe more complicated regions with boundaries consisting of different pieces. Thus the "first" quadrant of the unit circle is described by the system of simultaneous inequal ities : x2 + y 2 < J ,
y > 0.
X > 0,
(See Fig. 1 .2 1 .) y
Figure 1 .20
G"aph of x'
+ y'
< 1.
Sec. 1 . 2
The Concept of Function 3 1 y
Figure 1 .21
Graph of x 2
+ y2 <
1, x >
0, y > 0.
d. Continuity
Intuitive and Precise Explanation The functions and graphs just considered exhibit a property of greatest i mportance in the calculus, that of continuity. I ntuitively, corotinuity means that a small change in the independent variable x implies only a small change i n the dependent variable y = f(x) and excludes a jump in the value of y : thus the graph consists of one piece. In contrast, a graph y = f(x) consisting of pieces separated by a gap at an abscissa x0 exhibits there a jump discontinuity. For example, the function 1 f(x) = sgn x defined by .f(x) = + 1 for x > 0, by f(x) = - 1 for x < 0, and .f(O) = 0 has a "jump discontinuity" 2 at x0 = 0 (see Fig. 1 .22). The i dea of continuity is implicit in the everyday use of elementary mathematics. Whenever a functio n y = f(x) is described by tables, such as the logarithmic or trigonometric tables, the values of y can be l isted only for a "discrete" set of values of the independent variable x, say at i ntervals of 1 / 1 000 or I / 1 00,000. Yet, unlisted values of the function may be needed for i n termediate x. Then we tacitly assume that an unlisted value j(x0) is approximately the same as that of j(x) Pronounced "signum" or "sign" of x . ' Technically, the word "j ump" refers only to the particular kind of discontinuity in which the function approaches values from the right and left that do not both agree with f(x0). An "infinite" discontinuity is exhi bited by the function y = I fx for x # 0 and y = 0 for x = 0. Still other types of discontinuities will be discussed later.
1
32 Introduction
Ch.
1
y
0
-------1 - 1
Figure 1 .22
The function ((x)
=
sgn x.
for a neighboring x which appears in the table and that j(x0) can be approximated as precisely as we want if only the x-values i n the table are spaced sufficiently close to each other. Continuity of the function j(x) for a value x0 j u st means that j(x) differs arbitrarily little from the value j(x0) on ce x is sufficiently close to x0. The words "differs arbitrarily little" and "sufficiently close" are somewhat vague and must be explained precisely in quantitative te r m s Prescribe any "margin of precision" or "tolerance," that is, any positive real number E (however small). For continuity off at x0 we require that the difference between j(x) and j(x0) stay withi n this margin, that is, that IJ(x) - j(x0)! < E, for all values x which are sufficiently close to x0 (or for all val ues x lyi ng within some d istance {J from x0). We can visualize most easily what continuity means if we interpret ! as a mapping assigning to poi nts x on the x-axis i mages on the y-axis. Take any poi nt x0 on the x-axi s and its i mage y0 = f(x0) (see Fig. 1 .23). .
J Yo + •
YO - •
Figure 1 .23
y
Continuity of the mapping y = f(x) at the point x0 • I
X
The Concept of Function 33
Sec. 1.2
We mark off a1;1 arbitrary open interval J on the y-axis having the point Yo as center. If 2£ is the length of J, then the points y of J are those whose distance from y0 is less than e or for which IY - Y ol < e . The condition for continuity off(x) at x0 i s : All points x close enough to x0 have images lying in J ; or : I t is possible to mark off an interval / on the x-axis with center x0, say the i nterval x0 - CJ < x < x0 + CJ such that every point x of I has an i mage f(x) which lies in J a n d thus lf(x) - f(x0) 1 < £. Continuity of f(x) at the point x0 means that for an arbitrary £-neighborhood J of the point y0 = f(x0) on the y-axis a CJ-neighborhood I of the point x0 on the x-axis can be found, all of whose points are mapped into points of J. 1 Of course, this makes sense only for points on the x-axis at which the mapping is defined, that is, which belong to the domain of f Thus we are led to the following precise definition of continuity. The function f(x) is continuous at a point x0 of its domain if for et•ery positit•e £ we can find a positiue number CJ such that
if(x) - f(xo) l <
E
for all values x in the domain offfor which lx - x01 <
o.
M o st useful is the geometric i nterpretation of continuity when we represent the function f by its graph in the xy-plane (see Fig. 1 .24 ) . Let P0 = (x0, y0) be a point on the graph. The poi nts (x, y) with Yo - £ < y < Yo + e now form a horizontal "strip" J containing P0• Continuity of f at x0 means that given any such horizontal strip J, however thin, we can find a vertical strip I given by x0 - o < x < x0 + o so thin that every poi nt of the graph lying in I also falls into J . As an illu stration we consider the linear function f (x) = 5x + 3 ; we have lf(x) - f(x0) 1
=
l (5x + 3) - (5x0 + 3 ) 1
=
5 lx - X01 ,
which expresses that the mapping y = 5x + 3 magnifies d istances by the factor 5. Here obviously lf(x) - f(x0) 1 < e for all x for which 1 In this definition of continuity I and J are intervals having their centers respectively at the points x0 and y0• This is convenient for the analytic definition of continuity at x0 which refers to the distances l x - x0l and I Y - Yol . but it is somewhat artificial if we interpret f geometrically as a mapping. We could instead define continuity of y = j(x) at a point x0 just as well by the requirement that for every open interval J on the y-axis which contains the point Yo = j(xo) we can find an open interval I on the x-axis containing the point x0 such that the y-image of any point x in I for which the mapping is defined l ies in J. The proof of the equivalence of the two definitions is left to the reader as a simple exercise.
34 Introduction
Ch. 1
lx - xol < £/5 . Consequently, the condition for continuity of f(x) at the point x0 is satisfied if we choose (j = £/5 (but, of course, any positive number (j < £/5 is also a possible choice) ; the image of any point of the interval x0 - (j < x < x0 + (j will then lie in the interval Yo - E < y < Yo + £. In this example the statement that the distance I Y - Yo l is "arbitrarily small" for "sufficiently small" l x - x0 1 can be given a q u ite specific me anin g ; i ndeed lx - x0 1 is sufficiently small if it does not exceed one-fifth of the value of IY - Yol · y
xo - li xo xo + li
Figure 1.24
X
Conti nuity of y = .f(x) at the point x0 •
Another example is furnished by the function j(x) lx - x0 1 < (j
=
x2• Here we
have for
l f( x) - f (x6) i = l x2 - x�l = l x - Xol l 2x0 + (x - X0) 1 � l x - X01 (2 1 xo l + l x - Xoi) � b(2 l xo l + b). We verify immediately that the condition lf(x) - j(x0) 1 < £ i s satis fied if w e choose 0 = - l x0 1 + .J E + j x0 j2•
I ntuitively, the idea of continuity seems obvious without explanation, but the precise formulation may initially be somewhat difficult to grasp because of the permissiveness of words such as "one can find" or "arbitrarily chosen." Yet the reader who may at first be well satisfied with some i ntuitive notion of continuity will gradually learn to
appreciate the logical precision and generality of the analytic definition, the outcome of a l o n g and per sistent struggle for reconciliation of the
Sec. 1.2
The Concept of Function 35
need for intuitive u nderstanding with that of logical clarity. I n the long run a precise meaning for the word "conti nuity" i s i ndispensable ; the analytic definition given here is the compelling formulation of an i mportant property of functions. For the begi nner it should be emphasized agai n that "small" is not an absolute designation of a number ; rather the term "arbitrarily small" refers to a n umber that i s not fixed at the outset but for which then any positive value may be chosen , and which i s subject to a subse quent smaller choice for a refined approxi mation ofJ (x0). "Sufficiently small" refers to a number b that must be adjusted to suit a margin of tolerance set previously by another number £. Continuity and Discontinuity Explained by Examples. We can illumi nate the definition of continuity by contrast with examples of dis continuity, examples which do not fit the definition above. Recall the simple example of the fu nction f(x) = sgn x on p . 3 1 . Obviously, for any x0 � 0 this functi on is continuous according to the £, b-definition above, i n fact, with a constant {J = j x0 1 no matter how small £ i s chosen . But for x0 = 0 no b at a l l can b e found i f £ i s less than I since I J(x) - /(0) 1 = I J(x) i = 1 > £ for every x unequal to zero, however close x might be to zero . The fu nction sgn x illustrates the si mple type of discontinuity at a point � known as jump-discontinuity, in whichf(x) approaches limiting values from the right and left as x approaches �-limiting values, however, that differ either from each other or from the val ue of J at the point rl The graph at X = � then has a gap. Other curves with jump discontinuities are sketched in Fig. 1 . 25a and b ; the definition of these fu nctions should be clear from the figu res.2 I n discontinuities of this kind the l i mits from the right and the left both exist. We turn to discontinuities in which this i s not the case. The most important of these are the infinite discontinuities or i nfinities. 1 The precise definition of limit will be given in Section I .7 ; an intuitive idea is sufficient for the descriptive remarks made here. 2 In all these examples of j ump discontinuities the limits of the function at the poin t of d isco n tin uity from the right and l e ft have different values. The trivial example of the function f(x) defined by
f(x) = 0
for
x
rf 0,
f(x) = I
for x = 0
illustrates a jump discontinuity i n which the limits from both sides are equal to each other but differ from the value of .f at the point of discontinuity � itself. We have then a removable singularity. Here f can be made continuous by merely changing the value of f at .; so as to agree with the limits from both sides.
Ch. 1
36 Introduction y
0
0
y
0
(a)
(b) Figure 1 .25
y
Figure 1 .26
Graph of function with i nfinite discontinuity.
The Concept of Function 37
Sec. 1.2
These are discontinuities l ike those exhibited by the functions 1 /x or 1 /x2 at the point x = 0; as x -+ 0 the absolute value lf(x) l of the function i ncreases beyond all bounds. The function 1 /x increases numerically beyond all bounds through positive and negative values, respectively, as x approaches the origin from the right and from the left. On the other hand, the function 1 /x2 has for x = 0 an infinite discontinuity at which the value of the function increases beyond any positive bound as x approaches the origin from both sides (cf. Fig. 1 .26 .f
Figure 1 .27
Function with infinite discontinuities.
and Fig. 1 .27). The function 1 /(x2 - I) shown in Fig. 1 .27 has infinite discontinuities both at x = 1 and at x = - I . An example of another type of discontinuity i n which no l i mit from the right or from the left exists is the "piecewise linear" even function y = j(x) illustrated in Fig. 1 .28, which is defined as follows for all nonzero values of x. This function alternately takes the values + 1 and - I for the x-values of the form ± 1 /2", where n is any i nteger : /( ± 1 /2") = ( - I )". In every interval I /2" 11 < x < l /2" or - I /2" < x < - I /2 " 1 I the func tion f (x) is l i near and ranges over all values between - I and + 1 . Therefore the function swings backward and forward more and more rapidly between the values - I and + I as x approaches nearer and nearer to the point x = 0, and in the immediate neighborhood of that
Ch. I
38 Introduction y
-
Figure 1 .28
y
1
Piecewise linear oscillating function with discontinuity.
. I y = Sin 7
0
X
-I
Figure 1 .29
Oscillating function with discontinuity.
The Concept of Function 39
Sec. 1.2
point an infinite number of such oscillations occur. A similar behavior is exhibited by the smooth curve (Fig. 1 .29). [Here f(x) actually is given by an expression in closed form, namely, j (x) = sin ( l fx), with the sine-function defined appropriately as on p. 5 1 ] . y
/
/
/
/
/
/
/
/
Figure
1.30 Continuous oscillating function.
A contrast to this example is the piecewise linear function y = f(x) that takes the values/( ± 1 /2 n) = ( !Y for all integers n (see Fig. 1 .30) and is l inear for intermediate values of x. Here f(x) remains continuous at the point x = 0 if we assign to it the value 0 at that point. In the neighborhood of the origin the function oscillates backward and forward an i nfinite n umber of times, but the magnitude of these oscillations becomes arbitrarily small as the origin i s approached. The situation i s the same for the function y = x sin ( 1 /x) (see Fig. 1 .3 1 ). -
40
Ch. 1
Introduction
These examples show that continuity permits all sorts of remark able possibilities foreign to our naive i ntuition. •
Removable Discontinuities
As noted it may happen that at a certain point say x = 0, a function is not defi n ed by the original law, as, for example, i n the last examples discussed . We are then free to extend the definition of the function by y
X
Figure 1 .31
Continuous oscillatin g function.
assigning to i t any desired value at such a point. I n the last example we can choose the definition in such a way that the function becomes continuous at that point also, namely, by choosing y = 0 at x = 0. A similar continuous extension can be defined whenever the limits from the left and from the right both exist and are equal to one another ; then we need only make the value of the function at the point in question equal to these limits in order to make the function continuous there. Whatever discontinuity may be imposed by definition at x = 0, this discontinuity is "removable" by assigning a suitable value j(O). For the function y = sin 1 /x or for the function in Fig. 1 .28, this is, however, not possible : whatever value we assign to the function at x = 0, the extended function is discontinuous.
Sec. 1.2
The Concept of Function 4 1
Modulus of Continuity. Uniform Continuity. Our definition of con tinuity of the function f(x) at x0 requires that for every degree of precision E > 0 there exist quantities lJ > 0 (so-called moduli of con tinuity) such that l f(x) - f(x0) 1 < E for all x in the domain off for which lx - x0 1 < b. A modulus of continuity expresses information about the sensitivity off to changes in x. A modulus of continuity 6 is never unique ; it can always be replaced (for the same x0 and E) by any smaller positive quantity /J' since l x - x0 1 < 6' i mplies l x - x01 < lJ and thereby lf(x) - f(x0) 1 < E. For practical purposes, as in numer ical computations, we may be i nterested in a particular choice of /J ; for example, i n the largest value for 6. O n the other hand, if we merely want to establish the fact thatfi s continuous at x0, then we need only to exhibit any one modulus of continuity for every positive E . In general, as our examples show, this 6 = D ( E) depends not only on E but also on the value of x0• Of course, we need not consider all positive values E. We can always restrict considerations to sufficiently small E, say to E S E o for an arbitrarily chosen E0, since for E > E0 we can use the same modulus of conti nuity as for E = E0• Si milarly, we only have to take i nto account the points x of the domain off lying i n an arbitrary neighborhood of x0, say those with lx - x01 < <50, since we can always replace any modulus of continuity 6 by a smaller one which does not exceed 60• Conti nuity off at x0 is a local property, meaning a property which only depends on the values of f in some neighborhood of x0 however small. As we have seen, the function f may be continuous for some x 0 and discontinuous for others. A function is called continuous in an interval if it is continuous at each point of the interval. For each x0 of the i nterval we have then a modulus of continuity b = b(E) which can be expected to vary with x0 reflecting the different rates at which y changes with changing x near different points x0• We calif uniformly continuous in an interval if we can find a uniform modulus of continuity tJ = 6(E) for that i nterval, that is, one not depend ent on the particular point x0 of the interval. Thus f(x) is uniformly continuous in an interval 1 if for each positive E there exists a positive number b such that l f(x) - j tx0) 1 < E for any two points x and x0 of the i nterval for which l x - x01 < 6. For a uniformly conti nuous function y = f(x) the values of y differ "arbitrarily little" from each other for any values of x that are "suffi ciently close" regardless of their location in the interval. In some respects 1 In this definition the word interval can refer either to closed or open or infinite intervals.
42 Introduction
Ch. I
uniform continuity comes closer to intuitive notions than the mere local property of continuity. For example, the function J(x) = 5x + 3 is uniformly continuous for all values of the independent variable since here /j(x) - f(x0) / = 5 Jx - x0/ < £ for Jx - x0J < £/5, and thus CJ(e) = £/5 represents a uniform modulus of continuity . The function f(x) = x 2 for a n infinite x-interval i s definitely not uniformly continuous. It is clear that small changes in x can produce arbitrarily large changes in x2 if only x is large enough. A glance at a table of squares of integers x shows how successive squares are spaced further and further apart as x increases. If, however, we only consider pairs of values x and x0 belonging to a fixed finite closed i nterval [a, b ] , w e can find a uniform modulus o f continuity. Indeed, for Jx - x0 / < CJ we have /f(x) - f( x0) / = / x 2 - x� / = Jx - X0 / / x + x0 / :$; 2 / x - x0 /( / b / + / a / )
< 2 CJ(/ b / + / a /) = £
if we take CJ = £/2(/b/ + J a l ) . A similar situation prevails for the function f(x) = 1 /x for x ""' 0, j(O) = 0. Consider a closed bounded interval a :$; x :$; b throughout which the function is continuous. Such an interval cannot i nclude the origin, which is a point of discontinuity, so that a and b must have the same sign. Suppose a and b are both positive. Then for x and x0 belonging to the interval and for Jx - x0J < CJ we have
for (J = a2e. Thus the function is uniformly continuous in the i nterval [a, b]. Of course, this proves also that the function f(x) = l fx is con tinuous at every point x0 > 0. For every such value x0 can be enclosed in some interval a < x0 < b with positive a, b. The expression CJ = a2e i s then a modulus of continuity for the function at x0, if we restrict x to a neighborhood of x0 lying completely in the interval. The continuous functions of the preceding examples turn out to be uniformly continuous in any closed bounded interval belonging to their domain. They illustrate a general fact which will be proved in the Supplement, p. 100.
Anyfunction, continuous in a closed and bounded interval, automatically is uniformly continuous in that interval.
Sec. 1.2
The Concept of Function 43
The restriction to bounded intervals is essential as the example of the function x2 sh�ws. Similarly, we must stipulate that the interval be closed; for example, the function y = 1 /x is continuous in the open i nterval 0 < x < I but is not uniformly continuous there ; arbitrarily large changes in y can be produced by arbitrarily small changes in x if only x is sufficiently close to the origin. If there existed a uniform modulus of continuity 0(£) for the interval (0, 1), we could take, for example, x0 < o, x = ix0 ; obviously then 1 1 /x - l /x01 = l fx0 is greater than any preassigned £ whenever x0 is sufficiently small, so that the assumption of a u niform 0(£) leads to a contradiction.
Lipschitz-Continuity-Holder-Continuity. I n the precedi ng examples of functions uniformly continuous in an i nterval [a, b] we found a particularly simple modulus of conti nuity, namely 0(£) proportional to £. This most common situation is presented by the so-called Lipschitz-continuous functions, that is, by the functions f(x) which satisfy an i nequality of the form
(a so-called Lipschitz condition) for all x1, x2 in the interval with a fixed value L. Lipschitz-continuity means that the "difference quotient"
formed for any two d istinct points of the interval never exceeds a fixed finite value L in absolute value or that the mapping y = f(x) magnifies distances of points on the x-axis at most by the factor L. Clearly, for a Lipschitz-continuous function the expression 0(£) = £/L is a mod ulus of continuity since lf(x2 ) - j(x1) 1 < £ for lx2 - x1 1 < £ /L. Conversely, any function with a modulus of conti nuity proportional to £, say 6(£) = C£, is Lipschitz-continuous, with L = l j c .
As we shall see in Chapter 2 most of the functions encountered are Lipschitz-continuous except at isolated points, as a consequence of the fact that their derivatives are bounded in any closed interval which excludes these points. However, Lipschitz-continuity is only sufficient but not nec essary for uniform continuity. The simplest example of a function which is continuous without being Lipschitz-continuous is given by j(x) = v;; for x ;::: 0 and .c0 = 0. Here the difference quotient o
j(x) - j(O) x -
=
v;
44 Introduction
Ch. I
becomes arbitrarily large for sufficiently small x and hence cannot be bounded by a fixed constant L. Thus it is not possible to choose 6(•) proportional to •; but there exist other, nonlinear, moduli of continuity for this function, for example, 6(•) = •2 • The function .;; belongs to the general class of functions called "H older continuous," satisfying a "Holder-condition" for all x 1 , x2 of an interval, where L and IX are fixed constants, the "H older exponent" "' being restricted to values 0 < "' :;;; I . The Lipschitz-continuous functions arise for the special Holder-exponent IX = I . Obviously, o = L-11•• 11• is a possible modulus of continuity for a Holder continuous function f; here cl is propo rtional to ,• I• , and not to < itself.
The function J(x) = .; :;: is H older-continuous with e x ponent "' = !- Th is follows from the ineguality
1 v:;; - v�1
:;;; l x2
- x1 1 �.
which we obtai n by observing
1 v;;,2 - v ;:;.1 :;;; 1 vx;; + v�1
and mu ltiplying by I vx;;
- V'f;"\ .
o(•) = • 2 for .;:;: as mentioned before.
This yields the modulus of continuity
More generally, the fractional powers f(x) = .r• for 0 < "' :;;; I are H older continuous with Holder - exponent "' · The H older-conti nuous functions stil l do not exhaust the class of all uniformly con tinuous fu nctions. lt is not difficult to construct examples of conti nuous functions for which powers of • do not suffice as moduli of continuity. (See Problem 1 3, p. l l8 . )
e. The Intermediate Value Theorem. Inverse Functions
Intu itively there is no doubt that a function which is contin uous . and hence has n o "jumps," cannot vary from one value to another without passing through all intermediate values. This fact is expressed by the so-called i ntermediate value theorem (its precise proof is given in the Supplement, p. I 00).
INTERMEDIATE VALUE TH EO REM . Consider a function f(x) contin uous at erery point of an interz•al. Let a and b be any two points of the interz•al and let 'YJ be any number betweenf(a) and.f(b). Then there exists a value ; bet11·een a and b for whicf f(;) = 'YJ ·
Interpreted geometrically, the theorem states that if two points (a,.f(a)) and (b,.f(b)) of the g ra ?h of a continuous function f lie on
Sec. 1.2
The Conupt of Function 45
different sides of a parallel y = 7J to the x-axis, then the parallel intersects the "graph at some intermediate point (see Fig. 1 .32). There may, of course, be several intersections. In the important case where the fun ctionf(x) is monotonic increasing or monotonic decreasing through out the i nterval, there can be only one intersection for then f cannot have the same va l ue 7J for two different values of c;. As an example we take the function f(x) = x2 which is monotonic increasing and continuous i n the interval I � x � 2. Here f( l ) = I , f(2) = 4. Taking for 7J the val ue 2 intermediate between I and 4 we find y
Y = f(x)
f(a) 1------t;!'--�-- '11
0
a Figure
X
1.32 The intermediate value theorem.
that there exists a unique c; between J and 2 for which q 2 = 2. This is, , of course, the number denoted by y 2. Continuity of the Inverse Func tion
For any monotonic increasing continuous functi onf( x ) defined in an interval a � x � b, we found that for every r; with f( a) � 7J � f( b) there is exactly one c; with a � c; � b for which fW = 7] . 1 Let r1. = f(a), � = f(b) . Si nce c; is determined uniquely by 7] , it represents a function c; = g('¥}) defined for arguments 7J in the closed i nterval [r1., �] . We call this functi on g the inl'erse off Since larger c; correspond to larger r; = f( q), the function g is again monotonic increasi ng. It is easy to show that the i nverse function g is also continuous.
The intermed iate value theorem as stated assigns ; for ' I in the open i nterval f(r) < 'I < f(b). However, of course, for 'I = ((a ) or 'I = f(b) we have only to take � = a or � = b. 1
Ch. 1
46 Introduction
y
a
Figure 1 .33
�-
E
�+
<
b
X
Continuity of the inverse of a monotonic continuous function. y
Figure 1 .34
Inverse functions.
Sec. 1.3
The Elementary Functions 47
Indeed, let '1 be any value between ex and {J (see Fig. 1 . 3 3). Then ; = g(TJ) must lie between a = g( ex ) and b = g({J). Let • be a given positive number which we can assume to be so small that a < .; - • < .; + • < b. We must show that !g(y) - g(TJ)! < • for all y sufficiently close to '1 · Since f is in creasing, '1 = f(i;) lies between the values j( e - • ) = A and f(; + •) = B and we can find a 6 so small that
A
< 'I
-o
< '1
+o
<
B.
If y is any value with '1 - /J < y < '1 + o and x = g(y ), we have A < y < B and hence g(A) < g(y) < g(B), that is, .; - • < g(y) < .; + • or ! g(y) - g(>J)I < <. The same proof, modified slightly, applies when '1 is one of the end points ex or {J of the interval of definition of g.
The relations y = f(x) and x = g(y) are equivalent and are repre sented by the same graph in the x,y-plane ; the points (x, y) i n the plane for which y = f(x) are the same as those points for which x = g(y). If we represent the function g i n the customary way by y = g(x), we must i nterchange x and y ; then the graph of y = g(x) is obtained from the graph of y = j(x) by taking the mirror i mage with respect to the line y = x. An example is given by the graphs of the function f(x) = x2 for x � 0 and of the inverse function g (x) = .J; for x � 0 (see Fig. 1 . 34). 1.3
The Elementary Functions
a. Rational Functions
We turn to a brief review of the fami liar elementary functions. The simplest types of function are constructed by repeated application of the elementary operations, addition and multiplication. If we apply these operations to an independent variable x and to a set of real n umbers a1, , a, we obtain the polynomials .
•
.
Polynomials are the simplest functions of analysis and in a sense the basic ones. Quotients of such polynomials, of the form y=
a o + alx + . . . + a,x" b o + b l x + . . . + b m xm '
are the general rational functions; these are defined at all points where the denominator differs from zero.
Ch. I
48 Introduction
The simplest polynomial, the linear function
y = ax + b, is represented graphically by a straight line. Every q uadratic function y = ax 2 + bx + c is represented by a parabola. The graphs of polynomials of the third degree y = axa + bx2 + ex + d, are occasionally called parabolas of the third order, etc. y
Figure 1 .35
Powers of x.
The graphs of the function y = x" for the exponents n = I , 2, 3, 4 are given in Fig. 1 . 35. For even values of n the function y = x" satisfies the equationf( - x) = f(x ), and is therefore an even function, whereas for odd values of n the function satisfies the condition f( -x) = -f(x), and is therefore odd. The simplest example of a rational. function which is not a polynomial is the function y = I fx mentioned on p. 27 ; its graph is a rectangular hyperbola. Another example is the function y = 1 f x 2 (cf. Fig. 1 .26, p. 36).
Sec.
The Elementary Functions 49
1.3
b. Algebraic Functions
We are at once forced out of the set of rational functions by the problem of forming their i nverses. The typical example of this is the function -yl;;, the i nverse of x". The function y = x" for x � 0 is easily seen to be monotonic increasing and continuous. It therefore has a single-valued i nverse, which we denote by the symbol x = Viy, or, i nterchanging the letters used for the dependent and i ndependent variables, By definition this root is always nonnegative. For odd values of n the function x" is monotonic for all values of x, i ncluding negative values. Consequently, for odd values of n we may extend the definition of Vi; uniq uely to all values of x ; i n this case .;v:;: is negative for negative values of x. More generally, we may consider
y = Vi R (x), where R(x) is a rational function. F urther functions of similar type are formed by applying rational operations to one or more of these special functions . Thus, for example, we may form the functions
y = V":;: + Y"x2 + I ,
y = x + .Jx2 + 1 .
These fu nctions are special cases of algebraic functions . (The general concept of an algebraic function will be defined in Volume I I.) c. Trigonometric Functions
The rational functions and the algebraic functions are defined directly by the elementary operations of calculation, but geometry is the source from which we first draw examples of other functions, the so-called transcendental functions.1 Of these we consider here the elementary transcendental functions, namely, the trigonometric functions, the exponential function, and the logarithm. In analytical i nvestigations angles are not measured in degrees, minutes, and seconds, but in radians. We place the angle to be measured 1 The word "transcendental" does not mean anything particularly deep or myste rious ; it merely suggests that the definition of these functions transcends the elementary operations of calculations, "quod algebrae vires transcendit."
50 lnlroduclion
Ch. J D
B
cot x ____.: I '
� cos x
Figure 1 .36
�
The trigonometric funct ions.
with its vertex at the center of a circle of radius I , and measure the size of the angle by the length of the arc of the circumference cut out by the angle.' Thus an angle of 1 80° is the same as an angle of 1r radians (has radian measure 1r), an angle of 90° has radian measure 1rj2, an angle of 45° has radian measure 7T/4, an angle of 360° has radian measure 27T. Conversely, an angl e of I radian expressed in degrees is 1 80°
or approximately 57° 17' 45".
Henceforth, whenever we speak of an angle angle whose radian measure is x.
x,
we shall mean an
y
Figure 1 .37 The radian measure of an angle can al3o be defined as twice the sponding sector of the circle of radius e>ne. 1
area
of the corre
The Elementary Functions Sl
Sec. 1.3
We briefly recall the meaning of the trigonometric functions sin x, cos x, tan x, cot x. 1 They are shown in Fig. 1 .36, in which the angle x is measured from the segment OC (of length I ), angles being reckoned positive in the counterclockwise direction. The functions cos x and sin x are the rectangular coordinates of the point A. The graphs of the functions sin x, cos x, tan x, cot x are given i n Figs. 1 . 37 and 1 .38. Later (see p. 2 1 5) we will be able to replace the geometrical definitions by analytical ones. y
,. I -I 81 Il l '"' I I I \ \
..
<:
.!3 II ,..,
Figure 1 .38
d. The Exponential Function and the Logarithm
In addition to the trigonometric functions, the exponential function with the positive base a, and its i nverse, the logarithm to the base a, x =
lo�;a y,
' I t is also sometimes convenient to i ntroduce the functions sec x = ! /cos x,
cosec x = 1 /sin x.
52
Introduction
Ch. I
are also i ncluded among the elementary transcendental functions. In elementary mathematics it is customary to pass over certain i nherent difficulties in their definition, and we too s hall postpone the detailed discussion of them until we have better methods at our disposal (cf. Section 2.5, p. 1 45). We can, however, at least indicate here one "elementary" way of defining these functions. If x = p/q is a rational number (where p and q are positive integers), then-the number a being assumed positi ve-we define cf' as {/-;; = a•l•, where the root, accord ing to convention, is to be taken as positive. Since the rational values of x are everywhere dense, it is natural to extend this function cf' to a continuous function defined for irrational values of x as well, giving to cf' when x i s i rrational, values which are continuous with the values already defined when x is rational . This defines a continuous function y = a', the "exponential function," which for all rational values of x gives the value of cf' found above. That this extension is actually possible and can be carried out in only one way we take for granted at the moment ; but it must be borne in mind that we still have to prove that this is so.1 The function x = loga y can then be defined for y > 0 as the i nverse of the exponential function : = log, y is that number for which y = a" .
x
e. Compound Functions, Symbolic Products, Inverse Functions
New functions are frequently formed not only by combining known functions by rational operations but by the more general and basic process of forming functions offunctions or compound functions. Let u = cf>(x) be a function whose domain is in the interval a � x � b and whose range lies in the i nterval r1. � u � {J. Moreover, let y = g(u) be a function defined for r1. � u � {3. Then g(cf>(x)) = f(x) defines a functionffor a � x � b which is "compounded" or "composed" from g and cf>. For example, f(x) = 1/(1 + x2") is composed of the functions cf>(x) = I + x2" and g(u) = ! fu. Similarly, the function f(x) = sin( l /x) is composed of cf>(x) = 1 /x and g(u) = sin u. It is useful to interpret the compound functions in terms of mappings. The mapping cf> takes every point x of the i nterval [a, b] into a point u i n the interval [ r1., {3] ; the mapping g takes any value u i n [ r1., {3] into a point y. The mappingfis the "symbolic product" gc/> of the mappings 1
This is done on p. 1 52.
Sec. 1.3
The Elementary Functions 53
g and cp, that is, the mapping carrying out cp and g successively, i n that order ; for any x in [a, b) we form its map u under the mapping cp, and then apply g to the image u = cp(x), obtaining g(cf>(x)) = f(x) = y (see Fig. 1 . 39). Such a symbolic prod uct gcp is natural and meani ngful for any type of operation ; it signifies that we first perform cf>, and then, on the result, perform g.1 We must not confuse the symbolic product gc/> = g( cp) of two functions with the ordi nary algebra ic prod uct g(x) · cf>(x) of the functions, in which both g(x) and cp(x) are formed for the same argument x (the mappings applied to the same point) and the product of the mlues of the functions is formed . N aturally, symbolic products cannot be expected to be commutative. In general, K(c/>) and cp(g) are not the same, even where both are defined ;
u
Figure
1 .39
Symbolic p rod uct g4> = J of two mappings. X
the order i n which operations are performed matters very much. I f, for example, c/> stands for the operation of "adding I to a n umber" and K for the operation of "multiplying a n umber by 2," then f((c/>(x)) = 2(x + I ) = 2x + 2,
cp(g(x)) = (2x) + I = 2x + I .
(See Fig. 1 .40. ) I n order t o b e able t o form the symbolic product gc/> o f two mappi ngs, the "factors" g and cp must fit together i n the sense that the domai n of g must include the range of cf> ; thus we cannot form gc/> when ;-
g(u) = ....; u,
and
1 That the product g4> corresponds to
cp(x) = - I
first carrying
out
-
x2 •
(in
that
order)
seems unnat ural at first glance, but actual ly corresponds to the convention al ways the symbol f for the function . Thus, for example, in sin ( l og x) it is al ways under stood that we fi rst form the logarithm of x and then take the sine of that, and not the
adopted in mathematics of writing the argument x of a fu nction j(x) other way around.
to the right
of
54 Introduction
Ch. 1
It is useful to consider functions which are compounded more than once. Such a function is f(x)
=
.) l + tan (x2),
which can be built up by successive compositions
cf>(x)
=
x2 ,
lp(c/>)
= l
+ tan cf>,
We would write symbolically f = g1pcf>. 0
g(1p) 2x
+ I
=
.J; = f(x).
2 (x + I)
0 0
X
Figure 1.40
Noncommutativity of mappin g s.
Inverse Functions The notion of "i nverse function" becomes clearer in the context of product of mappings. Consider the mapping associating with a point x of the domain of cf> the image u = cf>(x). Assume that our mapping cf> is such that different x are always mapped i nto different u. The mapping is then called "one to one." Then a value u is the image of at most one value x. We can associate with every u in the range of the value x = g(u) of which u is the image under the mapping cf>. I n this way we have defined a mapping g whose domain is the range of cf> and which when applied to an image u = cf>(x) of the -mapping reproduces the original value x, that is, g(c/>(x)) = x . We call g the inverse of cf>. It is characterized by the symbolic equation gcf>x = x. The Identity Mapping
We define the identity mapp ing I as the one that maps every x into itself; for the inverse g of cf> then, gcf> = 1. 1 The mapping I plays the same role for symbolic multiplication as the number I i n ordinary multiplica tion ; multiplication by I does not change a mapping. Accordingly, the equati on gcf> = I suggests the notation g = cf>-1 for the inverse of cf>. For example, the i nverse x = arc sin u of the function u = sin x is often denoted by x = sin-1 u.2 with /, in the domain of . This must not be confused with the al �;ebraic reciprocal l /(sin u).
1 More precisely g agrees
2
Sec. 1.4
Sequences 55
From the definition of the inverse g of cp it follows immediately that also is the ii1Verse of g so that not only g(c/>) = x but also cp(g(u)) = u.
• A monotone function u = cf>(x) defined in an interval a ::;; x ::;; b clearly defines a 1 - 1 mapping of that interval. If, in addition, is continuous, then as we saw earlier as a consequence of the intermediate value theorem (p. 44), the range of is the interval with end points (a) and (b). In that case the inverse g of exists and is again monotone and continuous in that latter interval. As a matter of fact the monotone continuous functions are the only continuous functions that have inverses or define one-to-one mappings . Indeed, let u = cf>(x ) be a continuous function in the closed interval [a, b] mapping different x of the interval into different u . Then in particular the values (a) = ex and (b) = {J are distinct. We assume, say, that ex < {J. Then we can show that cf>(x) is monotonic increasing throughout the interval. For if that were not the case we could find two values c and d with a ::;; c < d ::;; b for which (d) < (c). If here also (d) � ( a ) it would follow from the intermediate value theorem that there exists a � in the interval [a, c] for which W = (d). This � would be different from d and our mapping could not be 1 - 1 . If, on the other hand, (d) < cf>(a) = ex it would follow that (a) is intermediate between (d) and (b) ; there would then be a � intermediate between d and b for which cf>{;) = cf>(a) , and this also contradicts the 1 - 1 nature of .
An important, almost obvious property of compound functions, is that g(cp(x)) = j(x) is contin uous (where defined) if g and cp are. Indeed, for given positive e we have lf(x) - f(x0) 1 = /g(cp(x)) - g(cp(x0)) 1 <
e
for
1 (x) - cp(x0) 1 < b
as a consequence of the continuity of the function g. Since, however, .p is also continuous, we certainly have 1 (x) - cp( x0)1 < b for all x satisfying l x - x0 1 < b ' with some suitable positive b'. Hence lf(x) - f(x0 ) 1 < E
for
l x - x0 1 < b '
which shows the continuity of f. I t is much easier to appeal to this general theorem i n proving continuity of compound functions like ,j 1 - x2 than to try to construct directly a modulus of continuity for the function. 1.4 Sequences
Hitherto we have considered functions of a continuous variable, or functions whose domains consist of one or more i ntervals. How ever, numerous cases occur in mathematics in which a quantity a
Ch. I
56 Introduction
depends on a pos1t1ve integer n. Such a function a(n) associates a value with every natural number n. The function a(n) is called a sequence, specifically, an infinite seq uence, if n ranges over all positive integers. Usually, we write a n 1 instead of a(n) for the "nth element" of the sequence, and think of the elements forming a sequence arranged in order of increasing subscripts n :
Here the dependence of the numbers a. on n may be defined by any law whatsoever, and in particular , the values an need not all be distinct from each other. The idea of a sequence will most easily be grasped by examples. ,
I . The sum of the first n integers
S(n) = I + 2 + 3 + · · · +
n
=
!n(n + I )
i s a function of n , giving rise t o the sequence I , 3, 6, 1 0, 1 5, . . . . 2. Another simple function of n is the expression "n-factorial," the prod u ct of the first n integers . n ! = l · 2 · 3 · · · · · n. 3 . Every integer n > I which is n o t a prime number is divisible by more than two positive integers, whereas the prime numbers are divisible only by themselves and by I . We can obviously consider the number T(n) of divisors of n as a function of n itself. For the first few numbers it is given by the tab l e : 11 = 1 T(n) = I
2 3 4 2 2 3
5 6 7 8 9 2 4 2 4 3
10 4
II
2
12 6
4. A sequence of great importance i n the Theory of Numbers i s 1T(n), t h e number o f p r i mes less than t h e number n . I t s detailed investigation is one of the most fascinating problems. The principal result is : The number 1T(n) is given asymptotically,2 for large values of n, by the function njlog n, where by l og n we mean the l ogarithm to the "natural base" e, to be defined later (p. 77). 1 Pronounced "a�sub-n . "
2 That i s , t h e quotient of t h e number 77{ •) b y t h e number n/Iog n differs arbitrarily
little from one, provided only that n is large enough .
Sec. 1.5
1.5
Mathematical induction 57
Mathematical Induction
We insert here a discussion of a very i mportant type of reasoning which permeates much of mathematical thought. The fact that the whole sequence of natural numbers i s generated by starting w i th the number I and passing from n to n + I leads to the fundamental "pri nciple of mathematical induction." In the natural sciences we derive by em pi ri cal inducti on" from a l a rge number of samples, a law which is expected to hold gen e ral ly . The degree of certainty of the law depends then on the number of times a sample or an "event" has been observed and the law confirmed . This type of induction can be overwhelmingly convincing, although it does not ca rry with it the logical certainty of a mathematical proof. Mathematical induction i s u sed to est a b li sh with l ogical certainty the correctness of a theorem for an infinite sequence of cases. Let A denote a statement referri ng to a n arbi t ra ry natural n umber n. For example, A m i g h t be the statement "The sum of t he i nterior angles i n a simp l e po lyg on o f n + 2 sides i s n ti mes 1 80°" or mr. To prove a statement of this type it is not sufficient to prove i t for the first 1 0 or th e first 1 00 or even the first 1 000 va l ue s of n . I nstead, we have to a p p ly a mathematical method which we expl a i n first for this e xamp l e . For n = I the polygon reduces to a t riangle, for which t he sum of the angles i s k n own to be 1 80° . For a q uadrangle corresponding to n = 2 we d r aw a di ago nal divi d i ng th e q u a d r a ngl e into two triangles. This shows that the sum of the angles of the q uadrangle i s equal to the combi ned sum of the angles of the two triangles, that is, 1 80° + 1 80° = 2 · 1 80°. Proceeding to the example of a pentagon we can divide this i n t o a quadrangle and a tri angle by drawing a suitable diagonal . This yields for the sum of the angles of the pen tagon the value 2 1 80° + I · 1 8 0° = 3 · 1 8 0° We can go on i n this manner and pr o v e the ge nera l theorem successively for n = 4, 5, etc. The correctness of the statement A for any n fol l o ws from its correctness for the prec e d i ng n ; in t hi s way its general vali d i ty i s established for al l n. "
·
.
General Formulation
What is essential i n the proof of statement A i n o u r example is that A pr o v e d s uccessively for the special cases A 1 , A 2 , A n, . The poss i b i l ity of do ing this depends on two fac tor s : ( I ) a general proof has to be given showing th a t the statement A ," 1 is correct whenever A , is correct and (2) t h e statement A 1 must be proved . That these two is
•
.
conditions are sufficient to prove the correctness of all
•
.
.
.
A 1 , A 2, A a , . . .
58 lntroduction
Ch. I
constitutes the principle of mathematical induction. In what follows we accept the vali dity of this principle as a basic fact o f logi c . The principle can be formulated in a more general abstract form. "Let S be any set consisting of natural numbers which has the following two properties : ( I ) whenever S contains a number r, then it also contains the number r + 1 and (2) S contains the number I . Then it is true that S is the set of all natural numbers . " The previous formulation of the principle of mathematical induction follows if we take for S the set of all natural numbers for which statement A is correct. Often the principle is applied without specific mention or its use is indicated only by the expression, "etc." This happens particularly often in elementary mathematics. However, in more complicated situations an explicit appeal to the principle is preferable.
Examples. Two applications follow as illustrations. First we prove a formula for the sum of the first n squares. By som e trial we find for s m al l n, (say n < 5), that the fol l o w ing formula,' denoted by An, holds : n(n + 1 )(2 n + 1 ) 1 2 + 2 2 + 32 + . . . + n 2 . =
6
We conjecture that this formula is correct for all n. For the proof we assume that r is any number for which the formula Ar is correct, that is, that
1 2 + 2 2 + 32 + . . . + ,2
adding ( r + 1 ) 2 to
=
both sides, we obtain
1 2 + 22 + . . . +
,2
+ (r + 1 i
=
�
r( r + 1) 2r + 1)
;
r( r + 1)( 2 r + 1 ) + ( r + 6
! )2
( r + l )(r + 2) [2 (r + 1 ) + 1 ] 6
This, however, is j ust the statement Ar+ I obtained by substituting r + I fo r n in An. Thus the truth of Ar i m pli e s that of Arw To complete the proof of An for general n we need only to verify the correctness of A 1 , that is, of
1
Incidentally, this result was used by the Greek mathematician Archimedes in his work on spirals.
Mathematical Induction 59
Sec. 1.5
Since this i s obviously correct, the formula An i s established for all natural n . ·
The reader should prove by a similar argument that
13
+ 23 + 3 3 + . . . + n3 = [n( n : l)r
As a further i llustration for the principle of induction we prove
THE BINOMIAL THEORE M .
A n of the theorem is repre (a + b) " = a " + � a "-1 b + n ( n -2 1 ) a "-2 b2 + n(n - 1 )(n - 2 ) a -3b3 + . . . + n(n - I )( n - 2 ) · 2 · 1 b ". The statement
sented by the formula I
1
·
·
"
1·2·3
·
·
·
1 · 2 · 3 · · · · · (n - l ) · n
I t i s customary to write the formula i n the form
where the binomial coefficient
() n.
k
for k = I , 2,
=
n (n -
...
, n
-
I )(n
( ;) i s defined by
- 2) · · · ( n k!
- k +
1)
=
n! k ! (n -
k) !
I and
( ;) = ( : ) = I .
( ;)
(If we define 0 ! = 1 , the general formula for applies also to the cases k = 0 and k = n.) I f A , holds for a certain n , we find by m ultiplying both sides with (a + b) that
(a + w+ l = (a + b{ ( �) a" + ( � )a "-1 b + . . . + ( : )b "] = ( ; ) a n+ l + [(;) + ( �)} " b + [(�) + ( ; )} n-lb2 + . . . + [( � J + (:)Jab n + ( : )bn+1•
60
Ch. I
Introduction
Now
= n(n - 1) k (n! - k + 1) n(n - 1) · · ·(k(n -1)k! + I )(n - k) = n(n - 1 )(n - 2)k !· · · (n - k + 1 ) ( 1 + nk - k) = (n + I )n(n -(k 1+) · · )!· ( n - k + l) = (kn ++ 11 ) . l + n n Smce . (0) = ( 0 ) = and (nn) = (nn ++ J \ = I , we have (a + w+ l = ( n � l ) a •+l + ( n ; l ) a " b + ( n � l )a ·-l b 2 + . . . + ( n � 1)ab" + (: : � ) b•+t, · · ·
+
+
+ 1
I
I
L
l
which is the formula A n-t-t · Since also for n = I
(a + b/ = (�)a + ( !)b = a + b, the binomial theorem holds for all natural numbers n . 1.6
The Limit of a Sequence
The fundamental concept on which the whole of mathematical analysis ultimately rests is that of the limit of an infinite sequence A number a is often described by an i nfinite sequence of approxi mations ; that is, the value a is given by the value a. with any desired degree of precision if we choose the index n sufficiently large. We have already encountered such representations of numbers a as "limits" of sequences in their representations as infinite decimal fractions ; the real numbers then appeared as limits for increasing n of the sequences of ordinary decimal fractions with n digits. In Section 1 .7 we shall give a precise general discussion of the limit concept ; at this point we illustrate the idea of limit by some significant examples. Sequences at> a2 , can be depicted conveniently by a succession of "blocks," the element a . corresponding to the rectangle in the xy-plane bounded by the lines x I, x = a., = 0, having l a. l
a.
•
•
•
=n-
= n, y
y
a•.
The Limit of a Sequence 61
Sec. 1.6
as area, 1 or equivalently, by the graph of a piecewise constant function
a(x) of a centinuous variable x with jump discontinuities at the points x = n. a.
an = n
We consider the sequence 1 1 2 3
1, - '
'
.
.
0
'
-
n
'
•
•
•
(See Fig. 1 .4 1 .) No number of this sequence is zero ; but as the number an approaches zero. Furthermore, if we take any
n grows larger,
The sequence a. =
Figure 1 .41
�. I
interval centered at the origin, no matter how small, then from a definite index onward all n umbers an will be i n this interval . This situation is expressed by saying that as n i ncreases the numbers an tend to zero or that they possess the limit zero or that the sequence a1, a 2 , a3, converges to zero. I f the numbers are represented as points on a l ine, this means that the points 1 /n crowd closer and closer to the point zero as n increases. The situation is similar for the sequence •
•
•
I I 2 3
I
1, - - , - , - - , . . .
4
,
(
-
on-1
---
n
(See Fig. 1 .42.) H ere too, the numbers an tend to zero as n increases ; the only difference is that the numbers an are sometimes greater and sometimes less than the limit zero ; as we say, the sequence oscillates about the limit. 1
We might j ust as well have chosen the rectangle bounded by the lines x = n,
x = n + 1, y = a., y = 0 to represent a
•.
62
Introduction
Ch.
Figure 1 .42
The sequence
an
( _ 0n-l
=
--
n
1
.
The convergence of the sequence to zero is usually expressed sym bolically by the equation lim a n = 0, or occasionally by the abbreviation a"
----i"
0.
In the preceding examples, the absolute value of the difference between an and the limit steadily becomes smaller as n increases. This i s not necessarily the case, as i s shown by the sequence
1 1 1 1 1 - ' 1, - ' - ' - ' - ' 2 4 2 6 3
.
.
.
'
-
' - '
2m m
.
.
.
'
(see Fig. 1 .43) given for even values n = 2m by a" = a2m = 1 /m ; for odd values n = 2m - I by a n = a2,._ 1 = ! /2m. This seq uence
Figure 1 .43
The sequence a,n
1
1
= � a,n-1 = 2,; . ,
Sec. 1.6
The Limit of a Sequt>nce 6 3
also has the limit zero ; for every interval about the origin, no matter how small, contains all the n umbers an from a certain value of n onward ; but it is not true that every number lies nearer to the limit zero than the preceding one. n n + l
c. an = --
We consider the sequence al =
1
-
2
' a2 =
2
3
-
n
' . . . ' a n = -- ' n + 1
.
. .
Writing a n = I - 1 /(n + 1 ) , we see that as n i ncreases the number an will approach the number 1 , in the sense that if we mark off any interval about the point 1 all the n umbers an following a certain aN must fall in that i nterval . W e write lim a , = 1 . The sequence
n2 - 1 n2 + n + I
a n = --:-----
behaves in a similar way . This sequence also tends to a limit as n i ncreases, i n fact to the limit one ; lim a, = 1 . We see this most n� oo readily i f we write n + 2 a n = 1 - ---'--n2 + n + 1 we need only show that the n umbers r n tend to zero as n increases. For all values of n greater than 2 we have n + 2 < 2n and n 2 + n + 1 > n 2 • Hence for the remainder r "' we have 0 <
r,
2n
2
< - = n n2
( n > 2) ,
fro m which w e see that r " tends t o zero a s n i ncreases. Our discussion at the same time gives an esti mate of the largest amount by which the n umber an (for n > 2) can differ from the limit one ; this difference cannot exceed 2/n . This example illustrates the fact, that for large values of n the terms with the highest exponents in the n umerator and denominator of the fraction for an predominate and determine the li mit.
Ch. I
64 Introduction
d. a n =
fiji
Let p be any fixed positive number. We consider the sequence ah a2, a3, an, . . , where ,
•
•
,
We assert that
,
an = flp.
lim an = lim yrp = 1 .
n - oo
n - oo
We shall prove thi s by using a lemma that we shall also fin d useful for other p urposes. LEMM A .
(I)
If h is a positive number and n a positive integer, then ( I + h)" � I + nh.
This inequality i s a trivial consequence of the binomial theorem (see p. 59) accord ing to which
( I + h )" = 1 + n h + n (n
; I ) h2 +
· · ·
+ h ",
i f we observe that all terms i n the expa n s i o n of ( I + h)" are n on nega ti ve. The same argument yields the str on ger inequality
n(n - I ) h 2 . ( 1 + h )" � I + n h + 2
Returning to our sequence, we distinguish between the cases p > I and p < I ( i f p = I , then Vlp is equal to 1 for every n, and our statement is certainly true). I f p > I , the n Vlp also i s greater than I ; we set \;'p I + h,, where h,. is a positive quantity depending on n; by the inequality ( I ) w e have p = (I + h n)" � 1 + nh ,, implying - I 0 < h ,. :<;; P .
=
e
--
n
As n i n c re a s s the number h,. must te n d to 0, w h i ch proves t h a t a n converges to the limit one, as stated . At the same t i me we have a means for estimating how close any an is to the l i m i t one, since the difference hn between a,. and one i s not greater t ha n (p 1 )/n. I fp < I , then i /p > I and V" l /p converges to the limit one. However, -
VIP = , 1 · y J fp As the reciprocal of a quantity tending to one
flp i tself tends to one.
The Limit of a Sequence 65
Sec. 1.6
We consider the sequence a n = oc n , where oc i s fixed and n runs through the sequence of positive integers. First, let oc be a positive n umber less than one. We then put oc = 1 / ( 1 + h), where h is positive, and the inequality ( I ) gives a
n
=
1 1 < < -. ( I + hf 1 + nh nh -
--
Since h, and consequently l fh, depends only on oc and does not change as n increases, we see that oc" tends to zero as n increases : lim oc n
=
0
(O < oc < l ) .
The same relationship holds when oc is zero, o r negative but greater than - 1 . This is immediately obvious, since i n any case lim l oc i " = 0. J f oc = I , then oc" always i s equal to one and we shall have to regard the num ber one as the limit of oc". If oc > 1 , we put oc = 1 + h, where h is positive, and at once see from our inequality that as n i ncreases oc" does not tend to any definite li mit, but i ncreases beyond all bounds. We say that oc" tends to infinity as n i ncreases m that oc" becomes infinite ; in symbols, (oc > 1 ) . We explicitly emphasize that the symbol ro does not denote a number and that we cannot calculate with it according to the usual rules ; state ments asserting that a quantity is or becomes infinite never have the same sense as an assertion involving definite q uantities. In spite of this, such modes of expression and the use of the symbol ro are extremely convenient, as we shall often see i n the following pages. I f oc = - I , the value of oc" does not tend to any limit, but as n runs through the seq uence of positive i n tegers oc" takes the values + I and - I alternately. Similarly, if oc < I the value of oc" i ncreases numerically beyond all bounds, but its sign is alternately positive and negative. -
f. Geometrical Illustration of the Limits of oc" and
\lp
n I f we consider the graphs of the functions y = x " and y = x1 1 = \1:;; and restrict ourselves for the sake of convenience to nonnegative values of x, the preceding limits are illustrated by Figs. 1 .44 and 1 .45 respec tively. We see that in the i nterval from 0 to 1 the c urves y = x " come
66 Introduction
Ch. 1
y ., .. II
Figure
1 .., , .. '" I" I I I I
1.44 x'1 as n i ncreases.
closer and closer to the x-axis as n increases, whereas outside that interval they climb more and more steeply and approach a line parallel to the y-axis. All the curves pass through the point with coordinates x = I , y = I and the origin. The graphs of the functions 11 = x11" = + \1;;, come closer and closer to the line parallel to the x-axis and at a distance I above i t ; again all the curves must pass through the origin and the point ( I , 1 ) . Hence i n the limit the curves approach the broken line consisting of the part of the y-axis between the points y = 0 and y = I and of the parallel to the x-axis y = I . Moreover, it is clear that the two figures are closely related , as one would expect from the fact that the functions y = \1;;; are the inverse functions of the nth powers, from which we infer that for each n the graph of y = x" is transformed into that of y = V"'x by reflection in the line 11 = x.
Sec. 1.6
The Limit of a Sequence 61 y
Figure 1 .45
x' '" as
g. The Geometric Series
11
i ncreases.
An example of a limit fam iliar from elementary mathematics Is furni shed by the geometric series J + + q" I = S ; q + q2 + n .
.
•
the n umber q is call ed the common ratio or q uotient of t h e series. The value of this sum may, as is well known, be ex pressed in the form q
1 Sn = 1
q"
-q -
---
provided that '# 1 ; we can derive this expression by multiplying the sum S,. by q and subtracting the equation thus obtained from the original equation or we may verify the formula by division . What becomes of the sum S,. when n increases i ndefinitely ? The answer i s : The sequence of sums S,. has a definite limit S if q lies
Ch. 1
68 Introduction
between - I and + I , these end values being excluded, and S =
lim S n n ; oo
=
1- .
1 -q
In order to verify this statement we write Sn as (I - q")/( 1 - q) = 1 /( 1 - q) - q"/( 1 - q). We have already shown that provided l q l < I the q uantity q " tends to zero as n i ncreases ; hence under this assumption q n /(1 - q) also tends to zero and S" tends to the limit 1 /( 1 - q) as n increases. The passage to the limit lim (I + q + q2 + · · · + qn-I) = 1 /( 1 - q)
is u sually expressed by saying that when lql < I the sum of the infinite geometric series is the expression 1 /( 1 - q). The sums Sn of the finite geometric series are also called the partial sums of the infinite geometric series I + q + q2 + . . . . (We must draw a distinction between the seq uen ce of numbers q" and the partial sums of the geometric series.) The fact that the partial sums Sn of the geometric series tend to the limit S = 1 /( 1 - q) as n i ncreases is also expressed by saying that the infinite geometric series I + q + q2 + · · · converges to the sum S = 1 /( 1 - q) when lql < I . I n passing i t should be noted i f q i s rational, for example, q = ! or q = !, then the sum of the infinite geometric series has a rational value (in the cases mentioned the values are 2 and l, respectively). This observation is behind the well-known fact that periodic decimalfractions always represent rational numbers. 1 The general proof of this fact will be clear from the example of the number X =
0.343434
• • •
which can be evaluated by writing
=
34 34 I = 100 I - 1 / 1 00 99
-
-
1 See Courant and Robbins, What Is M.:Ithematics ?,
p. 66.
The Limit of a Sequence 69
Sec. 1 .6
We show that the sequence of numbers aa
tends to 1 as n i ncreases :
= -{l/3, . . . ,
l i m Vn = 1 .
Since a n exceeds the value 1 , we set a n = 1 + h n , with h,. positive. Then (see p. 64)
n = (a ,.r = ( 1 + h ,.t
h > - 1 + n n +
It follows for n > I that hence
h n2 h, < -
We now have
n(n
-
2 <
1)
hn > 2
n(n
-
2
1)
h"
2
_
2_ .
- n - 1 ' _
J2 Jn -
1
--=
·
1 � an = I + hn � I +
..j2 - . Jn - 1
The right-hand side of this ineq ual i ty ob v iously tends to one, and therefore so does a n . I.
a n = v1n + 1
-
-
I
vn
I n this example the a n are differences of two terms, each of which i ncreases beyond all bounds. Attempting to pass to the limit separately with each of the two terms, we obtain the meaningless symbolic expression oo - oo . I n such a case the existence o f a l i mit and what its val ue may be depends completely on the special case. We assert that in our example lim (Jn + 1 - /n) = o. For the proof we need only write the expression in the form
"In + 1
-
Jn =
1 (Jil+l - Jn>CJii+l + Jn) = Jn + 1 + J n Jn + 1 + Jn
and see at once that it tends to zero as n increases.
70 Introduction j.
a, =
n oc"
-
, for
Ch. I
oc > I
Formally, the limit of the a, is of the indeterminate type ooj oo already encountered in Example c. We assert that in this example the sequence of numbers a, = njoc" tends to the limit zero. For the proof we put oc = I + h, where h > 0, and again make use of the inequality
Hence for n > I
n 2 . a n = --- < (1 + W (n - lW Since an is positive and the right-hand side of this inequality tends to zero, an must also tend to zero. 1.7
Discussion of the Concept of Limit
a. Definition of Convergence and Divergence
From the examples d iscussed in Section general concept of limit :
1 .6
we abstract the following
Suppose that for a given infinite sequence ofpoints al> a2 , a3, there is a number l such that every open interval, no matter how small, marked off about the point !, contains all the points a, except for at most a finite number. The number I is then called the limit of the sequence a1, a2 , , or we say that the sequence a1, a2 , is convergent and converges to I; in symbols, lim a, = I. .
•
•
•
•
•
•
•
•
The following definition of limit is equivalent :
To any positive number <, no matter how small, we can assign a sufficiently large integer N = N(<) such that from the index N onward [that is, for n > N(•)) we always have lan - II < • · Of course, it is true as a rule that N( •) will have to be chosen larger and larger for smaller and smaller values of the tolerance • ; in other words, N( •) will usually increase beyond all bounds as e tends to zew. The vague intuitive notion of limit suggests a picture of the an moving closer and closer to I. This picture is replaced here by the precise "static"
Sec. I . 7
Discussion of the Concept of Limit 1 1
definition : Any neighborhood of I contains all an with at most a finite number of exceptions. 1 Obviously, a sequence a 1 a2, • • • cannot have more than one limit /. If on the contrary two distinct numbers I and /' were limits of the same sequence a1 , a2, • . . we could mark off open intervals about each of the points I and /' which do not overlap. Since each interval contains all but a finite number of the an , the sequence could not be infinite. The limit of a convergent sequence is therefore uniquely determined. Another obvious but useful remark is : If from a convergent sequence we omit any number of terms the resulting sequence converges to the same limit as the original sequence. ,
,
A sequence which does not converge is said to be dhoergent. If as n i ncreases the numbers a ,. increase beyond all positive bounds, we say that the seq uence diverges to + co ; as we have already done occasionally, we write then lim a , = co. Similarly, we write lim a n = - co if, as n n � oo
i ncreases, the numbers - a n i ncrease beyond all bounds i n the positive direction. But divergence may manifest itself in other ways, as for the sequence a1 = - 1 , a 2 = + I , a 3 = - I , a4 = + I , . , whose terms swing back and forth between two different values. Clearly, neither divergence nor convergence of a sequence is affected by removing finitely many terms. A sequence a 1 a 2 , • . • is bounded if there is a finite i nterval containing all points of the sequence. Any finite interval is contained in some finite interval that has the origin as center. Hence the requirement that the sequence is bounded means that there exists a number M such that I a ,. I � M for all n. A convergent sequence a 1 , a 2 , • • • necessarily is also bounded. For let I be the limit of the sequence. Taking • = 1 we find from the definition of convergence that all a,. from a certain N onward lie in the interval of length 2 centered at I. The only terms an of the sequence that may lie outside that interval are a 1, • • • , aN-t· We can then, however, find a larger finite interval that also i ncludes a 1 , • • • , a s_1 • .
.
,
b. Rational Operations with Limits
From the definition of l imit it follows at once that we can perform the elementary operations of addition, multiplication, subtraction, and division of limits according to the following rules. 1 The reader will notice the analogy with the defin ition of continuity of a function f(x) at a point x0 • The role played there by the sufficiently small quantity
here by the sufficiently large integer N(•). We shall see indeed on p.
rl{£) is played 82
that con
tinuity of a function at a point can be formulated in terms of limits of sequences.
Ch. I
72 Introduction
If at> a2, is a sequence with the limit a and b 1 , b2, is a sequence with the limit b, then the sequence of numbers en = an + b n also has a limit c, and e = lim en = a + b. n � oo •
•
•
•
•
•
The sequence of numbers en = a.bn likewise converges and lim e n = ab. Similarly, the sequence en = an - bn converges and lim en = a - b. n � oo Provided the limit b differs from zero, the numbers en = anfbn like wise converge and have the limit lim en = f!. . n � oo b In words : We can interchange the rational operations of calculation with the process of forming the limit ; we obtain the same result whether we first perform a passage to the limit and then a rational operation or vice versa. The proofs of all these rules become clear if one of them is carried out. We consider the multiplication of limits. If the relations an ---+- a and bn -+ b hold, then for any positive number E, we can insure both and by choosing
n
l b - bn l <
E
sufficiently large, say n > N( E) . If we write
and recall that there is a positive bound M, independent of n, such that lan l < M, we obtain
Since the quantity (lbl + M)E can be made arbitrarily small by choosing E small enough, the difference between ab and anbn actually becomes as small as we please for all sufficiently large values of n ; this is precisely the statement made in the equation
ab = lim a n b n . n � oo
Sec. 1 . 7
Discussion of the Concept of Limit 73
Using this example as a model, the reader can prove the rules for the remaining .rational operations. By means of these rules many limits can be evaluated easily ; thus, we have . I 1m
n2 - 1 = hm 2 n � oo n + n + 1 n � oo .
1 1
-
1
.!_ n2
1
+ - + -2 n n
= 1,
since i n the second expression we can pass directly to the limit i n the numerator and denominator. The following simple rule is frequently useful : If lim an = a and lim bn = b, and if in addition an > bnfor every n, then a � b. We are, however, by n o means entitled to expect that a will always be greater than b, as is shown by the sequences an = 1 /n, bn = l /2n, for which a = b = 0. c. Intrinsic Convergence Tests. Monotone Sequences
I n all the examples given the limit of the sequence considered was a known n umber. I n fact, to apply the above definition of limit of a se quence we m ust know the limit before we can verify convergence. If the concept of limit of a sequence yielded nothing more than the recognition that some known n umbers can be approximated by certain sequences of other known n u mbers, we should have gained very little from it. The advantage of the concept of l imit i n analysis lies essentially o n the fact that i mportant problems often have numerical solutions which may not otherwise be directly known or expressible, but can be described as limits. The whole of higher analysis consists of a succession of examples of this fact which will become steadi ly clearer in the following chapters. The representation of the i rrational n umbers as limits of rational numbers may be regarded as the first and typical example. Any convergent sequence of known numbers a1, a2 , defines a n umber /, its limit. However, the only test for convergence that arises from the definition of convergence consists in estimating the differences J an / J , and this is applicable only if the number I is known already. It is essential to have "intrinsic" tests for convergence that do not require an a priori knowledge of the value of the limit but only i nvolve the terms of the sequence themselves. The simplest such test applies to a special class of sequences, the monotone sequences, and i ncludes most of the i mportant examples. .
-
.
•
74 Introduction
Ch. 1
Limits of Monotone Sequences A sequence a1, a2 , is called monotonically increasing if each term an is larger, or at least not smaller than the preceding one ; that is, •
•
•
Similarly, the sequence is monotonically decreasing if an � an-I for all n. A monotone sequence is one that is either monotonically increas ing or decreasing. With this definition we have the basic principle : 1 A sequence that is both monotone and bounded converges. This principle is convincingly suggested, but not proved, by intuition ; it is intimately related to the properties of real numbers and in fact is equivalent to the continuity axiom for real numbers. The axiom (see Section ! b) that every nested sequence of intervals contains a point is easily seen to be a consequence of the convergence of bounded monotone sequences. For let [a 1 , b1], [a2, b2], be a sequence of nested intervals. By the definition of nested sequences we have •
•
•
Obviously the infinite sequence a1, a2 , is monotonically increasing. It i s also bounded since a1 � an � b1 for all n. Hence I = lim an ,
•
•
•
exists. Moreover, for any m and for any number n > m we have Hence also
a m � an � b m .
Thus all i ntervals of the nested sequence contain one and the same point /. (That they have no other point in common follows from the further property lim (bn - an) = 0 of nested sequences of i ntervals.)
Cauchy's Criteria for Convergence
A convergent sequence is auto matically bounded but need not be monotone (see Example b, p. 62). Hence , in dealing with general sequences, it is desirable to have a test for convergence that is also 1 The assumption of boundedness is essential since no unbounded sequence can converge. Oberve that a monotonically increasing sequence a, a,, . . . is al ways "bounded from below " : a. � a 1 for all n. In order to prove that a monotonically :5: M for all n.
increasing sequence converges it is sufficient then to find a number a.
M
such that
Sec. 1 . 7
Discussion of the Concept of Limit 7 5
applicable to nonmonotone sequences. This need is satisfied by a simple condi-tion, the Cauchy test for convergence ; this criterion characterizes sequences of real numbers which have a limit ; most i mportantly i t does not require a priori knowledge of the value of the l imit : Necessary and sufficient for convergence of a sequence a1, a 2 , •
•
•
is that the elements an of the sequence with sufficiently large index n differ arbitrarily little from each other. Formulated precisely : a sequence a, is convergent if for every e > 0 there exists a natural number N = N(e) such that I an - ami < E whenever n > N and m > N.
Geometrically, the Cauchy condition states that a sequence converges if there exist arbitrarily small intervals outside of which there lie only a finite number of points of the sequence. The correctness of Cauchy's test for convergence will be proved and its significance discussed in the Supplement. d. Infinite Series and the Summation Symbol
A sequence is just an ordered infinite array of numbers a1, a 2 ,
An infinite series
•
•
•
•
· · ·
a 1 + a2 + aa +
requires the terms to be added i n the order i n which they appear. To arrive at a precise meaning of the sum of an infinite series we consider the nth partial sum that is, the sum of the first n terms of the series The partial sums s, for different n form a sequence
and so on. The sum s of the infinite series is then defined as
provided this limit exists. In that case we call the i nfinite series con vergent. If the sequence sn diverges, the infinite series is called divergent; For example, the sequence 1, q, q2 , q3, gives rise to the infinite geometric series •
whose partial sums are
•
•
Ch. I
76 Introduction
For lq l < I the sequence sn converges toward the limit 1 -, I -q
s =
which then represents the sum of the infinite series. For lql � I the partial sums s n have no limit and the series diverges (see p. 67) . It is customary to use for a1 a2 · · · a n the symbol
+ + +
which indicates that the sum of the ak is to be taken with k running through the i ntegers from k = I to k = n. For example,
4 1 I stands for i! 1"J:1 k!
whereas n
I akbZk
k =l
More generally, values m, m
stands for
1 2 a b
+
2 a b4
n
I ak means the sum
k=m
+ 1, m + 2, . . . , n.
I
I
+ 2! + 3!1 + 4i , +
of all
a3b6 ak
+···+
a nb z n .
obtained by giving k the
Thus
� -k, = -3, + -4, + -5, 0 k-3 s 1
I
1
°
0
1
0
0
In these examples we have used the letter k for the index of sum mation . Of course, the sum is i ndependent of the letter denoting this index. Thus = I ak n
sn
We use the symbol
k=l
n
= I a,. i=l
00
to denote the sum of the whole infinite series. Similarly, I ak would
stand for the sum of the infinite series a0 a1 a2 . . . , whose nth partial sum is s. = a0 a1 a2 · · · a n_ 1 . Many of our earlier results can be written more concisely in this summation notation. The formula of p. 58, for the sum of the first n squares becomes 1) n(n 1)(2n = . k -1 6
+ + + + + + +
�Y
+
+
k=O
Sec. 1 . 7
Discussion of the Concept of Limit 77
The formula for the sum of a geometric series is 1
00
2 q k = - k-O J -q
for j q j < 1 .
Finally, the binomial theorem is expressed by (a + W =
i ( " ) a n-kbk.
k-0
k
Since an infinite series is merely the limit of a sequence sm convergence can be decided on the basis of the convergence tests for sequences. For example, the convergence of the series 1 1 1 1 = -1 + - + - + . 23 kk 1 22 3 1 k 00
. .
follows immediately from the fact that the partial sums s
n 1 I 1 1 1 =2 = - + - + - + · · · + -n n 3 11 -1 3 kk 2 n 2 k
increase monotonically with < sn 1 -
n
and are bounded since
I 1 1 1 +� 1 +-++ ···+2n 2 2 23 24 1 1 - 1 j2n- 1 I 3 1 = 1 += 1 + - - -n < - . 4 1 ! 2 2 2 -
Later, in Chapter 7, we shall study infinite series more system atically. e. The Number e As a first example of a number which is generated as the limit of a sequence, we consider
1 1 1 e= 1 +-+-+-+ ···. I ! 2! 3!
Thus e stands for lim Sn , where
1 1 1 1 s = 1 + -+- + ·· · +- . n
1!
2!
n!
1 Remembering the convention defining 0 ! as l , we can write the first term of the series as 1 /0 ! in agreement with the law of formation of the following terms. Notice that in our notation s. is really the (n + l )st partial sum of the infinite series, instead of the nth . This is, however, of no significance .
78
Introduction
Ch. I
The numbers e and 1r are the most widely used transcendental constants in mathematical analysis. In order to prove the existence of the limit e we need only prove that the sequence Sn is bounded since the numbers Sn increase monotonically. For all values of n we have Sn =
I +
1
1 1 1 + - + - + -- + 2
2 · 3
2 · 3 · 4
· · ·
1
2 · 3 ·4· · · ·
+ -----
� 1 + I + ! + .!. + .!. + · · · + -1n2 2 2 23 2 -l
· n
1/2" _ < 3 I - l The numbers S,. th e refo re have the upper bound 3, an d since they form a monotonic i n creas ing sequence, they possess a limit which we denote by e . The expression for e a s a series permits us to compute e rapidly with great accuracy. The error committed in approximating e by a partial sum S"' can be estimated by the same method of comparison with a geometric series that furnished the upper bou n d 3 for e . We have for any n > m = I +
1 -
I
s n = sm + ---- +
(m + l ) !
1
(m + 2) !
[ [ 1 + m +1 l + (m + 1 ) !
1 +...+n!
1
I I_ + +.. <S + I +_ - m (m + l) ! m + 2 (m + 2)(m + 3) < Sm
-
=S
m
__ +
1
+
I
(m + 1 ) !
Hence for n > m
1
I
_ _I_ m+ I
I
(m + l ) 2
+
I 1 = S +--. "' m m !
..
·]
·]
Letting n increase beyond all b o u nds while holding m fixed we find also that 1 I Sm < e � Sm + - - .
m m!
Hence e differs from Sm by at most ( 1 /m)( l /m !). Since m ! i ncreases e xtrem ely rapi d l y with m, the number Sm is a good approximation for e already for fairly small m ; for e x a mple , S1 0 differs fr om e by less than I 0-7• In this way we fi n d that e = 2. 7 1 828 1 · . ·
·
Discussion of the Concept of Limit 79
Sec. 1 . 7
e is an irrational number. The estimate for e in terms of Sm can also be used to establish this fact. Indeed, if e were rational, we could write e in the form p/m with positive integers p, m; here, m � 2, since e, lying between 2 and 3, cannot be an integer. Comparing e with the partial sum sm. we would have
s m < E.. :::;; sm + ..!._ _!_ m m m!
.
If we here multiply both sides by m ! , we find that 1
m ! sm < p (m - 1 ) ! :::;; m ! sm + - < m ! sm + 1 . m
But
m1 m' m' m ! S m = m ! + m ! + _: + _: + · · · + _: m! 3! 2!
-
is an integer since each term i n the sum is. Thus, if e were rational, the integer p(m 1 ) ! would lie between two successive integers, which is impossible. 1 e As Limit of ( 1 + 1 /n)n. The number e that was defined here as the sum of an infinite series can also be obtained as the limit of the sequence
The proof is simple and at the same time an instructive example of operations with limits. According to the binomial theorem,
( �r
Tn = 1 + =
1
+ n
�
1
+
= 1 +
n
- 1 ) .!._ .!_ ( 1 - � ) n
+
n(n
2!
n2
n(n - t ) (n
-
2) · · ·
1 ..!._ nn
n!
+ . ..
2!
1 The i rrational i ty of the number
with rational coefficients
+ .. . +
a, b
+
�(1 - �)(1 - �) . ( n. .
n.
n
.
t -
� n
)
e means that there is no l i near equation ax + b = 0 a � 0 having e as a solution. A much stronger
and
a 0x" + a,x• -1 + · · · + a._,x + a. = 0 of any degree n whatsoever and with rational coefficients a0, a,, . . . , a. (with a0 � 0) with x = e as a root . One says that e i s a transcendental n umber in contrast to "algebraic" numbers l i ke v2 or �10 that are statement has been proved (by Hermite), that there exists no polynomial equation
roots of certain polynomial e q uations with rational coefficients.
Ch. I
80 Introduction
From this we see at once that Tn � Sn <3. Furthermore, since we obtain Tn+ I from Tn by replacing the factors 1 - 1 /n, 1 - 2/n, . . by the larger factors 1 1 /(n + 1 ) 1 - 2/(n + 1 ) , . . . and finally adding a positive term we see that the Tn's also form a monotonic increasing sequence, from which the existence of the limit lim Tn = T follows. .
-
,
To prove that T = e, we observe that for m > T,. > 1 + 1 +
1
2!
-
( ) 1
1 -- + ···+ m
1 -
n!
(
1
-
n
) (
-1 · · · 1 m
n
-
1
--
m
)
.
If we now keep n fixed and let m increase beyond all bounds, we obtain on the left the number T and on the right the expression Sno so that T � Sn. Thus T � Sn � Tn for every value of n. We now let n increase, so that T. tends to T; from the double inequality it follows that T = lim S" = e. This was the statement to be proved.
We shall later (Section 2.6, p. 1 49) be Jed to this number e again from still another point of view. f. The Number 7T
as a
Limit
A limiting process which in essence goes back to classical antiquity (Archimedes) is that by which the number 7T is defined. Geometrically, 7T means the area of the circle of radius one. We regard it as obvious that this area can be expressed by a (rational or irrational) number, denoted by 7T . However, this definition is not of much help to us if we wish to calculate the number with any accuracy. We then have no choice but to represent the number by means of a limiting process, namely, as the limit of a sequence of known and easily calculated numbers. Archimedes already used this process in his method of exhaustion, which consists of approxi mating the circle by means of regular polygons with an increasing number of sides fitting it more and more closely. If we let/,. denote the area of the regular m-gon (polygon of m sides) inscribed in the circle, the area of the inscribed 2m-gon is given by the formula [proved by elementary geometry or from the expression /" = (n/ 2) sin (h/n) (see Fig. 1 .46)] f2 m =
�)2 - 2) 1 - (2fmfm)2.
We now let m range, not through the sequence of all positive integers but through the sequence of powers of 2, that is, m = 2 n ; in other words, we form those regular polygons whose vertices are obtained by repeated
Sec. 1 .7
Discussion of the Concept of Limit 8 1
bisection of the circumference. It is clear from the geometric interpre tation that tile J;.. form an i ncreasing and bounded sequence and thus have a limit which is the area of the circle :
This representation of TT as a limit serves actually as a basis for numerical computations ; for, starting with the value f4 = 2, we can calculate in order the terms of our sequence tending to TT . An estimate of the accuracy with which any term /2 • represents TT can be obtained by constructing the lines touching the circle and parallel to the sides of the inscribed 2"-gon. These l ines form a circumscribed polygon similar to the inscribed 2"-gon and having larger di mensions in the ratio I : cos (TT/2"). Hence the area F2 n of the circum scri bed polygon may be found from the ratio given by
n
( 7T )2 .
f2 = cos 2" F2•
Figure 1 .46
Since the area of the circumscribed polygon is greater than that of the circle, we have
For example, .{.
=
2 /2 , so that we have the estimate
These are matters with which the reader will be more or less familiar. What we wish to point out, however, is that the calculation of areas by means of exhaustion by rectilinear figures whose areas can be calculated easily forms the basis for the concept of integral, to be introduced in Chapter 2. For the actual numerical computation of TT much more efficient methods are available, as we shall see in Section 6.26.
82 1.8
Introduction
Ch. I
The Concept of Limit for Functions of a Continuous Variable
Hitherto we have considered limits of sequences, that is, of functions of an integral variable n . The notion of limit, however, frequently occurs in connec tion with a function j(x) that i s defined for al l x i n some interval . We say that the value of the function f(x) tends to a l imit r1 as x tends to �. or in symbols, Iim j(x) = TJ x-�
if f(x) differs arbitrarily l ittle from TJ fo r all x for which f(x) is defined and which lie sufficiently near to U Expressed more precisely the definition of l i m f (x) i s as follows. Whenever an a r bitrary positive quantity • is assigned, we can mark off an interval lx - �I < b so small that for any x which belongs both to the domain off and to that interval the i ne q uality l f (x) - r1 l < • holds, then limj(x) = TJ. There i s a close connection between the concepts of l i mi t of a function and continuity. I f � belongs t o the domain o f j; that i s, if /(�) i s defined , then l i m .f(:r), i f it exists a t all, must have the v al ue /{�). .... 4 Indeed , the definition of?) = I i m f(x) i mplies in particular 1/W - TJ I < • X
for every positive • and hence TJ = j(;). Now, comparing the defin itions of l i m it and of continuity, we see that the relation lim f(x) = f(;) x-�
j u st expresses the continuity of the function fat th e point ;. Hence for ; in the domain off the existence of lim f(x) j ust signifies that ( is con x-•;
tinuous at ;. M ore gene ra l l y, if f(x) is not defined at ; but l i m f(x) x
-·�
exists and has the val ue TJ, we can assign to fat the poi nt ; the value TJ and the function f. thus completed, will be continuous at ;. (Removable Singularity. See p. 35.) The limit of a (unction c a n also be descri bed completely in terms of limits of sequences. The statement l i m j(:r) =
means that
x-�
rJ
n � oo
for every seq uence xn with limit � (where it is assumed, of course, that the xn belong to the domain of j) . For if l i m j(x) = "' and if l i m xn = �. then f(x) n- oo x -�
1 It is assumed here that arbitrarily close to ; there are points where f is defined.
Sec. 1.8
The Concept of Limit for Functions of a Co11tinuous Variable 83
is arbitrari ly close to 'I for x sufficiently close to $ ; but ·'"n is sufficiently close to $ if only '! is large enough, and consequently, l i m [(.rn ) = 'I · l f, on the other hand, l i m .f(x.) = 1J whenever
·" n
$, we must have lim [(.r) = 'I·
Otherwise there would exist a positive ' such that l[(a:) - •71 :<: ' for some x arbitrarily close to � ; there would then also exist a sequence ,.n converging to $ for which l .f(.r,) - •71 :<: '• but then lim [(J·.) could not be 'I · }J --+ C(I
___,.
:r; - t'- �
Continuity of the fu nction [(.r) at the point $ implies then : lim [(.rn l = [($), n - .. oo
for every seq uence ,." i n the domain of .f that converges to �. M ore generally, for a function continuous i n an interval the relation lim f(.r.,) n - oo
=
[( l i m J.·. ) n - oo
is val id for any sequence in the domain of{ which converges to a poi nt of the intervaL We see that for a continuous function the limit symbol can be i nterchanged (or, as one says, "commutes") with the symbols for the function.
Li m its of s ums, products, and q uotients of functions are fo und by the same rules as for seq uences (see p . 7 1 ) : I f l i m f(x) = 'IJ and x- · < l i m g(.r) = '· exist, then l i m (j(x) + g(x)) = "' + S,
and for s � 0 also
l i m ( f (:r)g( >:) ) = 'f/S
x _, �
The proofs are the same as for sequences . (The rules would also follow from th ose for sequences by writing l i m its of fu nctions as l i m its of seq uences.) Consequently, when � belongs to the domain of f and g. the sum , product, and quotient of t wo functions f(x) and g(x) which are con tinuous at a point E are again continuous (where for q uo tients we have to assume that g( �) r!' 0). The cases where � does not belong to the domain of.f w i l l t u rn out to be of particular i mportance for differential calculu s . As a first example we consider the rel ation n
tn
l i m � = n e'- 1 , x� < X - �
for n a positive i n tege r . Of course, [(x) = (x" - �")/(x - �) i s a function defi ned only for x � �. But for x � � the algebra ic identity
Ch. 1
84 Introduction
is valid as a consequence of the summation formula for the geometric series. To find the limit we only have to let x tend to � and to evaluate the limit of the right-hand side by the rules for limits of sums and quotients. Less obvious is the formula . sin x 1 I I m -- = z...,. O
X
(where, of course, the angle x is measured in "radians," as explained on p. 50). Again the quotient (sin x)jx is defined only for x � 0. c
0
Figure 1.47
But, if we define (sin x)jx = I for x = 0 we complete the quotient as a function which is continuous also at x = 0. For the proof of the limit formula we appeal here to a geometric argument. From Fig. 1 .47 we find by comparing the areas of the triangles 0 AB and OAC and the sector OAB1 of the unit circle that if 0 < x < 1rj2 ! sin x < !x < ! tan x.
From this it follows that if 0 < Jxj < 1rj2,
1 <
X
-
sin x
<
1 --
cos x
.
Hence the quotient (sin x)jx lies between the numbers 1 and cos x. We know that cos x tends to 1 as x � 0, and from this it follows that the quotient (sin x)jx can differ onl�r arbitrarily little from 1 , provided that 1 Of course, we could have defined the angle x in the first place as twice the area of sector OAB.
Sec. 1.8
The Concept of Limit for Functions of a Continuous Variable
85
x is near enough to
0. This is exactly what is meant by the equation which was t'O be proved . From the result just proved it follows that
I . tan x 1 . sin x 1 . I tm = tm -- tm -- = 1 x -+ 0 X x-+0 X :r -+ 0 COS X
and also
lim x -+ 0
-l
cos X
x
=O
,
.
This l ast follows from the formula, valid for 0 < lxl < TT/2,
- COS X
-
(I
X
=
1 - cos 2 x x(l + cos x)
cos x)(1 + cos x) x(l + cos x)
x . _ s t_ n_
X
1
.
___
+ COS X _
sin x.
For x 0 the first factor on the right tends to 1 , the second to t, and the th ird to 0 ; the product therefore tends to 0, as was stated. Dividing the same formula by x, we obtain �
- cos x
x2
=
from which lim 1 x�o
( x)2 sin X
- cos x
x2
_
___
1
=
+ COS X
! 2
Limits for x r:JJ . Finally we remark that it is just as well possible to consider l imiting processes in which the continuous variable x i ncreases beyond all bounds. For example, the meaning of the equation ___..,
x � oo
lim
x2 + 1 + l / x2 I = lim l = 2 x - 1 x� ao 1 1 / x 2
-
is clear. It signifies that the function on the left differs arbitrarily little from one, provided only that x is sufficiently large. The rules for forming the limits of this kind for sums, products, and quotients are the same as before. * There is one further result which is frequently useful i n the calculation of limits, the rule for obtaining the l imit of a compound function. The compound function f(g(z)) is defined for those values of z for which x = g{z) lies in the domain off(x). The function g(z) may be a function of a continuous variable or an integer variable, but j(x) must be a function of a continuous variable.
86
Introduction If lim g(z) ·--
=
Ch. I
� where � lies within an open interval of the domain of
! and if lim f(x) �-·
=
'1),
then lim f(g(z)) · --
=
1J ·
As a corollary we observe
that a continuous function of a continuous function is itself continuous (as already mentioned on p. 55). The result is obvious from the fact that we can make f(x) arbitrarily close to 1J by taking x sufficiently close to � and to make x = g(z) close enough to � we have only to take z sufficiently close to ' · With slight modifications, the same statements apply when any of the variables is allowed to increase beyond all bounds. a. Some Remarks about the Elementary Functions
So far we tacitly assumed that the elementary functions are con tinuous. The proof of this fact is very simple. First, the function j(x) = x is continuous ; therefore x2 = x · x is continuous, as the product of two continuous functions, and every power of x is likewise continuous. Thus every polynomi al is continuous, being the sum of contin uous functions. Every rational function, as a quotie nt of con tinuous functions, is likewise continuous in every interval in which the denominator does not vanish. The function xn is continuous and monotonic for x � 0. Hence the nth root, being the inverse function of the nth power, is continuous. From this fact i t is easy to conclude that the nth root of a rational functi on is continuous (except where the denominator vanishes). The continuity of the trigonometric functions could now be proved, using the concepts already developed. However, we omit the dis cussion here, since in Chapter 2 (p. 1 66), the continuity of all these functions will be seen to follow simply as a consequence of their differentiability. W e merely make a few remarks about the definition and continuity o f the exponential function a�, the general power function x•, and the logarithm . 'Ye assume, as in Section 1 . 3 (p. 5 1 ), that a is a positive number, say greater tlian one, and r = p/q is a positive rational number (p and q being integers) ; then a' = aPia is the positive number whose qth power is a1'. If IX is any irrational number and r1, r2 · · • r,., . . . is a sequence of rational numbers approaching IX, we assert that lim a'm exists ; we then call this limi t a•. m- oo
In order to prove the existence of this limit by Cauchy's test, we need show only that l ar• - a'm l is arbitrary small, provided that n and m are sufficiently large. We suppose, for example, that rn > r m • or that rn - r m = o, where tJ > 0. Then ar• - arm = a•m(� - 1 ).
Since the r converge to suffices to shew that m
Supplement
ex,
81
they are bounded and so are the a'"' ; thus it
I ll" - I I = a6 - I
is arbitrari ly small when the values of n and m are sufficiently large. However, the rational number <'l certainly may be made as small as we please provided the values of n and m are sufficiently large. Hence if I is an arbitrarily large positive integer, o < 1 // if n and m are large enough. Now the relations <'J < I f/ and a > 1 give1 1 < a� < a l lz,
and since a111 tends to one as f increases to infinity (cf. p. 64), our assertion follows i m mediately. It can be shown that the function ax extended to irrational values in this way is also continuous everywhere, and, moreover, that it is monotonic. For nega\ive values of x this function is naturally defined by the equation 1
ax = - . a-x
As x runs from oo to + oo , ax takes all val ues between zero and + oo . Consequently, it possesses a continuous and monotonic i nverse function, which we call the logarithm to the base a. I n like manner we can prove that t he general power x• is a continuous function of x, where ex is any fixed rational or i rrational number and x varies over the interval 0 < '� < oo, and is monotonic if ex ¢ 0. The "elementary" discussion of the exponential function, the logarithm, and the power x• outlined here will later (p. 1 49) be replaced by another discussion wfiich in principle is much simpler. -
Supplement
One of the great achievements of Greek mathematics was the reduction of mathematical statements and theorems in a l ogically coherent way to a small n umber of very simple postulates or axioms, the well-known axioms of geometry or the rules of arithmetic governing relations among a few basic objects, such as integers or geometrical points. The basic objects originate as abstractions or idealizations from physical reality. The axioms, whether considered as "evident" from a philosophical point of view or merely as overwhelmingly pla usible, are accepted without proof; on them the crystalized structure of mathematics rests. For many centuries t he axiomatic Euclidean ' This statement fol lows from the fact that for a > I the power a"' I • is greater than
one if mfn is positive. For
a = (a'l•)'•
one, and so is greater than one.
is the product of m factors all greater than
88 Introduction
Ch. I
mathematics was accepted as a model for mathematical style and even imitated for other i ntellectual endeavors. (For example, philosophers, such as Descartes and Spinoza, tried to make their speculations more convincing by presenting them axiomatically or, as they said, "more geometrico. ") The axiomatic method was discarded when after the stagnation during the Middle Ages mathematics in union with natural science started an explosively vigorous development based on the new calculus. I ngenious pioneers vastly extending the scope of mathematics could not be hampered by having to subject the new discoveries to consistent logical analysis and thus in the seventeenth century an invocation of i ntuitive evidence became a widely used substitute for deductive proof. Mathematicians of first rank operated with the new concepts guided by an unerring feeling for the correctness of the results, sometimes even with mystical associations as in references to "infinitesimals" or "infinitely small quantities." Faith in the sweeping power of the new manipulations of calculus carried the investigators far along paths i mpossible to travel if subjected to the limitations of complete rigor. Only the sure instinct of great masters could guard against gross errors. The uncritical but enormously fruitful enthusiasm of the early period gradually met with countercurrents which rose to full strength in the nineteenth century but did not impede the development of constructive analysis i nitiated earlier. Many of the great mathematicians of the nineteenth century, in particular Cauchy and Weierstrass, played a role in the effort toward critical reappraisal. The result was not only a new and firm foundation of analysis, but also increased l ucidity and simplicity as a basis for further remarkable progress. An important goal was to replace indiscriminate reliance on imprecise "intuition" by precise reasoning based on operations with numbers ; for naive geometric thinking leaves an undesirable margin of vagueness as we shall see time and again i n the following chapters. For example, the general concept of a continuous curve eludes geometrical i ntuition. A continuous curve, representing a continuous function, as defined earlier, need not have a definite direction at every point ; we can even construct continuous functions whose graphs nowhere have a direction, or to which no length can be assigned. Yet one must never forget that abstract deductive reasoning is merely one aspect of mathematics while the driving motivation and the great universal scope of analysis stem from physical reality and intuitive geometry. This supplement will provide a rigorous buttressing (with some repetitions) for basic concepts treated intuitively earlier in this chapter.
Sec. S. l S.l
Limits and the Number Concept 89
Limit.s and the Number Concept
We start with the ideas of Section 1 . 1 , analyzing fully the concept of real number and its connection with that of limit. We define the number continuum by a constructive procedure based on the natural numbers. We then prove that the extended number concept satisfies the rules of arithmetic and the other requirements, making it the adequate tool for measurement. Since a complete exposition would require a separate book,1 we shall ind icate only the main steps. In struggling through the somewhat tedious material the student will marvel at the fact that on the basis of the natural numbers the human mind could erect a logically consistent number system superbly suited to the task of scientific measurement. 2 a. The Rational Numbers
Limits Defined by Rational Intervals. We begin by accepting the system of rational numbers with all its usual properties, derived from the basic properties of natural numbers. Thus the rational numbers are ordered by magnitude, permi tti ng us to define "rational" i ntervals as sets of rational numbers lying between two given rational numbers (intervals includ ing the end points are called closed). The length of the interval with end points a, b is lb - a l . As observed in Section Ia the rational numbers are dense and every rational interval contains infinitely many rational numbers. For the time being, all quantities occurring are assumed to be rational numbers. Within the domain of rational numbers we define sequences and l i mits (see p. 70). Given an infinite sequence of rational numbers a1, a2, and a rational number r we say that •
•
•
lim a. = n � oo
1 960.
1 See for example, E . Lan dau,
r
Foundations of Analysis, 2nd E d . , Chelsea, New York,
' Real numbers can also be introd uced p urely axiomatically, with all their basic properties accepted as
axioms.
In the approach we shall take here we accept, in
principle, only the axioms for natu ral n umbers (i ncluding the principle of mathe matical i n d uction).
The rational n u mbers and real numbers are then
constructed theorems
on that basis. The "axioms" for real nu mbers are then, i n principle, merely about natural numbers for which proofs are required .
Actually, we shall start
already with the rational numbers as known elements, since the construction of the rational from the natural n umbers and the derivation of the basic properties of rational n umbers present no d ifficulties at all.
90
Ch. I
Introduction
if every rational interval contammg r in its interior also contains "almost all" an, that is, all an with at most afinite n umber of exceptions. It follows immediately that a sequence of rational numbers cannot have more than one rational limit and that the usual rules for limits of sums, differences, products, and quotients (see p. 7 1 ) are valid for sequences of rational numbers with rational limits. An entirely obvious consequence of this definition is that passing to the limit preserves order : if lim a,. = a, lim bn = b and for every n , an ::;: bn, then a ::;: b. Note that even assuming an < bn strictly, we cannot say more than a ::;: b, or exclude possible equality of the limits (for example, both sequences an = 1 - 2/n and bn = I - 1 /n > an have the limit 1 ) . Statements about limits can b e expressed in terms o f rational null sequences, that is, sequences a,, a2, of rational numbers for which •
•
•
lim a n = 0.
n � oo
One says a,. "becomes arbitrarily small a s n tends t o infinity," meaning that for any positive rational £, no matter how small, the inequality l a, l < £ holds for almost all n. Obviously the sequence a,. = 1 /n i s a null-sequence. Thus a sequence of rational numbers a,. has the rational limit r if and only if the numbers r - an form a null-sequence. b. Real Numbers Determined by Nested Sequences of Rational Intervals
We observed on p. 5 that intuitively the rational points are dense on the real axis and that there are always rational n umbers between any two real numbers. This suggests the possibility of rigorously defining a real number entirely in terms of order relations with respect to the rationals, a procedure we shall now follow. A nested sequence of rational intervals (see p. 8) is a sequence of closed intervals Jn with rational end points ano b., with each interval contained in the preceding one, whose lengths form a null-sequence and
an-1 ::;: an ::;: bn ::;: bn -1
Since each interval Jn = [an, bJ of a nested sequence contains all succeeding intervals, a rational number r lying outside any Jn also lies outside and on the same side of all succeeding intervals. Thus a nested
Sec. S.I
Limits and the Number Concept
91
sequence of rational intervals gives rise to a separation of all rational numbers iRto three classes. 1 The first class consists of the rational n umbers r lying to the left of the i ntervals Jn for sufficiently large n , or for which r < an for almost all n . The second class consists of the rational numbers r contained in all intervals Jn. This class contains at most one number, since the length of the interval J" shrinks to zero with i ncreasing n . The third class consists of the rational numbers r for which r > bn for al most all n. It is clear that any number of the first class is less than any of the second class, and any number of the second class is less than any of the third class. The points an themselves are either in the first or second class, and the numbers bn either in the second or third class. If the second class is not empty, it consists of a single rational number r. In this case the first class consists of the rational numbers less than r, the third class of the rational numbers greater than r. We say then that the nested sequence of i ntervals J" represents the rational n umber r. For example, the nested sequence of i ntervals [r I (n, r + 1 /n] represents the number r. If the second class is empty, then the nested sequence does not represent a rational number ; these nested sequences then serve to represent irrational numbers. The individual intervals [an, bnl of the sequence are for this purpose unimportant ; only the separation of the rational numbers i nto three classes generated by this sequence is essential, tel ling us where the i rrational number fi t s in among the rational ones. Thus we call two nested seq uences of rational intervals [an, bnl and [a ,/ , b/] equivalent if they give ri se to the same separation of the rational numbers i nto three classes. The reader should prove as an exercise that necessary and sufficient for equ ivalence is : a,' - an is a null-seq uence, or also : the inequalities -
hold for all n. We assign a real number to a nested sequence of rational intervals [an, b , ] . The real numbers determined by two different nested sequences will be considered to be equal if the sequences are equivalent. A real number then is represented by the separation of the rational numbers i nto three classes generated by equivalent nested seq uences of rational i ntervals. If the second class consists of a rational number r, we con sider the real number represented by this separation into c lasses as i dent ical with the rational number r. 1 A so-cal led "Dedek ind
Cut."
92
Introduction
Ch. 1
*c. Order, Limits, and Arithmetic Operations for Real Numbers
Having defined real numbers, we can now define the notions of order, sum, difference, product, limit, etc., for them and prove that they have the usual properties. To be consistent any definition concerning real numbers must : ( 1 ) have the ordinary meaning in case the real numbers are rational and (2) be independent of the individual nested sequences intervals used to represent the real numbers. •
Intervals with Real End Points
Although so far, even for the definition of irrational numbers, the end points of nested intervals were assumed to be rational, we must now remove such restrictions and show that we can operate with real numbers exactly as we do with rational numbers. In carrying out this program we have to be careful at each step to avoid reliance on facts not yet proved by logical deduction from our basis of departure, the rational numbers. We shall denote real numbers by letters x, y, . . . . If the real number x is given by the nested sequence of rational intervals [a,., b,.], we write x """ {Ia,., b .. ]}. From our definition of real number we draw a natural definition of order for a real number x """ {[a .. , b,.l} relative to a rational number r. We say that r < x, r = x, r > x according as r belongs to the first, second, or third class of the separation of the rational numbers generated by the sequence of nested i ntervals. This definition is obvi ously independent of the special nested sequence { [a.., b,.l} defining x and has the ordinary meaning when x is rational. Equivalently, we say that. r < x if r < a,. for almost all n, r = x if a,. ;:5; r � b .. for all n , and r > x if r > b .. for almost all n . By comparing real numbers with rational numbers we can compare real numbers with each other. Let x """ {[a.. , b .. ]}, y """ {(<X,., �.. ]}. We say x < y if there is a rational number r such that x < r < y. Clearly, this definition does not depend on the particular representations of x and y by nested sequences since comparisons with rational r are independent of such representations. Thus we say that x < y if there exists a rational r such that b 11 < r < <X11 for almost all n, or simply if b .. < <X11, for almost all n. The relation x < y precludes the possibility that y < x or x = y. Obviously x < y and y < z implies x < z. For any two real numbers x and y, one of the relations x < y, x = y, y < x must hold. For if x "" y and either number, say y, is rational, then y must be in the first or third class of the separation generated by x, that is, either & < x or x < y. If neither x nor y is
Limits and the Number Concept 93
Sec. S.l
rational, the second classes of the corresponding subdivisions are empty, a m;J there must be a rational number r in the first class with respect to one of the numbers and in the third class with respect to the other. Thus either x < y or y < x . Density. An i mmediate consequence of these definitions is the density of the rational numbers in the sense that between any two real numbers x, y there is always a rational number r. We also observe that if a real number x is represented by a nested sequence of rational i ntervals [am bnJ, then an � x � bn for all n. For i f x < am for some m, then bn < am for almost all n, contradicting the i nequality am < bn which holds for all n. Hence every real number can be confined to a rational i nterval [an, bnJ of arbitrarily small length. Once the real n umbers are ordered we can talk of intervals with real end points. The density of the rational numbers guarantees that every such i nterval i ncludes rational numbers . Limits. A real number x is called the limit of a sequence xi> x2, of real numbers if every open interval with real end points containing x also contains xn for almost all n. This definition is consistent with the definition in terms of rational intervals given earlier, in the sense that a rational limit of a rational sequence i s a l imit of the same sequence of numbers in the more general sense of a real limit. As a consequence of the definition of limit we find that for a real number x represented by a nested sequence of rational intervals [an, bnJ •
•
•
x = lim a,. = lim bn. n - oo n -+ co
*Arithmetic. W e next define the arithmetic operations for real numbers x ,....., {[a,., bn] } and y ,....., { [ocn, f1n l} : This is achieved most easily for the operations of addition and subtraction. We define X
+ y ,....., { [an + oc n, bn + fJ n J} ,
X - Y ,.._, { [an - fJ,. , bn - oc n l}.
To prove these definitions meaningful is a simple exercise whose details are left to the reader (see Problem 3 , p. 1 1 6). For example, for x - y it is necessary only to verify the i ntervals [an - f1n, bn - oc] form a nested sequence with lengths tending to zero, and hence that they represent a real number z . The fact that z does not depend on the special repre sentations of x and y is proved by characterizing the separation of rational numbers into three classes generated by z directly in terms of x and y ; for instance, the first class consists of the rational numbers r < z , or of the r which are exceeded by an - f1n for some n ; these r are easily seen to be the rational numbers of the form s t, where s and t are rational numbers for which s < x and t > y. -
Ch. I
94 Introduction
The product of the two real numbers x, y is for y > 0 defined by X • Y ,..._, { [an <X n , b n.Bn]},
where we have assumed that all ocn > 0; it is obvious what nested sequences are proper to use for xy in the case y < 0 and y = 0. When ever y is a positive rational number, the product x · y also is representable in the form For a natural number y = m, the product x · y = mx also can be obtained by repeated addition of x, that is, mx = x + (m I )x =
X+X+ ""
•
-
+ X.
The arithmetic operations obey the usual laws. In particular, the relation x < y is equivalent to 0 < y x. We can introd uce the absolute value of a real number and prove the triangle inequality lx + Yl � I xi + I YI · The notion of limit of a sequence of real numbers defined above in terms of order relations can then be given the equivalent formulation : x = lim x n if for every real positive € the relation -
lx
xnl < € holds for almost all n. We now verify the so-called -
AXIOM OF ARCHIMEDES. If x and y are real numbers and x is positive, then there exists a natural number m such that mx > y.
In essence this means a real number cannot be "infinitely small" or "infinitely large" compared with another (except if one of them is zero) . To prove the Axiom of Archimedes (which in our context is really a theorem) we observe that for rational numbers it is a consequence of the common properties of integers. If now x ,..._, { [an, b,]} and y { [ocn, .Bnl} are real numbers and x is positive, then a, > 0 for almost all n . Since an and f3n are rational numbers, we can then find an m so large that ma n > .Bn, whence mx > .Bn � y. d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria
Real numbers make possible limit operations with rational numbers, but they would be of little value if the corresponding limit operations carried out with them necessitated the introduction of some further kind of "unreal" numbers which would have to be fitted in between the real ones, and so on ad infinitum. Fortunately, the definition of real number is so comprehensive that no further extension of the
Sec. S. J
Limits and the Number Concept 95
number system is possible without discarding one of its essential properties.(as "order" must be discarded for complex numbers).
Principle of Continuity Thi s completeness of the real number continuum is expressed by the basic continuity principle (cf. p. 8) : Every nested sequence of intervals with real end points contains a real number. To prove this, consider closed intervals [x n , Ynl, each interval contained in the preceding one, whose lengths y,. - x n form a null-sequence. We claim there is a real x contained in all [x, Y n l : The sequences x,. and y,. will then h ave x as limit. To prove this we replace the nested sequence [xn, y,.] by a nested sequence of rational intervals [a m b,.], containing the [x, y,.]. This rational sequence will then define the desired real number x. For each n let a, be the largest rational number of the form p/2" less than x,, and b,. the smallest rational number of the form q/2" greater than y , w here p and q are integers. Clearly, the intervals [ a ,. , b,] form a nested sequence representing a real n umber x. I f x lay outside one of the intervals [xm, Yml• say x < xm, there would exist a rati onal r with x < r < x , . . whence for all sufficiently large n we would have which is impossible. Hence all intervals [x m, Yml contain the point x.
Weierstrass' Principle-CompacTness Several other versions of this principle of continuity are important. The first i s the Weierstrass principle of existence of limit points or accumulation points of bounded sequences. A point x is a limit point of a sequence x 1 , x2, if every open interval containing x also contains points x,. for infinitely many n. Notice the difference between this definition and the definition of limit, where the x,. for almost a11 n must lie in the open interval , or for all n with at most a finite number of exceptions or for all sufficiently large n. If a sequence has a limit, then this limit i s also a limit point of the sequence and is in fact the only one. There may be n o limit point (as i n the example of the sequence I , 2, 3, 4 , . . . ) or a si ngle limit point (as in a convergent sequence) or several limit points (for example, the sequence I , - 1 , I , - I . . . has the two limit points + I and - I ). The Weierstrass principle asserts : Every bounded sequence has at least one limit point. To prove this we observe that since the sequence x1 , x2 , is bounded, there exists an interval [y 1 , z 1 ) containing all x,.. Starting with [yh z1] we construct by induction over n a nested sequence of intervals [Ym z,.] each containing points x,. for infinitely many m . If [Yn , z,. ] contains •
•
•
,
•
•
•
96 Introduction
Ch. l
infinitely many xm, we divide [ym zn l into two equal parts by its mid point. At least one of the two resulting closed i ntervals must again contain infinitely many xm and can be taken as the interval [yn+l• zn+l l · It is clear that the [ym znl form a nested sequence representing a real number x. Every open interval containing x will contain the intervals [ym zn l for sufficiently large n and hence must contain infinitely many xm. Limit points can also be defined as limits of subsequences of the given infinite sequence x1 , x2 , A subsequence is any infinite sequence extracted from the given sequence, or of the form x" ' x n, • x"• ' . , • where n 1 < n 2 < n3 < · · · Obviously, a point x i s a limit point of the sequence xh x2 , if it is limit of some subsequence. Conversely, for any limit point x we can, by induction, construct a subsequence x" • ' x n, • . . converging to x. If x" • ' . . . , x" •- are defined already we • take for nk one of the infinitely many integers n for which n > n k-l and l xn - x l < z-k. We restate the Weierstrass principle in the form : •
•
•
•
.
.
•
•
.
•
.
THEOREM. Every bounded infinite sequence of real numbers has a convergent subsequence.
A set is called compact if every sequence of its elements contains a subsequence converging to an element of the set. Rephrasing our theorem we say that closed intervals of real numbers are compact sets. Monotone Sequences
A special consequence of this theorem is that every bounded be monotone sequence converges. Indeed, let the sequence x 1 , x2 , monotone, say monotonic i ncreasing. If the sequence is also bounded, it has a limit point x. Arbitrarily close to x there must be poi nts xn of the sequence, none exceeding x, since the subsequent terms increase, and if x n > x then xm � x n > x for m > n. It follows that every interval containing x contains almost all x n , or x is the limit of the sequence. •
•
•
Cauchy's Convergence Criterion
The condition that a sequence is bounded and monotone is sufficient for convergence. The significance of this statement is that it often permits us to prove existence of the limit of a sequence without requiring a priori knowledge of the value of the limit : in addition, boundedness and monotonicity of a sequence are properties usually easy to check in concrete applications. However, not every convergent sequence need be monotone (although it has to be bounded) and it i s i mportant t o have a more generally applicable criterion for convergence.
Sec. S. l
Limits and the Number Concept 97
Such is the intrinsic convergence test of Cauchy which is a necessary and sufficient condition for the existence of the limit of a sequence.
The sequence x1, x2, x3, converges if and only iffor ez•ery positive there exists an N such that l xn - xm l < £for all n and m exceeding N. •
•
.
£
In other words, a sequence converges if any two of its elements with sufficiently large i ndices differ by less than £ from each other. We claim that the condition is necessary for convergence. If x = lim x n then every x" with sufficiently large n differs from x by less than £/2, and hence by the triangle inequality every two such values xn and xm will differ from each other by less than £. Conversely, consider a sequence for which l x n - x m l < e for any £ > 0, for all sufficiently large n and m. Then there exists a value N such that almost all xn differ from xN by less than I . This means that almost all xn can be enclosed in an interval of length 2. We can then find an interval so large that i t includes also t h e finite number of x n which may l i e outside the interval about x.v . Thus the sequence is bounded and hence has a limit point x. Every open interval containing x will also contain some points x,. with arbitrarily large m. Since points xn differ arbitrarily little from each other for sufficiently large n, it follows that the open interval about x must contain almost all X11, and so x is the limit of the sequence. e. Least Upper Bound and Greatest Lower Bound
It is of great i mportance that a bounded set of real numbers has "best possible" upper and lower bounds. A set S of real n umbers x is bounded, if all numbers of S can be enclosed i n one and the same finite interval. There are then upper bounds of S, numbers B which are not exceeded by any number x of S : X�B
for all x in S.
Similarly, there are lower bounds A of S : A �X
for all
x
in S.
Thus for the set of reciprocals of natural n umbers I , !. !. !. . . . , any number B 2 I is an upper bound, any n umber A � 0, a lower bound ; here the number I , a member of the set is the least upper bound, and the number 0, a limit point of the elements of the set although not a member, is the greatest lower bound. The least upper bound of a set of real numbers is often called its supremum, the greatest lower bound its infimum. In general the supremum and infimum of a set are either members of the set or at least limits of sequences of members of the
98 Introduction
Ch.
1
set. For, if the least upper bound b of S does not belong to S, there must be some members of S lying arbitrarily close to b, since otherwise we could find upper bounds of S smaller than b ; thus we can select successively a sequence of numbers x�> x2, from S which lie closer and closer to b and converge to b. The existence of a least upper bound of a bounded set S follows immediately from the convergence of monotone bounded sequences. For any n we define Bn as the smallest rational upper bound of S with denominator 2 n . Clearly, for any x in S and any n •
•
•
X � Bn+l � Bn � Bl•
Thus the Bn form a monotonically decreasing and bounded sequence which must have a limit b. It is easy to see that b is an upper bound of S and that there exists no smaller upper bound. The existence of the greatest lower bound is proved in the same way. f. Denumerability of the Rational Numbers
A surprising discovery concerning the rational numbers was made late in the nineteenth century and stimulated the creation by Georg Cantor of the Theory of Sets after 1 872. Although the rational numbers are dense and cannot be ordered by size, they can be arranged never theless as an infinite sequence r�> r2, , rn• in which every rational number appears once. In this way the rational numbers can be enumerated, or counted off, as a first, second, . . . , nth, . . . rational number, where, of course, the order of the numbers in the sequence does not correspond at all to their order by magnitude. This result, which holds j ust as well for the rational numbers in any interval, is expressed by the statement : The rational numbers are denumerable, or they form a denumerable set. To prove this res ult we simply give a prescription for arranging the positive rational numbers as a sequence. Every such number can be written in the form p/q, where p and q are natural numbers. For each positive integer k there are exactly k I fractions p/q for which p + q = k. These are arranged in order of increasing p. Writing the different arrays of numbers for k = 2, 3, 4, . . successively, we obtain (see Fig. l .S . l ) a sequence which contains all positive rational numbers. Omitting fractions, in which numerator and denominator have a common factor greater than 1 , and thus represent the same rational number as a previous fraction, we obtain the sequence •
•
•
•
•
•
-
.
t, f, f, i, l, !, i, -�-, t, t, f, }, f, . . .
Sec. S.2
Theorems on Continuous Functions 99
in which every positive rational number occurs exactly once. A similar sequence containing all rational numbers or all rational numbers in some particular interval is easily constructed. This result i s seen in proper perspective only i n the light of another basic fact : that the set of all real n umbers is not denumerable. 1 This
Figure
l.S.l
Den umerab ility of the positive rationals.
is an i ndication that the set of real n umbers contains " many m ore" elements than that of the rational numbers, although both sets are infi nite ; th us denumerability is indeed a h ighly restrictive property of a set. The Theory of Sets plays an important clarifying role i n mathe matics, although its use in unrestricted generality has led to paradoxical results and controversies. Such paradoxes, however, do not affect the substance of constructive mathematics and are absent from the theory of sets of real numbers. 8.2
Theorems on Continuous Functions
I mportant properties of continuous functions are establi shed on the basis of the completeness property for real numbers. We recall the definition of continuity : the function f(x) is continuous at the point � if for any given positive € the inequality l f(x) - !W I < € holds for 1 For proof and a brief general d i scussion of the basic facts of set theory see
What is Mathematics ?
by Courant and Robbins, p.
81.
I 00
Ch. I
Introduction
all x sufficiently close to $, or, for all x differing from $ by less than a suitable quantity CJ, which generally depends on the choice of £ and $. It is understood in this definition that only values of x and $ for which f is defined are considered. A more concise definition of continuity in terms of convergence of sequences is : f(x) is continuous at the point $ if lim f(x.) = fW for n - oo
with limit ; (where again the values x. and ; every sequence x1, x2, are in the domain of f). The equivalence of the two definitions was proved in Section 1 . 8, p. 82. We call f continuous in an interval iff is continuous at each point of the interval. f(x) is uniformly continuous if for given £ > 0 we have l f(x) !W I < £ whenever x and ; are sufficiently close regardless of their location in the interval ; thus f is uniformly continuous if the quantity (J appearing in the definition of continuity can be chosen in dependently of ; : For every £ > 0 there exists a (J = Cl(£) > 0 such that lf(x) - !W I < £ whenever lx - ;1 < CJ. For practical purposes this means that if we subdivide the interval in which f is defined into a sufficiently large number of equal subintervals, then f will vary by less than a prescribed amount £ in each subinterval : At any point, f will then differ by less than £ fro m its value at any other point of the same subi nterval. •
•
•
-
We now prove : Every function continuous in a closed interval [a, b] is uniformly continuous in that interval.
Iff were not uniformly continuous in [a, b], there would exist a fixed > 0 and points x, $ in [a, b] arbitrar il y close to each other for which if(x) - !WI � £. It would then be possible for every n to choose points xno ; . in [a, b] for which l f(x.) - f($.) 1 � £ and l x. - ;. 1 < 1 /n. Since the x. form a bounded sequence of numbers we could find a subsequence converging to a point 'YJ of the inte rval (using the compact ness of closed intervals). The co rresponding values ; would then also converge to 'YJ : since f is continuous at 'YJ, we would find that 'YJ = limf(x ) = lim j( ; .) for n tending to infinity in the subsequence, which is impossible if lf(x.) - f(;.) l � £ for all n. The intermediate value theorem asserts : If for a function f(x) con tinuous in an interval a :=;; x :=;; b, y is any value between f(a) and f(b), th en f( ;) = y for some suitable ; between a and b. Thus the existence of a solution ; of the equation jt_;) = y is certain if one exhibits two values a and b for which f(a) < y and f(b) > y respectively. This i mmediately implies the existence of a uniquely determined inverse function i f f is continuous and monotonic, as we have seen (p. 44). £
n
n
Polar Coordinates 101
Sec. S.3
To prove the intermediate value theorem let a < b, f(a ) = oc , f(b) = {3, -and (X < r < fl. Let s be the set of points X of the interval [a, b] for which f(x) < y. s is bounded and has a least upper bound e also belonging to the closed i nterval [a, b]. Then f(x) � y for e < X � b. The point e either belongs to s or is the limit of a sequence of points x n of s. In the first case JW < r ; hence e < b, since j(b) > ")' , and there are points X between e and b, arbitrarily close tO e for which f(x) � y. This is impossible if f is continuous at e and JW < y . In the second case, JW � y, we find from f(x n) < y and lim xn = e that JW � y ; since we saw already that JW < r i s
i mpossible, w e must h ave J W = y . A third basic property o f a continuous function f(x) i n a closed interval [a, b J is the existence of a largest value (maximum), meaning that there exists a point e in the interval [a, b J such thatf(x) � JW for all x in the interval. Similarly, f will assume its least value (minimum) at some point 'YJ of the interval : f(x) � f('YJ) for all x in the interval . It is essential to have the interval closed: for example, the functions f(x) = x or f(x) = l fx are continuous, but they do not have a largest value in the open interval 0 < x < I ; the maximum may just occur at one of the end points or not exist at all iff is not continuous at the end points. To prove this principle we observe that a function f continuous in [a, b J i s necessarily bounded: that is, the valuesf(x) forming the "range" S off lie in some finite i nterval. I ndeed by the uniform continuity off we can find a finite n umber of points x1 , x2 , , xn in the interval such thatf(x) at any x of the i nterval differs by less than one from one of the numbers f(x1 ), f(x2) , , f(xn) which can all be fitted into a finite interval. Since then the set S of values f(x) is bounded, it has a least upper bound M. This M is the smallest n umber such that f(x) � M for all x in [a, b ]. Either M belongs to S or is the limit of a sequence of points of S. I n the first case, there exists a e in [a, b] with fee) = the second case, there exists a sequence of points xn in [a, b] with lim f(xn) = M ; thus we can find a subsequence of the xn which con•
•
•
•
•
•
M.
In
verges to a point e of [a, bJ and again fW = M by continuity off at e. Clearly, JW is the maximum off 8.3
Polar Coordinates
In Chapter I we have represented functions geometrically by curves. Analytical geometry follows the reverse procedure, beginning with a curve and representing it by a function, for example, by a function
I 02
Ch. I
Introduction
expressing one of the coordinates of a point of the curve in terms of the other. This point of view naturally leads us to consider, i n addition to the rectangular coordinates to which we restricted ourselves, other systems of coordinates possibly better suited for the representation of curves given geometrically. The most i mportant example is that of polar coordinates r, 11 connected with the rectangular coordinates x, y of a point P by the equations
x = r cos 11, y = r sin
11,
r' = x 2 + y2 ,
tan 11 = 1j_ , X
whose geometrical interpretation is made clear in Fig. 1 .5.2. 1 y
Figure 1 .8.2
Po lar coordinates.
We consider, for example, the lemniscate. This is geometrically defined as the locus of all points P for which the product of the distances r, and r2 from the fixed points F1 and F2 with the rectangular coordinates x = a, y = 0 and x = -a, y = 0 respectively, has the constant value a2 (cf. Fig. 1 .5.3). S ince
a simple calculation gives us the equation of the lemniscate in the form
Introducing polar coordinates, we obtain
1 The polar coordinates are not completely determined by the p o i nt P. In addition to IJ, any of the angles B ± 2rr, B ± 4rr, . . . can be considered a polar angle of P.
Sec. S.4
Remarks on Complex Numbers 1 03
y
Figure 1 .8.3
Lemniscate.
dividing by r2 and using a simple trigonometrical formula this becomes r2 = 2a2 cos 20.
Thus the equation of the lemniscate is simpler in polar coordinates than in rectangular. S.4
Remarks on Complex Numbers
Our studies will be based chiefly on the continuum of real numbers. Nevertheless, with a view to discussions in Chapters 7, 8, and 9, we remind the reader that the problems of algebra have led to a still wider extension of the concept of number, the complex numbers. The advance from the natural numbers to the real numbers arose from the desire to eliminate exceptional phenomena and to make certain operations, such as subtraction, division, and correspondence between points and numbers, always possible. Similarly, we are compelled by the requirement that every quadratic equation and in fact every algebraic equation shall h ave a solution, to introduce the complex numbers. If, for example, we wish the equation x2 + 1 = 0
to h ave roots, we are obliged to i ntroduce new symbols i and - i as the roots. (As i s shown in the theory of functions of a complex variable, this is sufficient to i nsure that every algebraic equation shall have a solution. 1) 1 An algebra ic equation is of the form
complex coefficients.
P(x)
= 0, where P is a polynomial with
104
Introduction
Ch. I
If a and b are ordinary real numbers, the complex number c = a + ib denotes a pair of numbers (a, b) with which calculations are performed according to the following general rule : We add, multiply, and divide complex n umbers (among which the real numbers are included as the special case b = 0), treating the symbol i as an undetermined quantity, and simplify all expressions using the equation i 2 = - I to remove all powers of i higher than the first, leaving only an expression of the form a + ib. We assume that the reader already has a certain degree of familiarity with the complex numbers. We nevertheless emphasize a particularly
X + yi
=C
-I
Figure l .S.4 conjugate.
.:r
- yi :: C
Geometric representation of a complex nu mber x + yi and of its
important relationship which we shall explai n in connection with the geometrical or trigonometrical representation of the complex n umbers. If c = x + iy is such a number, we represent i t in a rectangular co ordinate system by the point P with coordinates x and y. By means of the equation s x = r cos IJ, y = r sin IJ, we introduce the polar coordinates r and (} (cf. p. 101 ) instead of the rectangular coordinates x and y. Then r = ..) x2 + y 2 is the distance of the point P from the origin, and (} the angle between the positive x-axis and the segment OP. The complex number c is represented in the form
c = r (cos (} + i sin IJ).
The angle (} is called the amplitude of the complex number c, the quantity r its absolute value or modulus, for which we also write l e i . T o the "conjugate" complex number c = x - iy there obviously corresponds the same absolute value, but the amplitude - IJ (Fig. l . S.4).
Sec. S.4
Remarks on Complex Numbers 105
Clearly, lf we u se this trigonometrical representation, the multiplication of complex numbers takes a particularly simple form, for then c · c
'
= r(cos (} + i sin 0) · r'(cos ()' + i sin ()') = rr'[(cos () cos ()' - sin () sin 0') + i(cos () sin ()' + sin () cos ()')].
If we use the addition theorems for the trigonometric functions, this becomes c · c ' = rr'(cos (0 + 0') + i sin (0 + 0')).
Figure
l .S.S The nth roots of unity (for n
=
1 6).
We therefore multiply complex numbers by multiplying their absolute val ues and adding their amplitudes. The remarkable formula (cos 0 + i sin O)(cos ()' + i sin 0') = cos (0 + 0') + i sin (0 + 0')
is usually called De Moivre's theorem . It leads us to the relation (cos () + i sin 0)" = cos n O + i sin nO,
which , for example, at once enables us to solve the equation x" = 1 for positive integers n ; the roots (the so-called roots of unity) are 2 47T . . 47T 27T . 27T € 2 = € = COS - + I SI D = € = COS - + I. S I D - , n n n n 1 )7T 2(n . 2(n 1 )7T n- 1 + I SID = COS , " n- 1 = € n n
€1
(Fig. I .S . 5).
-
-
.
,
.
.
. ·,
Problems I 07
SECTION l . l c, page 9 1 . Let [x] den ote the integer part of x ; that is, [x] is the integer satisfying x
- I < [.r] :::;;
;·.
Set Co = (.r], and Cn = [ I O"(x - Co ) - w•-lcl - wn-2c2 - . . . - I Ocn-11 for n = I , 2, 3, . . . . Verify that the decimal representation if :r is x
=
c0 + 0 · c1c2c3
·
· ·
and that this construction excludes the possibil ity of an i nfinite string of 9 s 2. Define inequality x > y for two real numbers in terms of their decimal representations (see Supplement, p. 92). *3. Prove if p and q are integers, q > 0, that the expansion of p/q as a decimal either terminates (all the digits fol lowing the last place are zeros) or is periodic ; that is, from a certain point on the decimal expansion consists of the sequential repetition of a given string of digits. For example, ! = 0.25 is terminating, l1 = 0.090909 · · · is periodic. The length of the repeated string is called the period of the decimal ; for 1'1 the period is 2. In general, how large may the period of p/q be ? '
.
SECTION l . l e, page 1 2 I . Using signs of inequality alone ( not usi ng signs of absolute value) specify the values of x which satisfy the following relations. Discuss all cases. (a) lx - a l < l x - bl . (b) l x - a l < x - b. (c) l :c2 - ai < b. 2. An interval (see definitions i n text) may be defined as any connected part of the real continuum. A subset S of the real continuum is said to be conn ected if with every pair of points a, b in S, the set S contains the entire closed interval [a, b]. Aside from the open and closed intervals already mentioned, there are the "half-open" intervalS a :::;; X < b and a < X :::;; b (sometimes denoted by [a, b) and (a, b], respectively) and the unbounded intervals that may be either the whole real l i ne or a ray, that is, a "half-line" oo, a], x :::;; a, x < a, x > a, x � a (sometimes denoted by ( - oo , oo )] and ( ( oo, a), (a, ro ) , [a, ro), respectively) (see also footnote, p. 22). *(a) Prove that the cases of intervals specified above exhaust all possibilities for connected subsets of the number axis. (b) Determine the intervals in which the following inequalities are satisfied. (i) x2 - 3x + 2 < 0. (ii) (x - a)(x - b)(x - c) > 0, for a < b < c. (iii) I I x i - x � 0. x -a (iv) � 0. x +a -
-
-
--
I
(v) x +
� I � 6.
(vi) [x] :::;; xf2. See Problem I of this page. (vii) sin x � V Z/2. (c) Prove if a :::;; x :::;; b, then lxl :::;; lal + lbl .
108 Introduction
Ch. I
3. Derive the inequalities I x + � � 2, for x > 0,
(a)
(b)
(c)
x
I + - � -2, for x < 0,
�/
X
+
X
� 2, for x � 0.
4. The harmonic mean
� of two positive numbers a, b is defined by
�
=
H � �) +
·
Prove that the harmonic mean does not exceed the geometric mean ; that is, that � � v'Cib. When are the two means equal ? S. Derive the following inequalities : (a) x 2 + xy + y2 � 0, n 1 *(b) x2 n + x2n- y + x2n- 2y2 + . . . + y2 � 0, *(c) x� - 3x3 + 4x2 - 3 x + I � 0. When does equality hold ? *6. What is the geometrical interpretation of Cauchy 's inequality for n
= 2, 3 ?
7. Show that the equality sign holds i n Cauchy's inequality i f and only if the a. are proportional to the b, : that is, ca. + db, = 0 for all v where c and d do not depend on
v
and are not both zero.
(a) l x - a11 + lx - a 2 1 + lx - a31 � a3 - a1, for a1 For what value of x does equality hold ?
8.
*(b) Find the largest value of y for which for all x - al l + l x - a l + · 2
lx
·
·
<
a 2 < a3•
+ l x - an i � y ,
where a1 < a 2 < · · · < an. Under what conditions does equality hol d ? 9. Show that the following inequalities hold for positive a , b , c .
(a) a 2 + b2 + c2 � ab + be + ca. (b) (a + b)(b + c)(c + a) � 8abc. (c) a 2b2 + b2 c2 + c2a2 � abc(a + b + c) . 10. Assume that the numbers xl> x2 , x3 and a;k (i, k = I , 2, 3) are all positive, and in addition, a1k � M and x12 + x22 + x3 2 � I. Prove that a1 1 x1 2 + a1 x1x + · · · + a33x3 2 � 3M. 2 2
*1 1 . Prove the following inequality and give its geometrical interpretation for n � 3, V (al - b1) 2 + · · · + (an - bn) 2 � "1(a12 + · · · + an2) + V(b12 + · · · 12.
Prove, and interpret geometrically for n � 3,
V (al + bl + · · · + z 1)2 + · · · + (an + bn + · · · + Zn)2 � Va12
+
..
.
+ b,2).
. -+,-,_·-.zn"2 . + a n2 + Vf; 12 + . . . + bn2 + . . . + V z-1• 2 -+
Problems 109 13. Show that the geometric mean of n positive numbers is not greater than the·arithmetic mean ; that is, if a, > 0 (i = I , . . . , n), then
1 " ;' a1a2 · · · an � n (a l
+
a2
+ · · · +
On) .
(Hint: Suppose a1 � a2 � · · · � an. For the first step replace an by the geometric mean and adjust a1 so that the geometric mean is left unchanged.) SECTION 1 . 2 d , page 3 1
I . lf ((x) i s continuous a t x = a and [(a) > show that the domain of f contains an open interval about a where [(x) > 2. In the definition of continuity show that the centered intervals l [(x) - [(x0)1 < • and lx - x01 < 15
0,0.
may be replaced by an arbitrary open interval containing f(x0) and a suffi ciently small open interval containing x0, as indicated on p. 33. 3. Let f(x) be continuous for 0 � x � I . Suppose further that [(x) assumes rational values only and that f(x) = ! when x = t. Prove that [(x) = i everywhere. 4. (a) Let f(x) be defined for all values of x in the following manner : J(x) =
{
0, I,
x irrational x rational.
Prove that f(x) is everywhere discontinuous. (b) O n the other hand, consider g(x) =
(0�,'
x irrational
x = E. rational in lowest terms.
(The rational number p/q is said to be in lowest terms if the i ntegers p and q have no common factor larger than I , and q > Thus [( 1 6/29) = 1 /29.) Prove that g(x) is continuous for all irrational values and discontinuous for al l rational values. *5. If f(x) satisfies the functional equation q
0.
f( x + y) = /(x) + f (y) for all values of x and y, find the values of ((x) for rational values of x and prove if J(x) is continuous that J(x) = ex where c is a constant . 6. (a) If J(x) = x", find a o which may depend on � such that
l[(x)
-
/WI <
lx - �� < 15 .
whenever
*(b) Do the same if f(x) is any polynomial f(x) = OnXn + On-lxn-1 + . • where an '¢
0.
•
•
+
alx + Oo ,
1 10 Introduction
Ch. 1
SECTION 1 .2e, page 44
1. Prove that i f j'(:r) is monotonic on [a, b] and satisfies the intermediate value property, t hen f(x) is continuous. Can you draw the same conclusion iff is not monotonic ?
a
2. (a) Show that x" is monotonic for x > 0. As a consequence, show for > 0 that x " = a has a unique positive solution \:1�. (b) Let j' (x) be a polynomial
Show (i) i f n is odd, then j'(x) has at least one real root, (ii) i f an and a0 have opposite signs, then f(x) has at least one positive root, and, in addition, if n is even, n "" 0, then .f(x) has a negative root as well.
*3. (a) Prove that there exists a line i n each direction which bisects any given triangle, that is, divi des the triangle into two parts of equal area . (b) For any pair of triangles prove that there exists a line which bisects them simultaneously. SECTION 1 . 3b, page 49
1 . (a) Prove that v:;: is not a rational function . (Hint: Examine the possibility of representing V:;: as a rational fu nction for x = ?:". Use the fact that a nonzero polynomial can have at most finitely many roots.) (b) Prove \'':;: is not a rational function. SECTION 1 . 3c, page 49
1 . (a) Show that a straight l ine may intersect the graph of a polynomial higher than first degree i n at most finitely many points. (b) Obtain the same resu l t for general rational functions. (c) Verify that the trigonometric fu nctions are not rational.
SECTION 1 . 5 , page 57 1.
Prove the following properties of the binomial coefficients.
(a) 1 +
(�) (�) (�) (�) (;) (�)
(b) I (c)
(�) + (�) + " · + (n � l ) + (�) = 2". + . . · + ( - 1 )"
+
-
+ 2
+ . . · + 11
+ 3
(�)
(�)
= 0.
= 11(2"-1) .
(Hint:
Represent
the binomial coefficients in terms of factorials.)
(d) 1
°
z(�) (�) () () + 2 3 °
+
0
0
0
(�) (11)
+ (n - l )n
= n(n - 1 )2"-2•
1 2"+1 - 1 1 1 = --- . (e) 1 + - 111 + - n2 + . . . + 11 + 1 n n + 1 2 3 --
( r (;f (:f e:). () () () ()
* (n/> . �
+
of x i n (I + x)2".) •
(
(g) s..
=
Hint: Prove
-
n
0
=
+ · · · +
I n -3 I
+
Problems 1 1 1
(Hint: Consider the coefficient
I n 5 2
I 'n -7 3
+
, . . +
(2n-+I )n1
(n) n
4"(n !)2 (2n + I ) ! ·
�: � ; Sn = Sn-tl · )
Prove ( I + x)n :2 I + nx, for x > - 1 . + 3. Prove by induction that I + 2 · · · + n = 1 n(n + 1 ) . *4. Prove by induction t he fol l ow i ng :
2.
I - (n + l )q" + nq" 1 1 (a) 1 + 2q + 3q2 + . . . + ngn-1 = ---'-----;-:-_:..;....,-;;-�'- ( 1 - q)2
q2fl-f l - 1 (b) ( 1 + q)( l + q2) ' ' ' (I + q2") - -
I -q Prove for all natural numbers n greater than I that n is either a prime or can be expressed as a product of primes. ( Him : Let A,_1 be the assertion for all integers k with k ::;; n that k is either prime or a product of primes.) *6 . Consider the sequence of fractions I 3 7 Pn 5.
f • z · :s · · · · · q;, · · · · ·
where p,.1 1 = Pn + 2q, and 9n n = Pn + g.. . (a) Prove for all n that P n/q, is in lowest terms . (b) Show that the absolute difference between p .. /q.. and \1Z can be made arbitrari ly small . Prove also that the error of approximation to v2 alterna tes in sign . 7. Let a, b, a .. and bn be integers such that (a +
b VZ)"
= a
.. + bn v2,
wher� a is the integer closest to b vz. Prove that a, is the integer closest to b 11 \ 2 . *8. Let an and bn be defined by and b1 = 9,
For each value of n, determine the minimum value 9. If 11 is a natural number, show that (I +
VS)" - (1 - VS)" 2" vs
is a natural number.
m
such that
a111
:2
b..
.
Ch. l
1 1 2 Introduction
10. Determine the maximum number of pieces into which a plane may be cut by n straight l i nes. S h ow that the maximum occurs when no two of the lines are parallel and no three meet in a common point, and determine the number of pieces when concurrences and parallelisms are permitted. 1 1. Prove for each natural number n that there exists a natural number k such that c v2 - I )" = vk: vk - 1 .
-
12. Prove Cauch y s ine qu alit y inductively. '
-
SECTION 1 .6, page 60
1. Prove that lim ( v' n
2. Prove that lim ( V' n
+
1
+
I
v'n)( v' n
+
V'n) = 0.
!J
=
!·
3. Let a" = 1 0"/n ! . (a) To what limit does a, converge ? (b ) J s th e sequence monotonic ? (c) Is it monotonic from a certain n onward ? (d) G ive an estimate of the difference between a, a n d the limit. (e) Fr om what value of n o n wa rd is this difference less than I / 1 00 ? 4. Pro ve that l i m
;n = 0.
(�
S. (a) Prove that l im n � oo
n- C(;
(� n
(b) P ro ve that l i m
n
+
.;11 + 1
---2 (n + I)
+
the su m with its largest term .) ,,_. "'
(c) Pr ove that lim
n- • W
0(d)
(----= ' I
Prove that lim
\
(
,
n
+
I
+
--;= \• n + I
I
- + I
)
. . · + - -2 = 0. (2n)
1
.
+
I
v· 112
.
+
· +
.�
y 2n
)
2+··· +
-
=
v 11" +
00 ,
n2
I
---;= '\' + 11
6. Prove th a t every peri odi c decimal rep re se n ts a ,,_. "'
(Hint: Compare
rational
(Compa re Section J . l c, Problem 3 . ) 7. Prove that lim 11100 exists and determine its value. I . OFt 8. P r o ve that if a and b :S: a are positive, the sequence -;v a" to a. S i m i la r l y for any k fixed positive n umbers a1, a 2 , \' ' a1" + a2" + · · · + ak" conver ges and fi n d its l i mit.
,
r/ - - � oo
9.
Prove that the sequence
its l imit . 10. If
n
v(11) is n
2,
•
•
= I. n u m ber.
,
+ b" converges a" prove that
,h I 2, J 2.J 2 v 2, . . . , converges.
the number of prime factors of 11, pro ve that
lim •{ ) = 0.
n - oo
•
\
).
Find
11.
P ro ve that if lim an = �. then l i m an .
mean (a1 12.
(a)
+ + + (-1- + + + --1+-). 1+ � - � ) (-- + -- + n( + n- oo
a2
· · ·
Find
I) =
k (k
. (b ) l lm
13. If a0 n�· "'
2·3
l
n- oo
an)/n.
-1-
lim n- oo I · 2
( Hint :
Problems 1 1 3
·
··
k
�. where an is the arithmetic
l)
n(n
I ·
I
2·3.4
I ·2·3
=
+
0 • 0
I
.
n
)
l )(n + 2) ·
-o----:-777--=
+ a1 + · · · + aP = 0, prove that lim (ao v � + al v ' 1 1 + I + + an V n + p) = 0. n- oo
(Hint: Take
v � out as a fa cto r. )
0
0
1 4. Prove that lim < 2 n+ll v (n2 + n) = I . n- oo
+
*IS. Let an be a given sequence such that the sequence b , = pan qan+I • where l p l < q, i s convergent. Prove that an converges. If lp l > q > 0, show that a, need not converge. 1 6.
Prove the relation
hm � .
n - oo n
l
". ,. I L t. = k + I
- i '--- 1
for any nonnegative integer k . (Hint: Use induction with respect to k and use the relation
expanding (i
-
i; [i'·' ' 1
i -- 1
_
(i
_
I )" 1]
=
11k··n,
1 )"+1 in powers of i.)
SECTION 1 .7, page 70 * 1 . Let a1 and b1 be any two positive numbers, and let and b2 be defined by the equations
b2 Similarly, let
=
a1 + b1 2
a1
<
b1• Let
a2
0
a + b a = 2 b2 . 2
and, in general, On
= v' a n-I bn-l ,
b
n
_
-
an-1 + b n-1 2
Prove (a) that the sequence a 1 , a2, . • • , converges, (b) that the seq uence b1, b2, . . • , converges, and (c) that the two sequences have the same limit. (Th is limit is called t he arithmetic-geometric mean of a1 and b1.)
1 14 Introduction
Ch. 1
•2. Prove that the limit of the sequence
v2,
-h + v2, J 2
(a) exists and (b) i t is equal to 2.
+
.J2 + v2, . . .
03. Prove that the limit of the sequence
I --
I n
a. = - +
n + 1
1 2n
+ " · + -
exists. Show that the limit is less than 1 but not less than
4. Prove t h a t t h e limit of the sequence b = "
1
n + I
+
--
1.
. . · + _!_ 2n
exists, is equal to the l imit of t he previous example .
5. Obtain the fol lowing bounds for the l i mit L in the two previous examples : 37/60 < L < 57/60 . * 6. Let
a 1 , b1 be any two positive numbers, and let
a1 <;;; b1 . Let
and in general
Prove t hat the sequences a1, a2, same limit . "7. Show that
1 /e
=
I
-
•
•
and h1, b2 ,
•
I + 2!
I
- 3!I +
·
•
•
·
·
•
converge and have the ( - I )"
+ ----;;! +
· · · . (Hmt: .
Consider the product of the nth part ial sums of the expansions for e and 1 /e.)
8. (a) Without reference to the binomial t heorem show that an = (I + 1 /n)" is monotone i ncreasing and bn = ( I + 1 /n)"H is monotone decreasing. (Hint: Consider a,H /an and bnfbnn · Use the result of Section 1 . 5, Problem
2.) (b) Which is the larger number ( I ,OOO,OOO)l.Ooo , ooo or ( 1 ,000,001 )999. 999 ? 9. (a ) From the results of Problem Sa show that
(b) For
n
> 6 derive the sharper inequality
(II)"
n! < n e .
a *10. If an > 0, and lim nH = L, then lim n- � co · an
n- co
\'·;,; =
L.
Problems 1 1 5
1 1 . Use Problem 1 0 to evaluate the limits of the following seq uences : (a) ,.. , n ,
(b) >'Yn5 + n4 ,
(c)
12. Use Problem I I c to s how
J"iii _:
nn
n! = nne-nam
where a, is a number whose nth root tends to I . (See Appendix, Chapter 7.) 1 3. (a) Evaluate I I I - + - + · · · + --! ·3 2·4 n(n + 2) · ( Hint: Compare Section 1 .6, Problem 1 2a.) ( b) From the result above, prove that
00 1 2 2 converges. k�t n
14. Let p and q be arbitrary natural numbers. Evaluate I
( a)
2 k - dk 11
.
+ p )(k + p + q)
I . (b) k - dk + p)(k p + + q) 15. Evaluate
i
. (a) -- + -- + . . . + n(n + l )(n + 2) I 2 · 3 2 ·3 ·4 I (b) k(k + l )(k + 3) . I
I
I
·
n
k�l
(c) Evaluate the limit on each of the above expressions as 11 _,. OCJ . *(d) Let a1, a 2 , , a ... be nonnegative integers with a1 < a 2 < · · · < am. •
•
•
Show how to obtain a formula for Sn =
and how to find lim S11 •
11
k�l (k
I + a1)(k + a 2 ) · · · (k + a ,..)
�
16. If ak is monotone and ak converges, show that lim ka,. = 0. k=l k- oo 17. If a�: is monotone decreasing with limit 0 and bk = ak - 2alc+1 + ak+2 � 0
for all k, then show
� kbk
k�l
=
a1.
SECTION 1 .8, page 82
I . Prove that lim (cos 1rx) 2 m exists for each value of x and is equal to 1
or 0 according to whether x is an integer or not. 2 2. (a) Prove that lim [lim (cos n ! 1rx) "'] exists for each value of x and 00 is equal to I or 0 acc �;din to whether �' is rational or irrational. ( b) Discuss the continuity of these limit functions.
g
1 1 6 Introduction
Ch.
1
3. Let f(x) be continuous for 0 � x � I . Suppose further that .f(x) assumes rational values only, and that .f(x) = ! when x = }. Prove that f(x) = ! everywhere.
SECTION
l .S . I , page
89
1. Let r = p/q, s = mfn be arbitrary rational numbers where p, q, m, n are integers and q, n are positive. In terms of the integers p, q, m, n, define
(a) r + s,
(b) r - s,
(c) rs,
(d) � . s
(e) r
<
s.
2. Prove for nested sequences of rational numbers [an, bn l and [an '• bn'l that each of the following conditions is necessary and sufficient for equiv alence : (a) an' - a. is a null sequence, (b) an � bn' and an' � bn. 3. Given x - { [a., bnl } , y - { [o:n, Pn]}, (a) verify that the definitions of addition and subtraction,
X - Y = {[an -
{)., bn - o: n l} ,
are meaningful. Specifically, verify that (i) the given representations are, in fact, nested sets for x + y and x - y when x and y are rational ; (ii) if x < y, t hen x + z < y + z, where z is an arbitrary real number . (b) Define the product xy and verify specifically that your definition of product is meaningful. (i) that the given nested set is, in fact, a nested set for xy when x and y are rational . (ii) that if x < y and z > 0, then xz < yz . 4. Prove that the following principles are equivalent in the sense that any one can be derived as a consequence of any other. (a) Every nested sequence of intervals with real end points contains a real number. (b) Every bounded monotone sequence converges. (c) Every bounded infinite sequence has at least one accumulation or limit point. (d) Every Cauchy sequence converges. (e) Every bounded set of real numbers has an infimum and a supremum.
Miscellaneous Problems I . If w1, w2, , Wn > 0, prove that the weighted average •
•
.
WlXl + W2X2 + . . • + WnXn Wt + w2 + . . . w2
l ies between the greatest and the least of the x's. 2. Prove 2( v' n + I
-
I)
<
I +
+� +··· +� � v'2 v' 3
,; n
<
2 v'� .
3.
Prove for x, y > 0
xn + yn
2
�
( y)n + 2
X
Problems
117
.
I nterpret this result geometrically in terms of the graph of xn .
4. If a1 � a2 �
• • •
�
G11 and b1 � b2 � · ·
•
� bn , prove
n ,i a ,b, � ( i a , ) ( i b.-) . 1 1 t=l
..
l ---'-
can be written a s the sequence ( a) Show that the sequence a1 , a2, a3, of partial sums of the series u 1 , u2 , u3, . where u11 = a11 - a11_ 1 for n � I and u 1 = a 1 . (b) Write t he sequence a . = 113 as t h e sequence of partial sums of a series. (c) From the result obtain a formula for the nth partial sum of the series 5.
•
•
•
1 + 4 + 9 + · · · + n2 + · · · . (d) From the formula for 1 2 + 22 + · · · + n 2 , find a formula for J 2 + 3 2 + 5 2 + . . . + (2n + 1 )2 .
6. A sequence is called an arithmetic progression of the first order if the differences of successive terms are constant. It is called an arithmetic progression of the second order if the differences of successive terms form an ari thmetic progressio n of the first order ; and, in general, it is called an arithmetic progression of order k if the differences of successive terms form an arithmetic progression of order (k - I ). The numbers 4, 6, 1 3 , 27, 50, 84 are the first six terms of an arithmetic progression. What is its least possible order ? What is the eighth term of the progression of smallest order with these initial terms ? 7. Prove that the nth term of an arithmetic progression of the second order can be written in the form an 2 + bn + c, where a, b, c are independent of n. *8. Prove that the nth term of an arithmetic progression of order k can be written in the form ank + bn k-l + · · · + pn + q, where a, b, . . . , p, q are i ndependent of n. Find the nth term of the progression of smallest order i n Problem 6. 9. Find a formula for the nth term of the arithmetic progressions of smallest order for which the following are the i nitial terms : (a) I , 2, 4, 7, I I , 1 6, . . . . 1 0 - 9, 1 , 25, 68, . . . . (b) 7 * 10. Show that the sum of the first n terms of an arithmetic progression of order k is akSk + a1,, _1Sk_1 + · · · + a1S1 + aon, where S, represents the sum of the first nvth powers and the a; are i ndependent of n. Use this result to evaluate the sums for the arithmetic progressions of Problem 9. II. By summing v(v + J )(v + 2) · · · (v + k + I) (v - J )v(v + I ) · · · (v + k) -
, , -
-
1 18
from
Introduction v
= I to
v
I v(v + I )(v + 2)
v�t
12. Evaluate
13.
...
= n , show that
13
n n + I) (v + k ) = (
+ 23 + · · · +
j (x)
-
(n
k +2
+ k + I)
.
n3 by using the relation
v3 = v(v + l )(v + 2)
Show that the fu nction
...
Ch. I
(
= log: lxl ' 0.
3 v(v +
I)
+
v.
x rf O
X =0
=
is continuous but not Holder-continuous. (Hint: Show Holder continuity 1 /2"1�.) with exponent rx. fails at the origin by considering the values x 14. Let a, be a monotone decreasing sequence of nonnegative numbers.
Show that 1 a, converges if and only if 1 w
�
n -= 1
\1 = 0
2• a 2v does.
1 5 . I nvestigate for convergence and determine the limit when possible,
(a) n ! e - [n ! e] (b) a.f an H • where
a1
= 0,
a2
= I , and ak t- 2 = ak+l
+ a k.
2 The Fundamental Ideas of the Integral and Differential Calculus
The fu ndamental limiting processes of calculus are integration and differentiation. Isolated i nstances of these processes of calculus were considered even in antiquity (culminating in the work of Arch imedes), and with i ncreasing freq uency in the sixteenth and seventeenth centuries. However, the systematic development of calculus, started only in the seventeenth century, is usually cred ited to the two great pioneers of science, Newton and Leibnitz. The key to this systematic development is the insight that the two processes of differentiation and integration, which had been treated separately, are inti mately related by being reciprocal to each other. 1 A fair historical assessment of the merits cannot attribute the i nvention of calculus to sudden unexplainable flashes of genius on the part of one or two i ndividuals. M any people, such as Fermat, Galileo, and Kepler, stimulated by the revolutionary new ideas in science, contributed to the foundations of calculus. In fact, Newton's teacher, Barrow, was almost in full possession of the basic insight into the reciprocity between differentiation and integration, the cornerstone of the systematic calculus of Newton and Leibnitz. Newton has stated the concepts somewhat m ore clearly ; on the other hand, Leibnitz's i ngen ious n otation and methods of calculation are h ighly suggestive and remain indispensable. The work of these two men i mmediately stimulated the h igher branches of analysis including the calculus of variations and the theory of differential equations, and led to innumer· able applications in science. Curiously enough, although Newton, 1
This fact constitutes the "fundamental theorem of calculus."
119
120
The Fundamental Ideas of the Integral and Dijj'erential Calculus
Ch. 2
Leibnitz, and their i mmed iate successors made such varied uses of the powerful tool put i nto thei r hands, none succeeded in completely clarifying the basic concepts involved in their work. Their arguments employed "infinitely small quantities in ways which are logically i ndefensible and unconvi ncing Clarification came at last in the nine teenth century with the careful formulation of the concept of l i m it and with the analy s is of the number continuum as explained i n Chapter 1 . 1 W e begin with a discussion o f the fundamental concepts. They can be fully appreciated only through concrete illustrations and examples ; it is therefore recom mended here, as at many places in this book, that theoretical and general sections be carefully studied again after the reader h as absorbed more specific and concrete m aterial in subsequent section s "
.
.
2.1 a.
The Integral
Introduction
Only after a lengthly development the systematic procedures of i ntegration and differentiation met the need for precise mathematical descriptions of intuitive notions arising in geometry and natural science. Differen tiation is the concept needed for describing the notions of tangents to curves and of velocity of moving particles or m ore generally the concept of rate of change. The i ntuitive concept of area of a region with curved boundaries, finds i ts precise mathematical formulation in the process of integration Many other related concepts in geometry and physics also require integration as we shall see later. In this section we i ntroduce the concept of integral, in connection with the problem of measuring the area of a plane region bounded by curves. ,
,
.
,
Areas. We have an i ntuitive feeling that a region contained in a closed curve has an "area" which measures the number of square units inside the curve. Yet, the questi o n of how this measure for the area can be described in precise terms, necessitates a chain of mathe matical steps. The basic properties of area which intuition suggests are : area is a (positive) number (depending on the choice of the unit of length) ; this n umber is the same for congruent figures ; for all ,
1 The emergence of calculus extending over more than 2000 years represents one of the most fascinating chapters in the history of scientific discovery. Interested readers are referred to Carl B. Boyer, Concept of the Calculus, Hafner Publishing Company, 1 949. See also 0. Toeplitz, Calculus, A Genetic Approach, University of Chicago, 1 963.
Sec.
The Integral 1 2 1
2. 1
rectangles it i s t h e prod uct o f the lengths o f two adjacent sides ; and finally, fo r a region decomposed i nto parts, the area of the whole is equal to the sum of the areas of the parts. An immediate consequence is the fact : for a region A which is part of a region B, the area of A cannot exceed the area of B. These properties permi t the direct computation of the area of any figure that can be decomposed i nto a fi n ite n u m ber of rectangles . M ore generally, to assign a value F to the area of a region R we consider two other regions R' (inscribed) and R" (circumscri bed) decomposable
( \
f-- �
r---....
Figure
2.1
J J
v
R
'"""
R'
.
·-�-
R"
\
-- ---
I
J.-.--
7
Approximation of an area.
i nto rectangles, where R" contains R and R ' is contained in R (cf. Fig. 2. 1 ) . We know then at least that F has to lie between the areas of R' and R". The val ue of F is completely determ ined if we find seq uences of circumscri bed regions R," and i n scribed regions R ,' wh ich are both decomposable i n to rectangles and s uch that the areas of R , " and R , ' have the same l imit as n tends to i nfinity. Th i s is the method of "exhaustion", going back to antiquity which is used in elemen tary geometry to describe the area of a circle. 1 The precise form ulation of this i ntuitive idea now leads to the notion of in tegration. b. The Integral as an Area
Area under a Curve
The analytic notion of i ntegral arises when we associate areas with functions : We consider the area of a region bounded on the left and 1 Of course, we may use any k ind of inscribed and circumscribed polygon, since a polygon can be decomposed into right triangles and the area of a right triangle clearly is half that of a rectangle with the same sides.
1 22
The Fundamental Ideas of the Integral and Differential Calculus
Ch. "
right by vertical lines x = a and x = b, below by the x-axis and above by the graph of a positive continuous function f(x) (Fig. 2.2). This i� referred t o i n brief as t h e area "under t h e curve." For the moment we accept as intuitive the idea that the area of such a region is a definite n umber. We call this area F.b the integral of the function / between the
Figure
2.2
l imits1 a and b. In seeking the numerical value of F.b we make use of approximations by sums of areas of rectangles. For that purpose we divide the i nterval (a, b) of the x-axis i nto n (small) parts, not necessarily of the same size, which we shall call cells. At each point of division we d raw the line perpendicular to the x-axis up to the curve. The region with area Fa" is thus divided into n strips, each bounded by a portion of y
a
Figure
2.3
the graph of the function f(x) and by three straight line segments (Fig. 2. 3). Area or Integral as Limit of a Sum. Calculating the area of such strips precisely is not easier than calculating that of the original region. It is a step forward, however, to approximate the area of each strip from above and from bekw by the areas of the circumscribed and 1 N o confusion should arise from the use of the word "limit" for boundary points of the i nterval of integration.
The Integral 123
Sec. 2.1
inscri bed rectangles with the same base, where the curved bo u n da ry of the strip is replaced by a h orizontal line at a distance from the x-axis w hich i s either t h e greatest or the smallest value of j(x) i n t h e cell ( F i g . 2.4). More generally, we obta i n an intermed iate appr o x i m ati o n if we replace the strip by a rectangle of t h e same base and bounded on y
a
Figure 2.4
top by any horizontal l in e w h ich i n t ersect s t h e curved b o und a ry of the strip ( see F i g . 2.5). A n alytically, t h i s a m o u n t s t o replacing t h e function j(x) in each o f t h e cel l s by s o m e i n te r m ed i a te con s t a n t val u e . We de n o te by F, t h e s u m of the n rec tangul ar areas. I ntu ition tells us that the values F, tend t o F," if we make the subdivision finer and finer, that is, if w e l e t n increase w i t h out l imit w h ile the l argest length o f the
y
Figure
individual cell s ten d s t o zero . areas c o n s i st i n g of rectangles.
2.5
I n this way Fa"
is represented as a limit
of
c. A nalytic Definition of the Integral. Notations
Definition and Existence of Integrals I n the last paragraph we accepted the area under a curve as a quantity given int u i t i v ely and s u b seq u en t ly we represented it as a li m i t ing value. Now we shall reverse the p roced u re We n o longer i n v o ke .
1 24
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
intuition to assign an area to the region under a continuous curve ; on the contrary, we shall begin in a purely analytic way with the sums Fn defined previously, and we shall prove that these sums tend to a definite limit. This limit is then the precise definition of the integral and of the area. Let the function f(x) be continuous (but not necessarily positive) in the closed interval a ::;; x ::;; b . We d ivide the interval by (n - I ) y
--�or----&��L-L-L-����L-J-J--L��--� x Xn-1
Figure 2.6 T o illustrate t h e analytical definition o f integral .
points x., x2 ,
•
•
•
, xn _1 into n equal or unequal cells with the lengths (i = I, 2, . . . , n ) 1 ,
where in addition we put x0 = a, x n = b (cf. Fig. 2.6). In each closed subinterval [xi_1 , xi] or cell we choose any point �i whatever. We form the sum Fn
= /(�1)(x1 - Xo) + /( � 2)(x2 - X1) + · · · + f( �n)(x n - X n- 1) = /( � 1) �xl + /(�2) �X2 + · · · + f(�n) �xn.
1 The symbol Ll. m us t not be i n terpre te d as a factor but only as indicating a difference in values of the variable which follows. Thus the symbol Ll.x, means the difference x, - x,_, of consecutive values of x.
The Integral 125
Sec. 2.1
Using the s ? mmation symbol we write more concisely n F n = L / �; ) x; - X;- 1 ) i=l or F = L /a;) b. xi.
((
n
n
I f f(x) is positive, the value F, represents the area under the curve obtained by replacing f in each s ubinterval by the constant val ue /(� ; ). Of course, the sums Fn can be formed without assum ing / to be positive. It appears intuitively plausible t hat the sums F, must tend to a limit F.b as the number n of intervals increases indefinitely and at the same time the length of the largest subinterval tends to zero. This would i mply that the value of the limit Fab i s independent of the particular manner in which the points of division x 1 , x 2 , , x,_1 and the intermediate points �1 • � 2 , �n are chosen . We call Fab the integral of f(x) between the limits a and b. Geometric i ntuition, no matter how convincing, can only serve as a guide to our analytical l i m iting process ; therefore an analytic justification is needed, and we must furnish a proof for the existence of the integral as the limit described above. Furthermore, as already said, we need not at all insist on the assumption that the function f is positive in the interval. Thus we assert i= ]
•
•
•
•
•
•
.
THEOREM OF EXISTENCE. For any continuous function f(x) in a closed interval [a, b] the integral over this interval exists as the limit of the sums F, described above (independently of the choice of the points of subdivision x 1 , , x , _ 1 and of the intermediate points � 1, , � n as long as the largest of the lengths b.x; tends to zero). •
.
.
.
.
•
We shall first gai n some experience and insight before considering the existence proof for the integral i n the Supplement (p. 1 9 2).
Leibnitz's Notation for the Integral The definition of the integral as the limit of a sum led Leibnitz to express the integral by the following symbol :
ff(x)
dx.
The integral sign is a modification of the summation sign in the shape of a long S used at Leibnitz's time. The passage to the limit from a finite subdivision into portions b.x; i s indicated by the u se of the letter d in place of b. . In using this notation, however, we must not tolerate
1 26
The Fundamental Ideas of the Integral and Differential Calculus
Ch.
2
the eighteenth century mysticism of considering dx as an " i n fi n itely small" or "infin itesi mal quantity," or considering the integral as a "sum of an infin ite number of infin itely small quantities. " Such a conception is devoid of clear mean ing a nd obscures what we have previously formu lated w i th precision. From our presen t viewpoint the i n d ividual symbol dx has not been defined at all. The suggest ive combi nation of symbols
( 1'
J
x
/(x) d
is defined for a function f(x) i n the interval [a, b] by fo r m i ng
t h e o rdin ary sums F, and p as s i ng to the limit as n ex . T h e particular symbol w e use for t h e variable o f integratio n i s a matter of complete indifference (just as in the notation for sums it --+
y
X
Figure 2.7
did not m a tter what we called the i ndex of su mm a t i o n ) ; i nstead of
ft(:r)
dx we can eq ually well w r i te j)"( t) dt or ff(u) du. The
integrand denoted b y f i s a function of an i ndependent variable over the interval [a, h] and the name of the variable is irrelevant. O n l y the end
poi nt s of the interval of
i n tegration a ix.f(x) .C.f(a) da
and b affect the value of the
integral for given f Expressions like
dx or
in which
the same letter is used for the variable of i ntegration and an end· point of the i nterval are m i sle a d i n g under our definition and should, at fi rst, be avoided . lf the integrand f(x) i s positive i n the i n terval [a, b], we can
i m med iately identify
a,
i0.f(x)
dx with the area bounded by the graph off
and t h e lines x = x = h, and y = 0. The integral off, however, .is defined analytically as the l i m i t of sums F" i ndependent of any assump tion on the sign of f. l f.f(x) is negative in all or part of our i nterval, the only effect is to make the C ;) r r espon d i n g factors f(!;;) in o u r sum
Sec. 2. 1
The Integral
a
1 27
Figure 1.8
negative instead of positive. To the region bounded by the part of the curve below the x-axis we shall then naturally assign a negative area. The integral wi l l thus be the sum of positive and negative terms, corresponding respectively to porti ons of the curve above and below the x-axis 1 (see Fig. 2 . 7). It is i ntuitively convincing that our limit process converges even if the function f(x) is not everywhere conti nuous, but has j ump discon tinuities at one or several points like the function i ndicated by the cu rve in Fig. 2 . 8, where clearly an area under the c urve exists. 2
Figure
2.9
J
1
-1
1
2
x
sgn x dx
=
0.
Areas of regions bounded by arbitrary closed curves will be considered in Chapter4. As another example consider f(x) = sgn x on [ - 1 , I ] . We have f(x) = -1 for < 0 a n d f(x)
=
+I for x
> 0 (see Fig. 2.9). Then
f(x) dx = 0.
f+l -1
128
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
Thus the preceding limit process may well result in a definite limit of the sum Fn for functions having some discontinuities ; we indicate this possibility by calling such functions integrable. In the middle of the nineteenth century, the great Bernhard Riemann first analyzed the applicability of the process of i ntegration to general functions. More recently, various extensions of the concept of integration itself have been in troduced. Yet such refinements have less immediate importance for the calculus aimed at intuitively accessible phenomena, and it will not be necessary for us always to emphasize the integrability of our functions as a reminder that non integrable functions can be defined. In advanced courses the integral we have defined here is called the Riemann integral to distinguish it from various general ized concepts of integral ; the approximating sums Fn are called Riemann sums. 2.2
Elementary Examples of Integration
In a number of significant cases we are now able to calculate the integral of a function by carrying out the prescribed limiting process. This we shall do by an explicit evaluation of the sums F,. for a suitable choice of intermediate points ; ; (usually the left or right end point of the cells). The theorem on the existence of the integral of a continuous function assures that the limit of the F,. is the same for any other choice of the intermed iate points ;; and for any method of subdivision. a. Integration of a Linear Function
First we verify that the integral indeed gives the correct value of the area for some simple figures we know from geometry . Let f(x) = constant = To calculate the integral of f(x) between the limits of a and b we form the sums Fn (see Fig. 2. 1 0) . Since here /(0 = y, we find
y.
Fn =
I y Llx; = y ! Llx; = y(b - a) .
i=l
Hence, likewise lim F,. n -+ r::o
i=l
=i• a
y dx =
y(b - a) .
This is j ust the formula for the area of a rectangle of height b - a.
y and base
Sec. 2.2
Elementary Examples of Integration 1 29
f(x)
f(x) = "Y
Figure 2.10
Integral of a constant.
The integral of the function f(x) = x ,
lb
x dx,
(Fig. 2 . 1 1 ) , as we know from elementary geometry, has the value !(b - a)(b + a) = Hh2 - a2).
To confirm that our l i miting process leads analytically to the same result, we subdivide the i nterval fro m a to b into n equal parts by means of the points of division a + h, a + 2h, . . . , a + ( n - l )h, y
1+- b - a ---+i
Figure 2.1 1
1 30
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
where h = (b a)fn . Taking for �; the right-hand end point of each interval we find the integral as the limit as n -->- oo of the sum -
+ h)h + (a + 2h)h + · · · + (a + nh)h = n ah + ( I + 2 + 3 + · · · + n)h2 nah + !n(n + l )h 2 , where we have used the well-known formula for the sum of an arith metic progression (see p. I l l , Problem 3). Substituting h (b - a)jn, we see that F n = a(b - a) + H 1 + �)(b - a)2, Fn = (a
=
=
from which it follows immediately that lim F n =
a(b - a) + !(b - a)2 = Hb2 - a2).
b . Integration of x2
Elementary geometry does not so easily lead to the integration of the function f(x) = x2 , that is, to the determination of the area of the region1 bounded by a segment of a parabola, a segment of the x-axis, and two coordinates. A genuine limit process is needed. Assuming a < b we choose the same points of division and the same intermediate points as in the previous example (see Fig. 2 . 1 2) . It follows then that the integral of x2 between the limits a and b is the limit of the sums Fn =
(a + h)2h + (a + 2h)2h + · · · + (a + nh)2h + 2ah2( 1 + 2 + 3 + · · · + + h3( J ' + 22 + 3 2 + . . . + n2);
= na 2h
n)
by using the known values of the sums enclosed i n parentheses we find (see p. 58) Fn
na2h + n(n + 1)ah2 + l [n(n + 1 )(2n + l )W = a2(b - a) + (1 + �)a(b - a) 2 + Hl + �)(2 + �)(b - a)3•
=
Since lim n --� ro
6
�= n
0, we have
1 Sometimes referred to as "squar ing" the region.
Sec. 2.2
Elementary Examples of Integration 1 3 1
y
I I I I
Figure 1.11
Thus, for
a
Area under a parabolic arc by arithmetic subdivision.
< b,
*c. Integration of x� for Integers ot -¢ - I The next examples of this sectio n are i nstructive illustrations showing that in some cases the integration can be carried out by special ele mentary devices. Later i n Section 2.9d (p. 1 9 1 ) we shall achieve the same results more si mply by using general meth ods. The same kind of argument as u sed for x and x 2 , appl ied to the functions x3 , x4, . . , res ults in the relation .
(1)
Jbx• dx a
=
-1- (b"+l - a-+ 1 ),
ot + 1
I� �
w here ot is any positive i nteger ; this can be proved by finding appro priate formulas for the sums + 2 + · · · + n�, such as the relation lim
n - · oo
[o�
+ 2 � + . . . + n")
n
Ll]
=
1-
-
ot + 1
which can be proved by i nduction over ot (see Problem 16, p. 1 1 3) . In t h e following section, formula ( I ) will be proved i n a different way,
1 32
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
with greater generality and simplicity, indicating the power of the methods that we will develop. Its validity will be extended to all real values of r1. except r1. = - 1 Fortunately, the definition of the integral leaves us a great deal of latitude in the choice of subdivisions and furnishes a much sim pl er way to evaluate the integral . We do not have to use sums based on equi distant points of division . Instead, with the "quotient" �bja = q we .
y
Figure
2.13 Area under a parabolic arc by geometric subdivision.
subdivide the interval [a, b] by the points of a geometric progression
(Fig. 2. 1 3), a,
aq, aq2,
, aq•--1, aq" = b ;
we then need only to evaluate the sum o f a geometric seri e s . G i ve n the po i nts of div i si on xi = aqi the length of the ith cel l is given by •
•
•
llx; = aq' - aq•- = .
The largest Llx; is the last :
F n
6-x
. I
r.
=
aq i(q - 1 )
b( q - 1 ) q
q
.
.
o r ..... oo the number q ten ds toward the value one (see Example d, p. 64), and hence the length 6-x" of the largest cell, and then also the
Sec . 2.2
Elementary Examples of Integration 1 3 3
lengths o f all cells tend t o zero. For the intermediate points �. w e choose again the right-hand end points x, of each cell. The sum Fn
(2)
=
I C �;) " � x, = n
i�l
= a•+ 1
q
- 1i q
;� 1
1 I (aq ')"aq ' L=_ i- 1 q n
(q i +" )'
is known explicitly from the sum of the geometric progression with ratio qH•. Applying the well-known formula (p. 67), we find
=
a-+ 1(q
- 1)q"
(b/a)"+ 1 q •+ l -
l
1
=
(b"+ 1 - a-+1)q"
q-
q•+1
1
-
1
.
Since q -,= I , we can use once more the formula for the sum of a geo metric progression and write
q-
1
qi+•
q• + q•-1 + . . . + 1
1
For n -+ C1J all powers of q tend to one and it follows that lim F n
n � oo
=
1 -+- (bl+• - aH•) . OC 1
In this way we have verified the formula ( I ) for the integral of x� for 0 < a < b and any positive integer oc . The same method applies also for negative integers oc , provided that oc -,= - 1 . For the sum Fn we obtain as before Fn
=
=
q 1 q•+l - 1 q - 1 ( b•+ l - a•·H) q ( l - q - •- 1) (b"+ 1 - a•+l)q"
,
where we recall that - oc is positive and greater than one. Applying the formula for a geometric progression, we obtain
1( q - 1 ) -1
q
q - •- 1
which tends to 1 /( - oc
-
lim F
n�
oo
=
1
q - •- 1 + q - •-2 + . . . + q
1 ) as n -+ C1J. Consequently, as before,
n
=
-1-1 OC
+
(b"+ 1 - a•+ 1) ,
1 34
The Fundamental Ideas of the Integral and Diff'erential Calculus
Ch. 2
The i ntegral formula is meaningless for r1. = - I , since both n u mer ator and denominator on the right-hand side would then be zero. We find instead from our original expression (2) for F, for the case r1. = - I that F,. = n(q - 1 )/q. Consequently, observing that q =
1Ybja tends to one as n ___,.
(3)
l� I a
X
-
oo ,
we find
d x = l 1m ' n ( var,-;b/a - l ).
n -+ oo
Here the l i m i t on the right-hand side cannot be expressed in terms of powers of a and b but can be expressed in terms of logarithms of those q uantities as we shall see later (p. 1 45). *d.
Integration of x • for Rational rJ. Other Than
-l
The result obtained previously may be generalized considerably without essentially com plicating the proof. Let r1. = r/s be a positive rational n umber, r an d s bei n g positive i ntegers : then in the evaluation of the i n tegral given above nothing is changed except the evaluation of the limit (q - 1 )/(q• H - I) as q approaches one. This expression is now sim ply (q - 1 )/(q l r , s ) f s - 1 ) . Let us put q1 1' = .,. ( .,. # 1 ) : Then as q tends to one, .,. also tends to one. We have therefore to find the l i miting value of (T' - 1 )/(.,.' 1 ' - I) as .,. approaches one. If we d ivide both n umerator and denomi nator by .,. - I and transform them as before by the formula for geometric progress i ons, the l i m i t si mply becomes
Since both numerator and denomi nator are conti n uous i n .,., this l i m i t i s at once obtained b y substituting .,. = I, and thus equals sj(r + s ) = 1 /(rJ. + I ) ; hence for every positive rational value of r1. we obtain the integral formula
[ " x• d x = _I+_1 ( b"n - a• I I),
. a
(J.
just as with positive i ntegers. This formula remains valid for negative rational values of r1. = - r/s as well , provided we exclude the val ue r1. = - I ( for wh ich the . formula used above for the sum of the geometric progression loses its meani ng). For negative r1. we agai n evaluate the l i m i t of (q - 1 )/(q• H - I) by putting q-l/s = .,. for r1. = - r/s ; this is left as an exercise for the reader.
Elementary Examples of Integration 1 35
Sec. 2.2
It is natural to guess that the range of validity of our last formula extends also to irratjonal values of IX, We shall actually establish our integral formula for all real values of IX (except IX = - 1 ) in Section 2.7 (p. 1 54) i n a quite simple way as a consequence of the general theory. *e. Integration of
sin
x
and
cos
x
The last elementary example to be treated here by means of a special device is the integral of f(x) = sin :c. The integral
L
clearly is the limit of the sum
b sin x dJ:
sh = h [sin (a + h) + si n (a + 2h) + . . . + sin (a + n h)],
arising from division of the interval of integration into cells of size h = ( b - a)fn. We multiply the right-hand expression by 2 sin h/2 and recall the well-known trigonometrical formula 2 sin u sin
v
= cos (u - v) - cos ( u + v).
Provided h is not a multiple of 2", we obtain the formula S" =
� [cos ( 2 sm 2
a
n
+
( � h) =�[ ( �)
- cos a + 2 sm 2
+
· · ·
cos a +
- cos
(
a
+
( (
+ cos a +
- cos a +
� h)
2n - 1 h 2 2n +
2
(
+ cos a +
1 h
) ( ) J. - cos
� h) a
+
2n + 1
2
h
)J
Since a + n h = b, the i ntegral becomes the limit of
� [ ( �)
2 sm 2
cos a +
( �)]
- cos b +
as h - 0 .
Now we know from Chapter 1 (p. 84) that for h - 0, the expression (h/2)/(sin h/2) approaches the limit one. The desired limit is then simply cos a - cos b, and we arrive at the i n tegral
Similarly,
L
b
sin :r d.r
=
- (cos b - cos a).
Lbcos x dx = sin b - si n a
(see Problem
3, p. 196) .
1 36
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
Each of the preceding examples was treated with a special device. Yet the essential point of the systematic integral and differential calculus is the very fact that, i nstead of such special devices, we use general considerations which lead directly to the result. We shall arrive at these methods by first discussing some general rules concerning integrals and then introducing the concept of the derivative, and fin a lly establishing the connection between integral and derivative. 2.3 Fundamental Rules of Integration
The basic properties of the i n teg ral fo l l o w directly from its definition as t he limit of a sum :
f
J(x) dx =
l��
,t
J(;,) tix,,
where the in t erva l [a, b] i s broken up in to subintervals o r cells of length tix,, the number ; , stands for any value in the ith subinterval, and the largest tix, is re q u i re d to tend to zero fo r n � oo . a.
Additivity
Le t c be any value between a and b. I f we interpret integrals as areas and remember that the area of a region consisting of several parts is the sum of the areas of the parts (Fig. 2 . 1 4), we are led to the rul e
f J(x) dx = J.'f(x) dx + fJ(x) dx.
(4)
For an analytical proof we choose our s ub d ivi s i ons i n s uch a manner that the p o i nt c appears as a poi nt of division , say c = x,. ( w here m varies with n) . Then n
n
m
L J(;,) tix, = L J(;;) tix, + L J(;,) tix,,
where the first sum o n the ri g h t- hand side corre spo nds to a s ubd ivisio n of the interval [a, c] in m cells and the second sum to a subdivision of the interval [c, b ] . N o w for n ---+ oo we obtain our r ul e fo r inte gral s . i=l
i=m+l
i =-- 1
So far we have o n ly defined
J:
.r(x) dx when a
< b. For a = b or
> b we define the integral in such a way that the rule of add i t ivi ty is preserved. Therefore for c = a we must define
a
(5)
f
f(x) dx = 0,
Sec. 2.3
and then for b
Fundamental Rules of Integration =
a i t follows
137
that
fJ(x) dx + ff(x) dx ff(x) dx 0. This leads us to define icf(x) dx for c < a by the formula ( 6) ff(x) dx ff(x) dx, where the right side has the meaning originall established. Its geo metric meaning is that the area under the curve y f(x) is to be counted =
=
=
-
y
=
y
I
/
0
' \ \
b
a
Figure
2.14
as negative if the direction of moving from the lower limit of i ntegration t o the upper limit is that of decreasing A glance at the previous exam ples of i ntegrals confirms that i ndeed an interchange i n the l i m its of inte gration a and b results in changing the sign in the value of the integral.
x.
h. Integral of a Sum and of a Product with a Constant
f(x) g(x) ff( x) dx + fg(x) dx
(
)
If and are any two integ rable functions, the basic laws of operating with limits imply = = =
!��
[t f(!;i) �xi] + !�� [tg(!;;) �xi]
l�� [�J(!;i) � xi + i� g( !;;) �xi] l�� t�I [J(!;i) + g( !;i)] � xi} ;
Ch. 2
1 3 8 The Fundamental Ideas of the Integral and Differential Calculus
and hence the i mportant rule for the sum of two functions
r
(7)
f( x) dx +
r
g(x) d x =
similarly for the difference
r
j (x) dx -
b Ja
r
f " ��� i�
g(x) dx =
Furthermore, with any co n stant
and so
aj(x) dx =
( f (x ) + g(x)] dx ;
uc x) - g (x)] dx.
0(
aj(;,) t.xi
i=l n
= a l i m I j(;;) t.x;, n -+
J,a,
oo
aj(x) dx = a
(8)
r
J" a
j(x) dx.
The last two rules enable us to i ntegrate "linear combinations" of two or more functions that can be integrated individually. Thus for any quadratic function y = A:r 2 + Bx + C with any constants A , B, C, we h ave
J,, a
(A x2 + B x + C) dx
=
Jb Jb a
= A A
A x2 dx + a
f'' a
x2 dx + B
Bx dx +
J" a
B
i" a
C
C
X dx +
dx
J" a
1 dx
= - (b - a ) + - ( b - a 2 ) + C(b - a ) . 2 3 3
3
2
In the same way we integrate the general polynomial y
= A0x " + A1x "- 1 +
· ·
+ A ,_1x + A , :
1 A0(b"+1 1.( '' y d x = --
. a
c.
·
n +
J
Estimating Integrals
a "+
l
) + -l A 1 ( b" n
+ iAn-lW -
a
2
a"
)+
· · ·
) + A ,(b - a ) .
Another obvious observation concerning integrals is basic. Consider for a < b a function f(x) wh kh is positive or zero at each point of th e interval [a, b]. Then
(9)
Sec. 2.3
Fundamental Rules of Integration 1 39
This follows immediately if we write the integral as limit of a sum and notice that the sums contain only nonnegative terms. More generally, i f we h ave two functions f and with the property that f(x) for all i n the interval [a, then
(10)
z g(x)
x b], ff(x) dx z fg(x) dx.
g
For we have
ff(x) dx - rg(x) dx r [f(x) - g(x)] dx z 0, since f(x) - g(x) is never negative. We apply this result to a function f(x) which i s continuous i n the i nterval [a, b]. Let be the greatest value and m the least value off in that interval . Since �f(x) � M for all x i n [a, b], we have Ib m dx �Ibf(x) dx �IbM dx. =
M
m
a
a
a
Recalling that for any constant C C
fc dx = l\ dx
=
C(b - a),
we obtain the i nequal i ty
(I I )
m (b - a) � rf(x) dx � M(b - a),
which gives si mple upper and lower bounds for t h e definite i ntegral o f a n y conti n uous function. Again this esti mate is intuitively obvious. I f we thin k of the integral interpreted as an area, the a) and - a) represent areas of a circumscribed and an i n sc r i be d rectangle on the common base of length b a (see Fig. 2. 1 5) .
-
quantities M(b -
m(b
d. The Mean Value Theorem for Integrals
Integral as a Mean Value Significant is a slightly different interpretation of our inequalities in terms of the average of the function f in an interval [a, For a finite n umber of q uantities f1 , j�, . . . , f,. the average or arithmetic mean is the number
b].
n
140
The Fundamental Ideas of the Integral and Differential Calculus
y
a
Figure 2.15
y
M
- - - - -,-
'---,...., \ \
m
---r----0---------�------------�----- x a 0 b Figure 2.16
The mean value
1'-
of a function.
Ch. 2
Sec. 2.3
Fundamental Rules of Integration
1 41
If we want to assign a meaning to the average value of the infinitely many -quantities f(x) corresponding to arbitrary x in the i nterval [a, b ], it i s natural to pick out first a finite number n of values of f, say f(x 1),/(x2), , J(x11), to form thei r average •
•
•
f( x l ) +
··· n
+ f( x n )
and then to take the limit as n i ncreases beyond all bounds. The value of this limit, i f i t exists at all, w i l l depend very m uch on how the points X ; are spaced in the i nterval [a, b]. A definite value for the average off is attained if we take for the x; the points obtai ned when we d ivide the i nterval [a, b] into n equal parts of length �x; = (b - a)fn. We have then f( x l ) + . . . + f( x ") = _ 1 _ i. J ( x; ) tix;,
b-
n
and i t i s clear that in the l i m i t for n -+ oo the nth averages converge towards the value b b f ( x) d x b p, = -- f (x) dx = b - a
1
a ;�l
i
J
a
a
f.
dx
W e shall call p, the "arithmetic average" o r the mean value of f i n the i n terval [a, b]. Our inequalities then simply state that the mean value of a continuous function cannot be larger than the greatest value or less than the least va_lue of the function ( Fig. 2. 1 6). Since the function f(x) is conti n uous i n the i n terval [a, b], there m ust be points in the i n terval where f has the value M or the value m . B y the i n termediate value theorem for continuous functions there must then also be a poin t ; i n the i n terval where f actually assumes the i ntermediate value p, . We have proved then : MEAN VALUE THEOREM . Fo r a continuous function j(x) in the interval [a, b) there exists a value ; in the interval such that
( 1 2)
r
f( x) dx
=
f(�)(b - a)
.
This i s the simple but very i mportant mean t•alue theorem of integral calculus. I n words, it states that the mean value of a continuous function in an i n terval bel ongs to the range of the functi on. The theorem asserts only the existence of at least one � i n the i nterval for which f ( �) is equal to the average value off but gives no further i nformation about the location of �.
1 42 The Fundamental Ideas of the Integral and Differenial Calculus
Ch. 2
Note that the formula expressing the mean value theorem stays valid if the limits a and b are interchanged ; hence the mean value theorem is correct also when a > b. The Generalized Mean Value Theorem. Instead of the simple arithmetic average we often have to consider "weighted averages" of n quantities /I • . . . . fn given by p ,f, + Pdz + · · · + p ,.f,,
P I + P2 + · · · + Pn where the "weight factors" P i are any positive quantities. If, for example, , Pn are actually the weights of particles located respectively at the P �o p2, points /1 , [2, , .fn of the .r-axis, then I' will represen t the location of = p,
•
•
•
the center of gravity. I f all weights p1 are equal, the q uantity I' is just the arithmetic average defined above. For a function f(.t') we can form analogously the weighted average •
•
•
frc:r)p(.r) fp(.r)
( 1 3)
d:c
d.t
over the interval [a, b] where p(.t'), the weight function, is any positive function i n the i n terval. The assu mption that p is positive guarantees that the denominator does not vanish.
The weighted average I' also lies between the largest value M and the smallest value m <�{ the .fim c tio n f in the interval. For multiplying the inequality m
by the
S, .f{:e) s M,
positil•e number p(.c), we find that mp(1:)
I n tegration then yields
m
f
p(.r) d.�: s
S, f(:t )p(.c) S, Mp(:r).
J:f(x)p(x) i'�(a:)
Dividing by the positive q uantity
m S,
p
dx s M
r
p (:r) d.r.
d"', we i ndeed obtain the resu l t
s M.
If here f(x) is continuous, we conclude from the i n termediate value theorem (p. 44) that I' = f(�). where � is a sui table value i n the interval a s � s b. This leads to the following generalized mean value theorem of integral calculus :
Sec. 2.4
The Integral as Function of the Upper Limit (Indefinite Integral) 143
If j(:r) and p(x) are continuous in the interval [a, b] and moreover p(:r) is positive in that interval, then there exists a value ; in the interval such that
lb
( 1 4)
a
j()·)p(x) d.r = j(;)
ih a
p(.c) dx.
The special case p(x) = I leads to our earlier mean value theorem . 2.4 The Integral as a Function of the Upper Limit (Indefinite Integral)
Definition and Basic Formula The value of the integral of a fu nction f(x) depends on the limits of integrati on a and b : The i n tegral is a function of the two l i mits a and b. In order to study this dependence on the l i m its more closely we imagine the lower limit to be a fi xed number, say rt. , denote the variable of i n tegrati on no longer by x but by u (see p. 1 26), and denote the upper limit by x i nstead of by b i n order to i nd icate that we shall consider the upper l i m i t as the variable and that we wish to i nvestigate the value of the i n tegral as a function of t h i s upper l i m i t. Accord ingly, we write cp(x)
=
J:
t(u ) d u .
W e call the function cp(x) an indefinite integral o f the fu nctio n f(x). When we speak of an and not of the i ndefi n i te integral , w e suggest that i nstead of the l ower l i m i t rt. any other could be chosen, in which case we should ordinarily obtain a d i fferent value for the integral. Geometri cally, the i ndefinite i n tegral cp(x) is gi ven by the area (s hown by shad i ng i n Fig. 2. 1 7) u n der the curve y = f(u) and bounded by the u-axis, the ord i n ate u = rt. and the variable ordi nate u = x , the sign being determi ned by the rules d i scussed earlier (p. 1 26). Any particular definite i n tegral i s found from the i n defin i te integral cp(x) . Indeed, by our basic rules for i ntegrals,
ft
(u) du
= =
f
f (u) du +
-
J."
f
f( u ) du
f (u) d u +
f
f( u ) d u
=
cp( b) - cp( a ) .
I n particular, w e c a n express any other i ndefini te integral with a lower limit rt. ' in terms of cp(x) :
r
f ( u) d u
=
cp( x) - cp( rt.') .
144
Ch. 2
The Fundamental Ideas of the Integral and Differential Calculus y
Figure 2.17
The indefinite integral as an area.
As we see, any indefinite integral differs from the special indefinite integral rp(x) only by a constant.
Continuity of the Indefinite Integral
I f the function f(x) is continuous i n the i nterval [a, b J a n d point of that interval, then the i n definite integral rp(x) =
rx
is a
f
f(u) du
represents a fu n c t i o n of x w hi c h is again defined i n the same interval . As easily seen : The indefinite integral rf>(x) of a continuous function f(x) is likewise continuous. For if x and y are any two values in the i n terval we have by the mean value theorem that
(1 5)
rp(y) - r/>(x) =
J:
f(u) du = f(�)(y - x)
where ; i s s o m e value in the i n te rv al with end points x a n d y From the contin uity off we have then .
lim rp(y)
=
lim [rp(x) + f(;)(y - x )]
=
rf>(x) + .f( x ) · 0
=
rp( x),
which shows that ¢ is co n t i n uo us . More specifically, in any closed i nterval we have l ¢(y) ¢(:r) l � M IY xi , where M is t he maximum of I f I in the interval, so that ¢ is even Lipschitz-continuous. -
-
Sec. 2.5
Logarithm Defined by an Integral 1 45
Formula ( 1 5) for (y) - (x) shows : that (x) is an increasing func tion of ;t in case f is positive throughout the i n terval , namely, for y > x
(y) = (x) + f(�)(y - x) > (x).
Formi n g the i ndefinite i ntegral of a fu nction i s an i mportant way of generating new types of functions. I n Section 2 . 5 we shall apply this method to introduce the logarith m fu nction. This will also give u s a first gl i m pse of the fact that general theorems of mathematical analysis lead to the m ost remarkable specific formulas .
A s we shall see in Section 3 . 1 4a (p. 298 ), t h e defin i t i o n of new functions by means of i n tegrals of a l ready defined fu nctions is a satisfactory proced u re if we wish to put definitions (for e x a m ple, of the trigonometric fu nctions) on a p u rely analytical basis in stead of relying on i n t u i tive geom etrical explanat ions.
2.5 Logarithm Defined by an Integral a. Definition of the Logarithm Function In Section 2 . 2 we had succeeded i n expressing
f,,
x• dx for any rational
oc � - I in terms of powers of a and h. For oc = - I we were only able to represent the integral as l i m i t of a seq uence
{'' l du Ja U
t
= l im n( v' bja - 1 ) . n -+ rx)
Independently of the d i scussions of Section 2 . 2 we now introduce the function represented by the indefinite i ntegral
r x ! du , l Jr u
or, geometrically, by the area under a hyperbo la as ind icated i n Fig. 2. 1 8 . We cal l it the logarithm of x, or more accurately the natural logarithm of x, and write ( 1 6)
log x =
ix 1
1 -
u
du.
S ince y = 1 /u i s a continuous and positive function for all u > 0, the function log x is defined for all x > 0, is moreover continuous, and also is monotonical ly i ncreasing. The choice of I as the lower limit in In this section we again freely use the fact that the integral of a continuous function (here the function I /u) exists ; the general proof is given in the Supplement. 1
146
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
y
I I
I
\\ y = ?;I
I I
\
'
Figure 2.18
Log x represented by an area.
the i n defin i te integral for log x i s a m atter of convenience. It i mplies that log
( 1 7) and that l o g x is positive for
I
(Fig.
and
2. 1 9).
b can
1
x> I
=
0,
and negative for
x between
zero and
Any defi n i te i n tegral of 1 /u between positive l i m its
,
a
be expressed i n terms of l ogarithms by the for m ula (see p. 1 43)
(18) y
J
a
- du I
U
-2
Figure 2. 1 9
= log
b
-
log
a.
The natural logarithm.
Sec. 2.5
Lo!(arithm Defined by an Integral 1 4 7
Geometrically, this integral represents the area under the hyperbola lp; between the ordinates x = a and x = b.
y =
b. The Addition Theorem for Logarithms The fundamental property which j ustifies the trad i tional name for log x i s e x p re ssed by the
For any positiue
ADDITION THEOREM.
and y
l og (xy) = l og x + l og y
( 1 9) PROOF .
X
.
We write the addition theorem i n the form log (;ry)
fxy ]
or
y
-
- dv V
log 11 = log x =
ix 1 I
- du , U
where we h ave deli berately chosen different letters for the variables of i n tegrati o n i n the two i n tegrals. The equal ity of the two i ntegrals will follow from the fact that the approximating sums have the same value for su itable choices of subdivisions and of i n termediate points. Assume at first x > 1 . Then
fx
n 1 1 - du = l i m 2 6-ui, 1 U 1 --'-- 1 �i n --+ r:D
-
where u0 = I, u 1 , u2, , u " = x represent the points ari sing i n a sub division of the i n terval [ I , :r] and �i l ies in the it h cell. Putting l' ; = yui, 11i = y�, we see that the points r11, l'� > . . . , 1 ' , correspond to a sub d ivision of the interval [y, xy] with i n termed iate points 'YJi = �,y. Obviously, •
•
•
so that
For n tending to i n fi n i ty we obta i n the des i red i d entity between i n teg ra ls for the case x > I . For x = 1 the addition theorem holds trivially, si nce log 1 = 0. To prove the th eo rem also for the case 0 < x < 1, we observe that then
148
The Fundamental Ideas of the Integral and Differential Calculus
l fx > 1 , and hence
Ch.
2
(� )
Jog x + log y = log x + Jog
xy
1
= log x + Jog - + log (xy) X
1
X
= log - + Jog x + log (x y) = log
(� )
x + log (xy)
= log 1 + l og (xy) = Jog (xy). This completes the proof of the addition theorem .
A proof of the addition t heorem can also be based on formula (3) (p. 1 34), according to which log x = l i m n( \�1� 1 ).
Then log (xy) = lim n( 1"1 xy = lim [n( \1�
-
-
I)
I ) ·\' y + n( \:
l
= [lim n( v � - I )] ( i m n -� oo
since lim
\Vy
=
= log x + log y,
I (see p. 64).
\Vy)
-
y
I )]
1 + lim n( \ y - I )
Applying the addition theorem to the special case y
I
log 1 = Jog x + log or
l fx leads to
X
1
(20)
=
log - = - log x.
More generally then
(2 1)
log
-= y
X
X
log y + log
1 -
X
=
log y - log x.
Repeated application of the addition theorem to a product of factors yields
1
log (x x2 · • • xn) = log x1 + log x2 + · · · + log xn.
n
Sec. 2.6
Exponential Function and Powers 1 49
In particular, we find that for any positive integer n
( 22)
log (x")
=
n log x.
This identity also holds for n = 0, si nce x0 to negative integers n by observing that
(� ) ·
log ( x " ) = log
X
I,
=
and can be extended
= - log ( x-") = - ( - n ) log x =
n
log x.
n
For any rational r1. = m/n and any positive a we can form a• a m / n = X. We have then
1
m
1
log x = - log x " = - log a "' = - log a = n n n
r1.
log
=
a.
Thus the identity
(23)
log (a") =
r1.
log a
holds for any positive real a and any rational 2.6
r1. .
Exponential Function and Powers
a. The Logarithm of the Number
e
The constant e obtained on p. 79 as the l i m i t of ( I + 1 /n)" plays a d i stinguished role for the function log x. I ndeed , the n u mber e i s characterized b y t h e eq uat ion' log e = 1 . For the proof we observe that the continuity of the function log implies log e = log
[!�n: ( 1 + �n !�n: [ ( 1 �n ( 1 + I) . +
log
=
= l i m n log n -+ 00
x
n
Now by the mean value theorem of i ntegral calcul us log
( 1 1) il+l/ n +
-
n
=
1
J
-
u
1 1
du = -
�
n
-
,
1 This means geometrically that the area bounded by the hyperbola lines y = 0, x = I , and x = e has the value one (see Fig. 2. 1 8).
y = l fx and the
Ch. 2
1 50 The Fundamental Ideas of the Integral and Differential Calculus
where $ is some number between I and I + 1 /n which depends on the choice of n . Obviously, lim $ = I so that (24)
log e
=
lim
n � oo
!
$
=
1.
b. The Inverse Function of the Logarithm. The Exponential Function
From the relation log e
=
I it follows that for any rational
log (e")
= oc
log e
= oc.
oc
This shows that every rational number oc occurs as a value of log x for some positive x. Since log x is continuous, it assumes then any value intermediate between two rational values ; this means all real values. It follows that for x varying over all positive values the val ues of y = log x range over all numbers y. Since log x is monotonically inc reasing there exists for any real y exactly one positive x such that l og x = y . The solution x of the equation y = log x is given by the inverse function of the lo ga r it h m which w e shal l denote b y x = E(y). We know then that E(y) (Fig. 2.20) is defined and positive for all y, and again continuous and increasing (see p. 45) ,
y
Figure 2.20
The exponential function.
Sec. 2.6
Exponential Function and Powers
151
Since the equations y = log x and x = E(y) stand for the same relation between ·x and y, we can write the equation :x = log (e•), which is valid for rational :x, also in the form E(:x)
=
e".
We see : for any rational :x the value of E(:x) is the :xth power of the n u mber e. For rational :x = mfn the power e• is defined directly as \Y e"'. For i rrational :x the ex pression e• is defi ned most naturally by representing :x as the l i m i t of a sequence of rational numbers :x and putting e• = l i m (e" n). Si nce e • n = E(:x , ) a n d si nce th e function E(y) ..
depends continuously on y, we can be s ure that the l im it of the e•• exists and that it has the value E(:x) independently of the special sequence used to approxi mate :x. This proves that the equation E(:x) = e• holds for irrational :x as well . For all real :x we can now write e• i nstead of E(:x). We cal l e' the exponential function. This function is defined and continuous for all x , i s increasing, and positive everywhere. Since the equations y = log .1: and x = e• are two ways of expressing the same relation between the numbers x and y, we see that log x, the "natural logarithm" of x (as defi ned here by an integral ) stands for the logarithm to the base e, as that term would be used in elementary mathematics ; that is, log :r is the exponent of that power of e which is equal to x or u -- ... oo
(25)
e log r = x .
(26)
log e• =
We can write1 log x = loge x. Similarly, x = e• is that n umber whose logarithm is y, or y.
From the point of view of calculus it is real ly easier to introduce natural logarithms first as i ntegrals of the si mple function y = l fx, as we did here, and to define powers of e by taking the i nverse of the l ogarithm function. I n this way the continuity and monotonicity of the functions log x and ex arise j ust as consequences of general theorems and require no special arguments. 1 The reader may feel that the name "natural logarithm" should have been reserved rather for logarithms to the base 1 0 . However, historically the first table of log arithms published by Napier in 1 6 1 4 essentially gave logarithms to the base e. Logarithms to the base 10 were introduced only subsequently by Briggs because of their obvious computational advantages.
1 52 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
c. The Exponential Function as Limit of Powers
Originally we obtained the number e as the limit n -+ co
e = lim
(t + I)". n
A more general formula represents e• for any n e • = lim 1 + � . (2 7) n -+ n 00
( )
x
as a limit
For the proof it is sufficient to show that the sequence s n = log
( + �r 1
has the limit x. For then the sequence of values
must tend to
e•
( �r ( + x) (I+x/n , � a;. ; e'• =
1 +
since the exponential function is continuous. Now
s,. = n log
1
= n
1
J
By the mean value theorem of i ntegral calculus we have s,.
= n
.l[( �) ] ; ,.
1 +
n
-
1
=
�,
!; ,.
where ; n is some value between one and 1 + xjn. Since obviously ; n tends to one for n tending to oo, we have i ndeed lim s,. = x. d. Definition of Arbitrary Powers of Positive Numbers
Arbitrary powers of any positive numbers can now be expressed terms of t h e exponential a n d logarithmic functions. 1 We found for rational oc and any positive x that the relation log (x") =
oc
in
log x
holds. We write this equation i n the form X
ll = erJ. lOg �r
.
1 This obviates the more clumsy "elementary" definition and justification of these processes by passage to the limit from rational exponents indicated on p. 86.
Sec. 2.6
Exponential Function and Powers ! 53
For i rrational rx we again represent rx as limit of a sequence of rational numbers rx',. and define "" l "' x • = lim x"" = lim e og . n -+ co
The continuity of the exponential function i mplies again that the l i m i t exists and that it has the value e • Iog "' , since al g el e o = Jm ( a li } O g .2') = li m ean ) Og X
X
•
Hence the equation
(28)
holds quite generally for any rx and any positive x. Putting log x = fJ or, what is the same, x = e P we i nfer
(29 )
and m ore generally then for any positive x ( x")p = ( e• I og x)p = e•P i og x = x• P.
Another rule for working with powers which i s easily established in complete general ity, i s the multiplication law where x is a positive number and ot and {J are arbitrary. It is sufficient to prove the corresponding formula obtained by tak ing the logarithms of both sides : Now by the rules ( 1 9), (26) , and (28) al ready established it follows that log ( x • xP ) = l og x• + log x P = l og ( e" 10g "') + log ( eP J og "') = rx l og x + {J log x = (rx + fJ) log x log ( e<•+P> l og "') = log (x•+P) . =
e. Logarithms to A ny Base
I t is easy to express logarithms to a base other than e in terms of n atural logarithms. I f for a positive number a the equation x = a • is satisfied , we write y = log. x. Now a • = eY log a, so that x = eY log a or (30)
y
log a = log x. I t follows that
log x log. x = -- , log a
1 54
The Fundamental Ideas of the Integral and Differential Calculus
where log x is the natural logarithm to the base common logarithms to the base 10 are given by
e.
Ch. 2
l n particular, the
l og x log10 x = . --
log 1 0
Since logarithms to any base a are proportional to natural log arithms, they satisfy the same addition theorem : log. x + log. y = log, (xy).
2.7
The Integral of an Arbitrary Power of
x
ln Section 2.2 we obtained the formula
u J u" du = '
b"+l - a• · I o: + I
----
for any rational o: ¥ - I . (The case o: = - 1 was seen to lead to the logarithm.) To eval uate the i ntegral when o: is an irrational number, it is sufficient to discuss the indefinite i ntegral
cp(x) =
fu• du
from which all definite integrals with positive limits a and b can be obtained . Assume x > I (the case x < I can be handled in the same fashion after interchanging the l imits). We have then by (28) where log u � 0 for u i n the interval of integration . Let f3 and y be any two rational numbers different from - 1 for which Then also
f3 log u :::;; o: log u :::;; y log u.
Since the exponential function is increasi ng, this i mplies that is, We have then
Sec. 2.8
The Derivative
1 55
The integrals of u P and u1 were evaluated before, leading to I I __ (x P +l - I ) �
(x) � __ (xY+ l - 1 ).
/1 + 1
y+ l
lf we now let the rational numbers {1 and y converge to IX, we obtain i n the li mit I (x) = (x•+ l - ! ), IX + I since xP+I = e iP+ I )Io� • and xl'+1 = ei;+I >log x tend to e i • +I>Iog • = x"+ l because of the continuity of the exponential function. The same result follows for x between zero and one. Thus generally for positive a , b --
J "u • du a
=
(b) - (a )
=
1 - (b" 1 1 - a "+I) IX + 1 -
j ust as for rational rx . When IX is a positive integer, the formula remains valid even when the l i m its a or b become zero or negative ; it is easy to extend the formula directly to those cases. 2.8
The Derivative
The concept of the derivative, like that of the i n tegral, has an immediate i ntuitive origin and is easy to grasp. Yet it opens the door to an enormous wealth of mathematical facts and i nsights ; the student wi ll only gradually become aware of the variety of significant appli cations and of the power of the techniques which we shall develop i n t h i s book. The concept of derivative is first suggested by the intuitive notion of the tangent to a smooth curve y = f(x) at a point P with the coordinates x and y. This tangent is characterized by the angle IX between its direction and the positive x-axis. But how does one obtain this angle fro m the analytical description of the function f(x) ? The knowledge of the values of x and y at the point P does not suffice to determine the angle rx since there are infinitely many different lines besides the tangent passing through P. On the other hand, to determine IX one does not n eed to know the function f(x) in i ts total over-all behavior ; the knowledge of the function in an arbitrary neighborhood of the point P must be sufficient to determine the direction IX, no matter how tiny a neighborhood is chosen . This indicates that we should define the direction of the tangent to a curve y = j(x) by a limiting process, as we shall presently do.
1 56 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
The problem of calculating the direction of tangents, or of "differen tiation," was impressed on mathematicians as early as the sixteenth century by optimization problems, that is, questions of maxima and minima arising in geometry, mechanics and optics. (See the discussion in Section 3.6.) Another problem of paramount importance which leads to differen tiation is that of giving a precise mathematical meaning to the intuitive notion of velocity in an arbitrary nonuniform motion (see p. 1 62). We shall start with the problem of describing the tangent to a curve analytically by a limit process. a. The Derivative and the Tangent Geometric Definition. In conformity with naive intuition, we define the tangent to the given curve y = f(x) at one of its points P by means y
0 Figure 2.21
Secant and tangent.
of the following geometrical limiting process (Fig. 2.21). We consider a second point P 1 near P on the curve. Through the two points P, P1 we draw a straight line, a secant of the curve. If now the point P1 moves along the curve towards the point P, then the secant is expected to approach a limiting position which is i ndependent of the side from which P1 tends to P. This limiting position of the secant is the tangent ; the statement that such a ltmiting position o f the secant exists i s equivalent t o the assumption that the curve has a definite tangent o r a
Sec. 2.8
The Derivative 1 57
definite direction at the point P. (We have u sed the word "assumption" because we have actually made one. The hypothesis that the tangent exists at every point is by no means true for all curves representing simple fu nctions. For example, any curve with a corner or vertex at a point P does not have a uniquely determined direction there, such as the curve defined by y = \xi at (0, 0) . (See the discussion on p. 1 66.) y
--,0�----��----�- % Figure
2.22
Since our curve is represented by means of a function y = f(x), we must formulate the geomet r ic limiting process analytically, with reference to f(x) . This analytical limit process is called differentiation off(x). Consider the angle which a straight line makes with the x-axis as the one through which the positive x-axis must be turned in the positive direction or counterclockwise• in order to become for the first time parallel to the line. (This would be an angle (X in the i nterval 0 � IX < 1T. ) Let 1X1 be the angle which the secant PP1 forms with the positive x-axis (cf. F ig. 2.22) and IX the angle which the tangent forms with the positi ve x-axis. Then
1 That is, in such a direction that a rotation of -rr/2 brings it into coincidence with the positive y-axis.
1 58 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
where the meaning of the symbols is obvious. Let x, y and x 1 y1 be the coordinates of the points P and P1 respectively. Then we immediately have1 - Y f(xl ) - f(x) tan o-:1 = Y1 = ; ,
---
X1 - X
X1 - X
·
thus our l imiting process (disregarding the case dicular tangent) is represented by the equation lim x1-x
Notation.
a:
= TT/ 2 of a perpen
f(x1) - f(x) = lim tan o-:1 = tan x1 - x x 1 --+ x
a:.
The expression
we call the d!!Jerence quotient of the function y = f(x) where the symbols 6-y and 6-x denote the differences of the function y = f(x) and of the i ndependent variable x. (Here, as on p. 1 24, the symbol 6. is an abbreviation for difference, and is not a factor.) The trigonometric tangent of IX , the "slope" of the curve,• is therefore equal to the limit to which the difference quotient of our function tends when x1 tends to x. We call this limit of the difference q uotient the derivative3 of the function y = f(x) at the point x. We shall general ly use ei ther the notati on of Lagrange, y' = .f'(x) , to denote the derivative, or, as Leibnitz did, the symbol• dyfdx or df(x)fdx or (dfdx)j(x) . On p. 1 7 1 we shall discuss the meaning o f Leibnitz's notation in more detail ; here we point out : The notation f'(x) ind icates the fact that the derivatiz•e is itse(f a function of x since a value of f'(x) corresponds to each value of x in the interval considered. This fact is sometimes emphasized by the use of the terms derived function, derived curve. The definition of the derivative appears in several different forms : f'(x ) = lim x1-x
f(x l) - f(x) x1 - x
= lim
h -+ O
f(x + h) - f(x) h
,
1 In order that this equation may have a meaning, we must assume that both x and x1 lie in the domain of f. In what follows, corresponding assumptions will often be made tacitly in the steps leading up to l imiting processes. 2 The word gradient or direction coefficient is used occasionally. 3 The term dijj'erentia/ coefficient ;s also used in older textbooks . Cauchy 's notation Df(x) and Newton's notation y are also used. •
Sec. 2.8
The Derivative
1 59
where in the second expression x1 1s re pl aced by x + h, or in Leibnitz's ·notation, dy dx
=
f). y f(xi ) - f(x) . = f ' (x) = lim = lim dx z1 - z XI - x t. z - o 6.x
df(x)
If f is defined in a nei ghborh ood of the p oi n t x, then the quotient # 0 [f(x + h) - f(x) ] / h i s defined as a funct ion of for for which J h l i s sufficiently small to ensure that x + i s in the interval u nder consideration. The definition of f'(x) as a limit requires that f(x + - f(x) f'( ) - x IS ar b 1' tran· 1y sma II ,or ·· " a I I h # 0 ( pos1t1ve
I
h all values h h
h)
I
·
h or n egati v e) for which J h l i s sufficiently small.
A nalytic Calculation of Derivatives. The i ntuitive concept and the general analytic notion of derivative are simple and straightforward. Less obvious i s the procedure of actually carrying out such li m i t ing processes. I t is i mpossible to find the derivative mere ly by putting x1 = x i n t h e expression for t h e d i fference quotient, for then the n u merator and denominator would both be equal to zero and we would be led to the m e a n ingless expressi o n 0/0. T h u s the passage to the limit in each case depends o n certai n pre lim i na ry steps (transformation of the differe nce quotient). For example, for th e function f(x) = x2 we have xl 2 - x 2 f(xi) - f(x) -'--"'---"-- -'--- = --- = xi + x XI - X
XI - X
whenever x # xi .
This functio n xi + x does not have exactly the same domain as (xi2 - x2)j(xi - x) : The fun ction x1 + x i s defined a t the one poin t x 1 = x , where the q u o t i e n t (xi2 - x2)/(xi - x) i s undefined . For all other values of xi the two functions are equal to one another ; hence in the passage to the l i m i t , for which we specifically require that xi # x, w e obta i n the same value for lim (x12 - x2)/(x i - x) as for lim (x1 + x) . .X J -+ X
.Xt -+ X
However, since the function xi + x is defined a n d continuous a t the point xi = x, we can do with it what we could not do with the quotient, namely, pass to the limit by s i mply putting xi = x. For the derivative we then obtain
f'(x) =
d(x2) = 2x. dx
1 60 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
As another example we differentiate, that is, calculate the derivative of the function y = ,_;; for x > 0. We have for x1 ¢ x
+ = ,j"i; - ,_;; =
J(xl) - J(x)
X1 - X
Hence (for x > 0)
1
,jx1 + ,jx ·
d,j; - 1 -- = lim ,jx1 + ,jx 2.J; . dx x,-,
For x = 0 we h ave a singularity : The derivative is infinite, smce
(,J;,;_ - O)j(x1 - 0) = 1 /,j� --+ oo for x1 --+ 0. Analytic Definition
It is extremely sign ificant that the process of differentiating a function has a definite analytic meaning quite apart from the geometric intuitive conception of the tangent. The analytic definition of the integral , freed from the geometric visualization of area, allowed us to base the notion of area on that of integral. In a similar spirit, independently of the geometrical representation of a function y = j(x) by means of a curve, we define the derivative of the function y = j(x) as the new function y' = j'(x) given by the li mit of the difference quotient l'lyj !'lx provided that the limit exists. Here the differences l'ly = y1 - y = f(x1) - f(x) and l'lx = x1 - x are "corresponding changes" in the variables y and x. The ratio l'lyj !'lx can be called the "average rate of change" of y with respect to x in the i nterval (x, x + !'lx). The limit f'(x) = dyjdx represents then the "instantaneous rate of change" or simply the "rate of change" of y with respect to x. If this limit exists, we say that the function f(x) is differentiable. We shall always assume that every function dealt with is differentiable unless specific mention is made to the contrary.1 We emphasize that if the function f(x) is to be differentiable at the point x the limit as h --+ 0 of the quotient [j(x + h) - .f(x)]/h must exist, where h can have any val ue ¢ 0 for which x + h belongs to the domain of f If, in particular, .f is defined in a whole interval containing the point x in its interior, then the limit must exist independently of the manner in 1 Examples in which this assumption is not satisfied will be given later (see p . 1 67). Such examples j ustify mentioning differentiability as an assumption if the context warrants it.
Sec. l.8
The Derivative 1 6 1
which h tends to zero, whether it be through positive values or through negative "alues, without restriction upon sign . Having now an analytic definition for the derivative .f'(x), we take the direction angle 'Y. to the positive x-axis given by the equation tan 'Y. = f'(x) as the direction of the tangent to the curve at the point (x, y).1 By thus basing the geometric definition on the analytic one we avoid the difficulties which might arise from the vagueness of the geometric visualization. In fact, we have now defined precisely what we mean by a tangent to the graph of y = f(x) at a point (x, y), and we have an analytic criterion for deciding whether or not a curve has a tangent at a given point (x, y).
Monotone Functions
N evertheless, the visual interpretation of the derivative as the slope of the tangent to the curve is a h ighly usefu l aid to understanding, even i n purely analytic discussions. A case in q uestion is the following statement based on geometric intuition : Thefunctionf(x) is monotonically increasinK whenf'(x) > 0 and mono tonica1ly decreasing when f'(x) < 0. y
y
f(x)
--(�)t-------� x Figure
f(x)
--0�------�� x
2.23 Tangents to graphs of increasing and decreasing functions.
I ndeed, if .f'(x) is positive and the curve is traversed in the direction of i ncreasing x, then the tangent slants upwards, that is, toward increasing y ( ry_ is an "acute angle") ; therefore at the point i n question the curve 1
e<
The angle e< is not determined quite uniquely but can be replaced by ± 2.,, etc., unless we specify as above that 0 � e< < "·
e<
± "•
1 62 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
rises as x increases ; if, on the other hand, f'(x) is negative, the tangent slants downwards (oc is an "obtuse angle") and the curve falls as x increases (see Fig. 2. 23). Analytically this will be proved on p. 1 77. b. The Derivative as a Velocity
The need to replace the intuitive concept of velocity or speed by a precise definition leads once again to exactly the same l i m iti n g process we have already called differentiation. Consider the example of a poi n t moving on a straight line, the directed y-axis, the position of the point being determined by a single coordinate y. This coordinate y is the d istance, with its proper sign, of our moving po i n t from a fixed initial poi n t on the line. The m otion i s given if w e k n ow y as a fu n ct ion of t h e time t : y = f(t). I f t h i s function i s a linear function f(t) = ct + b, we speak of a uniform motion with the velocity c, and for every p ai r of disti n ct values t and 1 1 we can obtain the velo ci ty by di v i d in g the distance traversed in a time i nterval by the length of that time i n terv a l : c =
f( tl) 11
- j (l) -
t
.
The velocity is therefo re the difference quotient o f th e func t i on ct + b, and this difference q u ot ie n t i s indepen dent of the p a rti cu l a r p a ir of instants which we fi x upon . But what are we to understand by the ve l ocity of motion at an i nstant t if the motion is no longer uniform ? To answer t h i s q u e stion we consider the difference quotient [/(11) - f(t)]/(tl - 1),
which we shall call the average velocity in the time interval between t1 and 1 . Now i f th i s a ver age v el oc i ty tends to a defini te l imi t when we let 11 tend to t, we shall define th i s l i m i t as t he v e l o ci ty at the time t. ln o the r words : the velocity, that is, th e i nstantaneous rate of change of distance w i th respect to time at the time t, is the derivative
f'(t)
=
li m
,,�,
f (tl) - f( t) .
11 - t
Newton emphasized the i nte rp reta t ion of derivatives1 as velocity, and wrote y or j(x) instead off ' ( t ) , a notation which we shall occasionally use. Again, the d i fferenti a bility of the function is a necessary assump tion i f the notion of velocity i s to have a m e an i n g . 1
Called by him "fluxions. "
Sec. 2.8
The Derivative
1 63
A simple example is the motion of freely fal ling bodies. We start from the 'experi mentally established law that the d i stance traversed i n t i m e t b y a freely fall i n g body starti ng fro m rest a t t = 0 i s proporti onal t o t2 ; it i s therefore represen ted by a function of the form Y = f( t ) = atz
with constant a. A s on p . 1 59, the velocity then is given by the expression f'(t) = 2at ; thus : the l'elocity of a fi"eely falling body increases in proportion to the time. c.
Examples of Differentiation
We now i l l u strate the tech n i q ue of d i fferentiation by a number of typical examples. Linear Functions
For the function y = .f(.r) = c with constant c we see for all x , f( x + h) - f(x) = c - c = 0, so that l i m [j(x + h) - f(:r)]/h = 0 ;
h--.. (1
that i s , t he derimtiz·e of a constant function is zero . For a l i near fu nction y = f(x) = ex + b, we find
j( x + h ) - f(:r ) = l i m ch = h h�o h The derimtil'e of a linear fun c t io n is constant. h�o
f'( .r) = lim
c.
Po wers of x Next, we d i fferenti ate the power function
-
j( x)
" x ,
i s a positive i n teger. Provided x 1 ¥ x, we have " 1 x" = 1 - x = x; - + x �-zx + . . . + x • -1 x1 - x
at f1rst assuming that
j( x1 )
y = f( x) =
IX
x 1 - :r where we divide directly or use the formula for the sum of a geometric progressio n . Th is si mple algebraic mani pulat i on i s the key to the passage to the l i m i t ; for the last expression on the right-hand side of the equation i s a cont i n uous function of x h i n particular for x1 = x , a n d so we can carry out the passage to the l i m i t x1 --->- x for t h i s expres sion simply by replacing x1 everywhere by x. Each term then takes the value x"-1, and si nce the n umber of terms is exactly IX , we obtain d(x" ) y ' = f '(x) = -- = IXX"-1. dx
1 64 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
We arrive at the same result if oc is a negative integer - (3 ; we must, however, assume that x is not zero. We then find f(x1 ) - f( x ) X1
- X
=
x/
xP
�=
-
xP - x/ 1 . xPx/ X - x1
= - xP-l + xP-•xl
+ . . . + x:-l
x/xP
Once again we can carry out the passage to the limit simply by sub s ti tuti ng x for x1• Then just as before we obtain for the limit
' y =
-
: = - (3x-P-1.
f3 xP ' X
Hence for negative i ntegral values oc = given by the formula y' = cx:xa:-t.
-
fJ the derivative Is again
F i nally , we shall prove the same formula where x is positive and oc any rational number. We suppose that oc = pfq, where p and q are both i ntege rs and, moreover, positive. (If one of them were negative, no essential ch anges in the proof would be needed ; for oc = 0 t h e result is already known, since x" is then constant.) We now h ave f (xl) - f(x)
If we now put x'/q
= ; and xi ' •
xfl• .- x";"
= ; , , we obtain
+ f (x,) - f ( x) - e = ; r- ' + ;r- •; . . . + ;v-' = ;,• • ;, - ;• ;�-1 + ;�-2, + . . . + ;q- 1 X - X I
After thi s last transformation we can i mmediately perform the passage to th e limit x1 -+ x (or what amounts to th e sam e t h i n g , ; , -+ t), and thus obtai n for th e lim i ti n g value the expression y
or finally,
' - = E. ; •-• = qE. ; q- 1 q tv- 1
j'(x)
= !!.q x (v-olfq = !!.q x ( v/oi-I ,
= y' = oc x•- ' ,
which is formally the same result as before. We leave it for the reader to prove for himself that the same differentiation formula h olds also for negative rational exponents .
Sec. 2.8
The Derivative 1 65
We shall come back (p. 1 86) to the differentiation of powers and prove the· general validity of the preceding formula for arbitrary exponents ex.
Trigonometric Functions As a last example we consider the differentiation of the trigonometric x and cos x. We use the elementary trigonometric addition formula to transform the difference quotient
functions sin
x
sin (x + h) - sin h
sin x cos h + cos x sin h - sin h . cos h - 1 sin h Sin X + COS X h h
=
Recalling the relations of Section 1 . 8, pp. 84-85, . II m 11 � 0
sin h h
-- =
we i m medi ately obtai n
1
. I lm
,
11 � 0
, d(si n :r.) '1/ = dx
---
·
The function !I = cos x can Starting with
be
cos h - 1 h
=
x
-- •
0
'
= COS X .
differentiated i n exactly the same way.
cos ( .r. + h ) - cos x cos h - 1 . s in h = cos x - sm x h h h
--
and tak ing the limit as h y
,
---+
0, we obtain the derivative 1
d(cos x) dx
= --- =
.
- sm x.
d. Some Fundamental Rules for Differentiation
J ust as in the case of the integral, there exist certain basic rules for d ifferentiation that follow immediately from the definition and suffice for forming the derivative for many functions.
1 . If rf>(x)
2 . If 1p(x)
=
f(x) + g(x), then
=
cf(x) (where
c
' = f'( x) + g (x ) . is a constant), then 1p'(x)
r/>'( x)
=
cf'(x).
' If x is interpreted as an angle, then t hese simple formulas for the derivatives of sin x and cos x presupposes, of course, that the angle x is measured in radians.
1 66
The Fundamental Ideas of the Integral and Differential Calculus
Ch.
2
We have cp( x + h ) - cf>( x )
h
and
=
f( x + h ) - f( x ) g( x + h) + h h
1p( x + h) - 1p( x ) h
=
c
- g( x)
f( xh)'----"f(x ::... ..:... +'-'-) h
and our statements follow directly by passage to the limit.
Thus, for example, the derivative of the function cf>(x) = f(x) + + b (where a and b are constants) is given by the equation
ax
cf>'(x)
=
f'(x) + a.
With the help of these rules and of the formula for the derivative of a power we can i m mediately differentiate any polynomial 11 = a0xn + a 1x "- 1 + · · · + a, and find
y'
=
_x
t7a0X" 1 + (n - l )a1x"- 2 + · · · + 2 a , 2 + a ,_1.
e. Differentiability and Continuity of Functions
It is useful to know that differentiability i s a stronger condition than continuity :
rt a function is differentiable it is automatically continuous.
For if the difference q uotient [J(x + h) - f(x)]/h approaches a definite l i m i t as h tends to zero, the n umerator of the fraction, that is, f(x + h) - f(x) must' tend to zero with h; this j ust expresses the continuity of the function f(x) at the point x. Hence, separate cumber some continuity proofs are unnecessary for functi ons that can be shown to be differentiable (that is, for most functions we shall encounter) . Discominuities of the Derh•a th•e- Comers
The converse, however, is false ; it is not true that every conti n uous function has a derivative at every point . The si mplest counter-example is the function f( x) = l x l , that i s , J(x) = - x for x � 0 and j(x) = x for x � 0 ; i t s graph i s shown i n Fig. 2.24. At the point x = 0 this function is contin uous, but has no deri vative. The limit of [J(x + h ) - f(x)]/h is equal to I if h tends to zero through positive
1 Since then
lim [j(x + h)
h-+0
- [(x)] =
l,-
lim
h---.. 0
((x + h) - ·((x) h
J
(lim h)
h -0
=
['(x) · 0 = 0.
The Derivative 1 67
Sec. 2.8
values, and is equal to - I if h ten d s to zero through negative values ; if we do not restrict the sign of h, no limit exists. We say that our function has di fferent forward and backward derizoatit•es at the point x = 0, where by forward derivative and backward d e ri vati v e we mean respectively the limiting values of [j(x + h) - f(:r)]/h as h appr oache s zero through positive values only and negative values only. The differentiability of a function defined in an interval about the point y
f(x) = x
f(x) = - x
Figure 2.24
f(x)
=
[ .r [ .
con si d ered t h u s r eq u i res not merely that the forward and b ack w ard derivatives exist, but that t h ey are eq ual . Geometrically the i nequality of th e two derivati ves means that the c urve h as a corner. Differenti abi l i ty expresses i n a precise way what i n tu itively would b e called smoothness of the grap h of the function. Infinite Discontinuities
As fu rther examples of p o i nts where a continuous function is not d iffere n ti a b le we consider the p o i n ts where the derivative becomes i n fi n i te, that i s , the poi nts at which there exists neither a for w a rd nor a backward derivative, the difference q uotient [j(x + h) - f(x)]/h inc re as ing beyond all bounds as h --+ 0. For example, the function y = j(x) = �;: = x� " i s defined and continuous for all values of x. For all nonzero v a lue s of x its derivative is given (p. 1 64) by the formula ' y = ?,:c ' i . A t the p oin t x = 0 we h ave [j(x + h) - f(x)]/h = h'"'fh = lr ' a, and we see at once that as h --+ 0 the expression has no limiting val ue, but, on the contrary, tends to infinity. This state of affairs is often briefly described by saying that the function possesses an infinite derivati ve, or the derivative infinity, at the point in question ; as we should remember, however, this merely means that as h tends to zero the difference quotient i ncreases beyond all bounds, and that the
1 68 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
y
Y "'
x l>
Figure 2.25
derivative in the sense in which we have defined it really does not exist. The geometrical meaning of an infinite derivative is that the tangent to the curve is vertical (cf. Fig. 2.25). The function y = f(x) = _;;, which is defined and continuous for x � 0, is also not differentiable at the point x = 0. Since y is not defi n ed for negative values of x, we here consider the right-hand derivative only. The equation [j(h) - f(O)]/h = l j-/h shows that this deri vative is infinite ; the curve touches the y-axis at the origin (Fig. 2.26). Finally, in the function y = {/-;;:2 = x% we have a case in which the r igh t-han d derivative at the point x = 0 is positive and infinite, whereas the left-hand derivative is negative and i nfinite, as fol lows from the relation f(h) - f(O) 1 = h t/h .
As a matter of fact, the continuous curve y = x%, the so-called semi cubical p arabola or Neil's parabola, has at the origi n a cusp with a tangent perpen d ic u l ar to the x-axis (cf. Fig. 2.27). y
y "' " "
Figure 2.26
Sec. 2.8
The Derivative
y
1 69
y = x "i
--------�0�--� x Figure
2.27
f. Higher Derivatives and Their Significance
The graph of the derivative f'(x) of a function is called the derived curve of the graph of f(x). For example, the derived curve of the parabola y = x2 is a straight line, represented by the function y = 2x.
The derived curve of the sine curve y = sin x is the cosine curve y = cos x ; similarly, the derived curve of the curve y = cos x is the curve y = - si n x. (These latter curves can be obtained from each other by translation i n the direction of the x-axis, as is shown in Fig. 2.28.) It i s quite natural to form the derived curves of the derived curves, that is, to form the derivative of the function f'(x) = cp(x). This derivative f'(x + h) - f'(x) , cp'(x) = lim h� o
h
provided that it exists, is called the second deriva t iv e of the function
f(x) ; we shall denote it by f"(x).
Similarly, we may attempt to form the derivative of f (x) , the so called third derivative off(x), which we then denote by f"'(x). For most functions that concern us there is nothing to h inder us from repeating the process of differentiation as many times as we like, thus defining an nth derivative f (nl(x) . 1 Occasionally, it will be convenient to call the function f(x) its own Oth derivative. If the i ndependent variable is i nterpreted as the time t and the motion of a point i s represented as previously by the function f(t), the physical meaning of the second derivative is the rate of change of the velocity f'(t) with respect to time, or, as it is usually called, the acceleration. In the example of the freely falling body the distance traveled in the time t was given by the function y = f(t) = at 2 • We found f'(t) = 2at for the velocity at the time t. The acceleration has then the constant "
The terms second, third, . . . , nth differential coefficient are also used , or D'f, . . . , D"f (cf. footnote 3, p. 1 58).
1
170 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
value f"(t) = 2a (which is usually identified with the gravitational constant g). Later (p. 236), we shall discuss the geometrical interpre tation of the second derivative in detail. Here, however, we take note of the following facts : At a point where j"(x) is positive, f'(x) i n crease s y
fix) • sin
f' (x) = cos :x
:x
y f (x) = cos x
'
' '
0 '
f' (x) = - sin x
I
'
\
I
/
/
/
..... - ......
,
'
'
'
'
......
_ .....
Figure 2.28
/
Derived curves of sin
x
and cos x .
as x i ncreases ; i f here j'(x) is p o si ti ve , the curve becomes steeper for i n cre a s i ng x. If, on the oth er hand, j"(x) is negative, f'(x) decreases as x i ncreases, and if j'(x) is positive, the curve becomes less steep as x i ncreases. Finally, we observe that the higher derivatives may be used to define a function. Thus one can characterize the trigonometric func tions by a so-called differential equation involving the function and its second derivative. From the formulas (d cos x)fdx = - sin x, (d sin x)fd:�; = cos x we obtain immediately by differentiating ag ai n , d2 -2 cos x = - cos x, dx
d2 . d x2
.
- Sill X = - Sill X .
Sec. 2.8
The Derivative 1 7 1
Hence if the symbol u stands for either of the functions sin x or cos x, we have the relation (differential equation)
u"
=
- u.
This differential equation is also clearly satisfied by any linear com bination u = a cos x + b sin x with constant coefficients a, b. We shall see on p. 3 1 2 that s uch linear combinations, with arbitrary constants " a and b, are the only functions u for which u = - u. In all types of applications involving oscillations or wave phenomena, such as motions of springs or waves on the surface of water, we are led directly from physical considerations to a differential equation of the type u" = -u for the physically significant variable u (usually the independent variable is time). It is therefore important to recognize that u can be represented simply in terms of trigonometric functions (see Chapter 9). g. Derivative and Difference Quotient.
Leibnitt's Notation
In Leibnitz's notation the passage to the limit in the process of differentiation is symbolically expressed by replacing the symbol Ll by the symbol d, motivating Leibnitz's symbol for the derivative defined by the equation dy
dx
Ll y . Ax-oLlx
= li m
I f w e w i s h t o obtain a clear grasp o f t h e meaning o f t h e differential calculus, we must beware of the old fallacy of i magining the derivative as the quotient of two "quantities" dy and dx which are actually "infinitely small." The diffe rence quotient Ll yfllx has a meaning only for differences Llx which are not equal to zero. After forming this genuine di fference quotient we must perform the passage to the limit by means of a transformation or some other device which also in the limit avoids division by zero. It does not make sense to s uppose that first Llx and Lly go through something like a limiting process and reach values which are infinitesi mally small but still not zero, so that Llx and Lly are replaced by "infinitely small quantities" or "infinitesimals" dx and dy, and that the quotient of these quantities is then formed. Such a conception of the derivative is incompatible with mathematical clarity ; in fact, it is entirely meaningless. For many people it un doubtedly has a certain charm of mystery, always associated with the word "infinite" ; in the early days of the differential calculus even Leibnitz h imself was capable of combining these vague mystical ideas
172 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
with a thoroughly clear handling of the limiting process. But today the mysticism of infinitely small quantities has no place in the calculus. The notation of Leibnitz, however, is not merely suggestive in itself, but it is actually extremely flexible and useful. The reason is that in many calculations and formal transformations we can deal with the symbols dy and dx exactly as if they were ordinary numbers. By treating dx and dy like numbers we can give neater expression to many calculations which can admittedly be carried out without their use. In the following chapters we shall see this fact verified over and over again and shall find ourselves justified in making free and repeated use of it, provided we do not lose sight of the symbolical character of the signs dy and dx. " For the second and higher derivatives too, Leibnitz devised a sugges tive notation. He considered the second derivative as the limit of the "second difference q uotient" i n the following manner : T n addition to the variable x we consider x1 = x + h and x2 = x + 2h. We then take the second difference quotient, meaning the first difference quotient of the first difference quotient, that is, the expression
!(Y • -h Yt h
Yt
Y)
= _!_2 (y2
Zy1 + y) ' h where y = f(x), y1 = f(x1), and y2 = f(x2). Writing h = �x, Y2 - Y1 = �y1, and y1 - y = �y, we may appropriately call the expression in the last parentheses the difference of the difference of y or the second difference of y and write symbolically1 Y2 - 2y t + = �Yt - �y = � ( �y) = � 2Y· _
-
h
_
Y
In this symbolic notation the second difference q uotient is then written � 2yf( �x) 2 , where the denominator is really the square of �x, whereas in the numerator the superscript 2 symbolically denotes the repetition of the difference process. The second derivative is then expressed by
f"(x) = lim �·t 2 ( � x) •, - .
-- •
This symbolism for the difference quotient2 led Leibnitz to introduce
1 Here .:l.:l = il' is merely a symbol for "difference of difference" or "second difference." • As we must emphasize, the statement that the second derivative may be represented as the limit of the second difference quotient requires proof. We previously defined the second derivative, not in this way, but as the limit of the first difference quotient of the first derivative . The two definitions are equivalent, provided the second derivative is continuous ; the proof, however, will be given only later (see Chapter 5, Appendix I I since we have no pa"ticular need of the result.
Sec. 2.8
The Derivative 1 73
the notation y
"
=
f" (x)
=
�� ,
y
"'
=
f"'(x )
=
�� ,
etc . ,
for t h e second and higher derivatives, and w e shall find that this notation also stands the test of usefulness. 1 h. The Mean Value Theorem of Differential Calculus
The difference quotient involves the values of a function for distinct values of x, whereas the derivative at a point tells u s nothing about the function at any other point ; the difference quotient reflects properties of the function " in-the-large," while the derivative reflects a l ocal property or a property "i n-the-small . " W e shall often need t o derive over-all or "global" properties of a function from the local properties given by its derivative. For this p urpose we utilize a fundamental relation between the difference quotient and derivative known as "the mean value theorem of di fferential calculus." The mean value theorem is easily appreciated i ntuitively. We consider the difference quotient f( xl ) - f( x2) = !:!f x1 - x2 !:! x
of a function f(x), and assume that the derivative exists everywhere i n the closed interval x1 ::::;; x ::::;; x2 , so that the graph of the curve has a tangent everywhere. The difference quotient i s the tangent of the angle rx of i nclination of the secant, shown i n Fig. 2.29. I magine this secant shi fted parallel to itself. At least once it will reach a position in w hich it i s a tangent to the curve at a point between x1 and x2 , certainly at that point x = � of the curve which i s at the greatest distance from the secant say at x = � . Hence there exists an intermediate value � i n the i nterval such that f( x l ) - f (x2) = '(�) . f XI - X2
This statement i s cal led the mean value theorem of the differential calculus.2 We can also express it somewhat differently by noticing that 1 This is the customary notation. Writing y" = d'yf(dx)', y• = d'yf(dx)' with parentheses, would be somewhat clearer, but is not done ordinarily. 2 A more appropriate name would be the intermediate value theorem of differential calculus.
1 74 The Fundamental Ideas of the Integral and Differential Calculus
the number
� may be written in the form �
=
Ch. 2
x 1 + 8(x2 - x 1),
where all we know about 8 is that it lies between 0 and I . Although 8 (or �) generally cannot be specified more exactly, the theorem is extremely powerful in application. Consider, for example, the case where x is the time and y = f(x) the distance of a car from its starting point along a certain road. Then y
--O �----�X� t ------�----------� X2�------� x Figure 2.29
f'(x) is the velocity of the car at the time x. If, say, during the first two hours (�x = 2) the d river has covered a d istance �f = 1 20 miles, we can conclude from the mean val ue theorem that at least at one moment � during those two hours the driver had a speed of exactly 60 miles per hour (provided the velocity exists at every moment). The driver cannot clai m , for instance, to have traveled all the time at less than 50 miles per hour. On the other hand , there is nothing to indicate what the time � was at which the precise speed of 60 miles per hour was attained ; it might have been at some time during the first hour or during the second hour or on several occasions. A preci se statement of the mean value theorem is the following : Jff(x) is continuous in the closed interval x1 S x S x2 and differentiable at every point of the open interval x1 < x < x2, then there exists at least one value 0, where 0 < (} < I , such that
Sec. 2.8
The Derivative
1 75
If we replace x1 by x and x2 by x + h, we can express the mean value tR.eorem by the formula
�
f( x + h
-
f(x)
=
f'W
=
f'( x + Oh) ,
X < � < X + h.
Although i t i s essential that f(x) should be continuous for all points of the i nterval, including the end points, we need not assume that the derivative exists at the end points. lf at any point in the interior of the interval the derivative fails to exist, the mean value theorem is not necessarily true. It is easy to see this from the example off(x) lxl . =
i. Proof of the Theorem
The mean value theorem is usually derived by reduction to a special case which we establish first.
RoL LE's THEOREM. If a function rp(x) is continuous in the closed interval x 1 � x � x 2 and differentiable in the open interml x1 < x < x2 , 0, then there exists at least one and if in addition rp(x1) 0 and rp(x2) point � in the interior of the interval at which '(�) 0. =
=
=
Interpreted geometrically, this means that if a curve reaches the x-axis at two points, then it must have a horizontal tangent at some intermediate point (Fig. 2 . 30) . ci> (x )
Figure 2.30
1 76 The Fundamental Ideas of the Integral and Dijjiwntial Calculus
Ch. 2
I ndeed, since c/>(x) is continuous in the closed interval [xi> x2] there exists a greatest value M of c/>(x) and a smallest value m in that i nterval (see p. 1 0 1 ) . Since c/> vanishes in the end points, we must have m :::;; 0 :::; M. If these greatest and least values should be equal, then necessarily m = M = 0 and c/>(x) = 0 at all points of the i nterval ; then also c/>'(x) = 0 in the interval , and hence c/>'(�) = 0 for every � i n the i nterval. Thus we only have to consider the case where m and M are not both zero . I f, in particular, M is not zero, then M must be positive. There exists a point � of the i nterval [xi> x2] where c/>(�) = M. Since c/> vanishes i n the end points of the interval , the point � must be an i nterior point. Furthermore, c/>(x) :::;; c/>(�) = M for all x i n [x1, x2] . Consequently, for every number h whose absolute value l h l is small enough, the inequality c/>(� + h) - c/>W :::; 0 holds. This impl ies that the q uotient c/>(� + h ) - c/>(�) h
is negative or zero for h > 0 and positive or zero for h < 0. If we let h tend to zero th rough positive val ues, we find that c/>'(�) :::; 0, whereas for h tend ing to zero through negative values i t follows that c/>'(�) � 0. H ence c/>'(�) = 0 and we have proved Rolle's theorem in the case M # 0. The same argument holds for m # 0. To prove the mean value theorem we apply Rolle's theorem to a function which represents the vertical distance between the point (x, f(x)) of the graph and its secant :
( - ( c/>( x) = f(x ) - f (x, ) - f x. ) f x,) (x - x, ) . x2 - x l This fu nction' obviously satisfies the condition c/>(x1) = c/>(x2) = 0, and is of the form c/>(x) = .f(x) + ax + b with constant coefficients a = [j(x2) - .f(x1)]/(x2 - x1) and b. From p. 1 66 we know that -
c/>'(x) = f'(:r) +
a,
and thus by Rolle's theorem
0 = c/>'(�) = .f'W + a
This function also is p roportional to the d istance of the point (x, _f(x)) of t he curve from the secan t ; the reader can easily verify t h i s for himself, for example, by using the fact from elementary analyt ic a l geometry that the expression (.'/ - m.r - b)/ V l + m' rep re se n ts the (signed) distance of t he point (x, y) fro m the l i ne with the equation y - mx - b = 0. In this way w e find that i ndeed at the points of the curve having greatest d istance from the secant the tangent is parallel to th e sec a n t . 1
Sec. 2.8
The Derivative 1 77
for a suitably chosen intermediate value � ; hence f'( �)
= - a = f ( x2)
- f ( x l)
x 2 - xl
;
thus the mean value theorem is proved . Significance of the Theorem
The derivative of a function had been defined as the l i m i t of difference q uotients for an i n terval as the end points approach each other. The mean value theorem establishes a connection between d i fference quotients and derivatives of a d i fferentiable fu nction which does not i n volve the shrinking of the interval. Each difference quotient i s eq ual to the derivative at a suitable i n termed iate point �Examples. Just as in the mean value theorem of integral calcu lus there i s noth ing speci fic asserted i n the intermed iate val ue theorem about the location of � beyond the fact that � lies in the i nterior of the interval . For the example of the quad ratic fu nction y f(x) = x 2 with d erivative f'(x) = 2x we find =
f ( x2) - f ( x l ) x 2 - xl
= xi +
:r 2
= f (�) , '
where � = (x1 + x2) / 2 i s the m idpoint of the i nterval [x1, x 2 ] . I n genera l , h owever, � m i ght lie anywhere else between x 1 and x 2 • For example, i f f(x) = x3, we h ave [j( l ) - f( 0) ]/( 1 - 0) = I = f'(� ) = 3 � 2 , w here � = 1 / .Ji Monotonic Functions. As one of many appl ications of the mean value theorem of d i fferential calculus we prove that i f the derivative of f(x) has a constant sign, then f is monotonic. Specifically, we assu me f(x) to be continuous in the closed interval [a, b] and differentiable at each point of the open interval ( a, h). If then f'(x) > 0 for x in (a, h), then the function f(x ) is monotonic increasing; similarly, if f'(x) < 0, the function is monotonic decreasing. The proof i s obvious : Let x1 and x 2 be any two values in the closed i n terval [a, b]. Then there exi sts a � between x1 and x2, and hence also between a and b, such that f (x2) - f (x1)
= f'( �)(x2 - xl).
I f .f'(x) > 0 everywhere i n (a, b ) w e h ave i n particular f'W > 0 . H ence f(x 2) - f(x1) is positive for x 2 > x1 ; that i s , f( x) is i ncreasing. S i m i larly , / i s decreasing i f f'(x) < 0 i n ( a , h) . ln the same way we show that a function f(x) continuous in [a, b]
1 78 The Fundamental Ideas of the Integral and Differential Calculus
and differentiable in the open interval (a, b) must be a constant iff '(x) everywhere in (a, b) . For then
f ( x 2) - f ( x , )
= f'($) (x2
- x1)
=
Ch. 2 =
0
0.
This important statement corresponds to the intuitively obvious fact that a curve whose tangent at every point is parallel to the x-axis must be a straight line which is parallel to the x-axis. Lipschitz- Continuity of Differentiable Functions. I t was mentioned earlier that a function f(x) having a derivative is necessarily continuous. The mean value theorem of differential calculus furnishes much more precise quantitative information, namely, a modulus of continuity. We consider a function f(x) which is defined in the closed i nterval [a, b] and has a derivative .f' (x) at each point of that interval. Assume that f ' (x) is bounded in the interval (this is certainly the case provided f'(x) is defined and continuous in the closed interval [a, b] ) ; there exists then a number M such that /f'(x) / � M. For any two values X� o x 2 in (a, b) we infer from the mean value theorem /f(x2) - f(x,) / = / .f '($)(x2 - x1 ) / � M /x2 - x1 / .
For given • > 0 we have thus produced a simple modulus of con tinuity o = •I M such that / f(x2) - .f(x1) / �
•
for / x2 - x, / � b.
Take, for example, the functionf(x) = x2 in the interval -a � x � +a. Since /f'(x) / = / 2x / � 2a we see that here for / x2 - xd � •/2a. We said that a function f(x) "satisfies a Lipschitz-condition" or i s "Lipschitz-continuous" if there i s a constant M such that / f(x2) - f( x1) / � M / x2 - x1 /
for all x" x2 in question. This means that all difference quotients f(x2)
- f(x,)
have the same upper bound M for their absolute value. We see that every function f with continuous derivative f' on a closed interval is Lipschitz-continuous. H owever, even functions that do not have a derivative at every point can be Lipschitz-continuous, as the example
Sec. 2.8
The Derivative 1 19
f(x) = J xl shows. The reader can verify for hi mself that for this functio,n always l f( x2) - f(x1 ) l � J x 2 - X 1 l .
O n the other hand, not every contin u ous function i s Lipschitz continuous. This is shown by the example of f(x) = x�i ; here j(x) - f(O)
x-0
=
x -%
is not bounded for small x ; hence f(x) is not Lipschitz-continuous at = 0. This is consistent with the fact that the derivative f'(x) = 1 f3x'A does not remain bounded as x tends to zero. The functions which are Lipschitz-continuous form an i m portant class intermed iate between those that are merely continuous and t hose that have a continuous derivative.
x
j. The Approximation of Functions by Linear Functions. Definition of Differentials
Definition .
T he
j ' (x)
where ax • by
= h. E(IJ )
derivative of a function =
lim
h�O
y
= j(x)
f (x + h ) - f ( x) = l i m h
was defined by
/). y ,
&�� o a x
I f for a fixed x and a v a r iabl e h, w e define a quantity
= j( x + h ) - j ( x) h
_
f '(x)
= aa yx
_
j'(x),
then the fact that f ' ( x) is the derivative off at the point x amounts to the equation lim <( h ) = 0. h-0
The q uantity a y = f(x + h) - f(x) represents t h e change or increment in the value of the dependent variable y that results when the value x of the i ndependent variable is changed by the amount /).x = h. Since /).y
= f'(x) ax +
•
/).x,
the q uantity /).y appears as the sum of two parts, namely, a part f'(.r) /).x which is proporti onal to /).x and a part • ax which can be made as small as we please compared to /).x by making /).x itself small enough . The dominant, l i near p art in the expression for ay we shall call the diffe rential dy of y and write for it
dy
=
df(x)
= f'(x)
/).x,
1 80 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
For any differentiable function f and for a fixed x this differential is a well-defined linear function of = t;x. For example, for the function y = x2 we have dy = d(x2) = 2x t;x = For the particular function y = x whose derivative has the constant value one, we simply have dx = t;x. It is then consistent with our definition to write dx for t;x when x is the i ndependent variable ; hence the differential of any function y = f(x) can also be written as
h
dy = dj(x)
2xh.
= f '(x)
dx.
The increment of the dependent variable
t;y
= f'(x)
dx + • dx = dy + • dx
differs from the differential dy by the amount • dx, which in general is not zero. In the example of the function y = x2 we have dy = 2x dx, whereas
t;y = (x + dx)2 - x2 = 2x dx + (dx)2 = dy + • dx, where • = dx. Earlier we used the symbol dyfdx purely symbolically to denote the limit of the quotient t;yjt;x for t;x tending to zero. With our present definition of the differentials dy and dx the derivative dyfdx can actually be considered as the ordinary quotient of dy and dx. Here, however, dy and dx are now not in any sense "infinitely small" quantities or "infinitesimals ;" such an interpretation would be devoid of meaning. I n stead d.'IJ and dx are well-defined l i near functions of h = t;x which for large t;x may have large numerical values. There is nothing remarkable in the fact that the quotient dyfdx of those quantities has the same value as the derivativej'(x). This is merely a tautology restating the definition of dy as f (x) dx.l Rewriting the relation between i ncrement and differential of f i n the form '
f(x +
h) = f(x)
+
hf'(x) + •h ,
we see that the expression f(x + h) considered as a function of h i s represented b y t h e linear function f(x) + hf'(x) wth an error • h which is arbitrarily small compared to h i f h is sufficiently small. This approxi mate representation of f(x + by the linear function f(x) + hf'(x) means geometrically that we replace the curve by its tangent at the point x (see Fig. 2.3 1).
h)
d'y
= ['(.r)(dx)', etc . , in agreement with Leibnitz' notation for the hi gher derivatives.
1 Similarly, hi g her-order differentials could b e defined b y d'y
= .f"(x)h' =.f"(x)(dx)',
The Derivative 1 8 1
Sec. 2.8 y
Figure
2.31 Increment t!.y and differential dy.
Linear Approximation A more precise estimate of the magnitude of the "error," that is, of the deviation of the functionf( x) from the l inear function representing the tangent, is given by the mean value theorem of differential calculus. We h ave for a suitable ; between x and x + h f(x + h ) - f( x) = hf'(;),
so that E
f(x) f'(x ) h = f(x + ) = / ' ( ;) - f '(x). -
h
If, as usually i n applications, the function f'(x) itself has a derivative f "(x) , we find by applying the mean value theorem a second time that f' W - f' (x) = ( � - x) J" ( rJ ) .
where 17 is a value intermediate between x and � and hence also between x and x + h. It follows that
1 €1 = I (; - x)J "( tJ) I
=
1; - x l l /" ( tJ) I � h M ,
where M i s any upper bound for the absolute value of the second derivative off in the interval [x, x + h]. Then I Eh l , which measures the deviation of f(x + h) from the l i near function f(x) + hf'(x), is at most Mh2• For sufficiently small h the expression Mh2 is, of course, much smaller than f'(x)h, u nless f'(x) h appens to have the value zero. Thi s approximation of a function in a small interval by a linear function
Ch. 2
1 82 The Fundamental Ideas of the Integral and Differential Calculus
is of greatest significance both for practical applications and for advanced mathematical analysis. We shall return to this topic in later chapters, and i ncidentally derive then the better estimate j £h l s !Mh2• Interpolation
* When a function f(x) i s described n u m erically by a table of values, J i s ordinarily determined by linear interpolation for arguments x intermediate between those for which f is listed. This procedure also corresponds to replacing the function f by a linear function in an interval. In this case the graph of the linear function is given by a y
--0 �-------
Linear i n terpolation.
r a t her th an by a tangent to the curve representing f. I f, say, the values off are known at two points a and b, we replace J(x) for inter mediate x by the expression
secant
- a cf>(x) = f ( a ) + ( x - a/( b ) f ( ) , b - a
which is linear in x and gives the correct values off at the end points x = a and x = b of the i nterval (see Fig. 2. 32). Here again by use of the mean value theorem we can estimate the error in this approxi mation . We have f ( x)
[
_ cf>(x ) = ( x _ a ) J ( x) - f ( a ) _ f( b ) - f(a ) b - a x - a = ( x - a )[/'( � 1) - / ( � 2) ] '
.
J
Since �. lies between a and x , it also lies between a and b, as does � 2 • A second application of t h e mean value theorem of differential
Sec. 2.8
The Derivative 1 83
calculus then yields
f(x) - rf>(x) = (x - a)j"(ry)( ; 1
;2),
where 7J is between ; 1 and ; 2 and hence also between a and b. Con sequently, denoting by M an upper bound for 1 /"1 in the interval [a, b], we find that -
lf(x) - rf>(x) l ::::;; l x - a l l ; 1 - ; 2 1 1/"( tJ)I ::::;; M(b - a)2• Once again the deviation of f from its li near approximation can be estimated by the square of the length of the i nterval. As a numerical example we take from a table of trigonometric fu nctions in rad ian measure the val ues
sin 0. 75 = 0.68 I 6,
sin 0.76 = 0. 6889,
where the errors do not exceed 0.00005. lf we want to deduce the val ue of the sine function for the i ntermediate argument 0.754, we find by linear i nterpolation that sin 0. 754
�
0.68 1 6 + ,''0(0.6889 - 0.68 1 6)
�
0.6845.
For the function f(x) = sin x the first derivative i s f'(x) = cos x, the second derivative f"(x) = sin x. Obvi ously, IJ"(x) l ::::;; I , so that the error in the value found for sin 0.754 as a result of the l i near in ter polation proced ure does not exceed 1 x (0. 0 1 )2 = 0.000 1 . To this error estimate we must add possi ble errors due to round-off in the tabulated val ues and i n the i n terpolation . We can compare thi s val ue obtained by l i near i nterpolation with the value we would obtain by repl acing the sine curve by i ts tangent at the point x = 0 . 7 5 . Taking f'(0.75) = cos 0.75 = 0.73 1 7 from the table, we find -
sin 0 . 754
�
/(0. 75) + /'(0.75)(0.004) = sin 0.75 + 0.004 cos 0.75 �
0.6845.
I ncidentally, the true value of sin 0.754 correct to six si gnificant digits is 0. 684560. k. Remarks on Applications to the Natural Sciences
I n applying mathematics to natural phenomena we never deal with precise�)' knoll'n quantities. Whether a length is exactly a meter is a
question which can not be decided by any experiment and which con sequently has no physical mean ing. M oreover, there is no immediate physical meaning i n saying that the length of a material rod is rational or i rrational ; we can always measure i t with any desired degree of
1 84 The Fundamental Ideas of the Integral and Dif!erential Calculus
Ch. 2
accuracy by rational numbers, and the only meaningful question is whether we can manage to perform such a measurement u s ing rational num bers with relatively small denominators. Just as the question of rationality or irra ti onali ty in the rigorous sense of "exact mathematics" has no physical meaning, carrying out limiting processes i n applications is usually not more than a mathematical idealization. The p ract i cal-an d o v e r w he l mi n g-sign i ficance of such i dea l izatio n s lies in the fact that through the i deal i z a ti o n s analytical expressions become e ssen t ia l l y simpler and more manageable. For example, i t is vastly simpler and more convenient to work with the notion of in st an t a neou s velocity, which i s a fu n ctio n of o n ly one d efi n it e instant of time, than with the notion of average velocity between two different i n stants. Without s u ch i dealizat ion every sc i e n tific i nv e s t ig a t i o n of nature would be condemned to h o pel e s s c o m p l ica t i on s a n d wo uld b og down at th e o utse t . We d o n o t i n ten d t o e n t er i nt o a p h i l o s o p h ica l discussion of the re lat i o n s hi p of mathematics to rea l i ty . For t he sake of better u n d er s t a n d i n g of t h e t he ory , it should be emphasized t h a t i n a p p l i ca ti o ns we h ave the r i gh t t o r ep l a ce a d e r i v a t i v e by a difference quotient and vice versa, pro v i d e d o n ly that the d i ffer e n ce s a re s m a l l e n o ug h to guarantee a suffic i en t ly close ap p r o ximat i o n . T h e phys i c ist , the biologist, the e ng i n ee r , or anyone else who has to deal with these ideas i n practice will therefore h a v e t he r ight to i d e n t i fy the difference q u o t i e n t with the derivative within his limits of accuracy. The smaller the increment h = dx of the i ndependent variable, the more accurately can he re p resent the increment !:J.y = .f(x + h) - .f(x) by the diffe renti al dy = h.f'(x) . As long a s he keeps k n o w i ng l y withi n t h e l i m its of accu racy re qu i red by the p r o b l e m , he mi gh t even be p e rmi tt ed to s pe ak of the q u an ti t i e s dx = h and dy = h.f'(x) as "infinitesi mals." These " phy s i cally i n fi n i tesim al" q ua n t i t ie s have a precise me a n i ng . They are v aria bl e s with values which ar e fi n i te , un e q ua l to zero, and c h o s en small enough for the given investigation, for e xa m pl e , smaller than a fractional part of a wa v elengt h or s m a l l e r than the d i s t an ce between two electrons in an atom ; i n gen e r al , smaller than t h e d e g ree of accuracy req u i red.
2.9 The Integral, the Primitive Function, and the Fundamental Theorems of the Calculus
a. The Derivative of the Integral
As already stated , the connection between integration and differen tiation is the cornerstone of the differential and i n t e gra l ca lc u l u s .
Sec. 2.9
Fundamental Theorems of the Calculus 1 85
We recall from Section 2 .4 th at an indefinite i ntegral of a con tinuous function f(x) is defined as a function cf>(.r) of the upper end point of i ntegration by the formula cf>(x) = rf ( u) du,
where
oc
was any point i n the domain o f f We s h al l now prove
FUNDAMENTA L TH EOR EM OF C A L C U LU S ( P a rt One ) . The indefinite integral cf>(x) of a continuous function f(x) always possesses a derivative cf>'(x), and moreorer c/>'(x) = f(x) .
That i s , differentiation of t h e indefinite in teg r al of a continuous function a lway s re p r oduce s the in tegrand dx
d
f� a
f(u) du = f(x) .
This inrerse character of the operations of d!fferentiation and integration is the b a si c fact of calculus. The proof is an immediate consequence of the m ea n value theorem of in tegra l cal c u lus . Accor ding to that theorem we h ave for any values x and x + h of the domain off cf>(x + h) - c/>( x ) =
rx f h x f(u ) d u J
= hf( �),
where � is so m e value in the interval with end points x and x + h. F o r h ten din g to zero the val ue � m u s t tend to x so that lim
h�O
cf>(x + h ) - cf>( x ) = l i m f ( $ ) = f(x),
h�O
h
since f i s continuous. H ence cf>'(x) = f(x) a s stated b y t h e theore m . Applications. (a) We c a n u s e the theorem t o find derivatives for some of the functions introduced earlier. The natural l ogarithm was defined for x > 0 by the i ndefinite in tegral log x = I t follows i m medi ately that
d l og
x
---
dx
ix
- du . 1
1 u
=
-
x
1 86
The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
(b) More general logarithms to an arbitrary base a were expressible in the form log x log " x = -log a ·
Applying the rule for the derivative of the product of a constant and of a function we find that
d 1 - log x = --- . dx a X log a
(c) We found that
..!!__ x• = IXX"-1 dx
i n t h e case where the exponent IX i s an integer or more generally a rational number. We can now extend this formula to arbitrary IX . For that pu rp ose we recall the i ntegration fo r mula
1 (bP 1 1 - a P+l ) , iba u P du = -+1 ,3
which we had proved for any positive numbers a , b, and any p � - 1 . If we replace here the upper limit b by the variable x and differentiate both sides with respect to x, it follows that for x > 0 I ( P+1 xP = d -x dx ,3 + 1 _
_
aP+l).
Us i ng the rules for the d eri va t i ve of a sum and of a constant times a function, we can write this result i n the form
S ubstituting
IX
for ,3
x P = _1_ ..!!__ x P+l. ,3 + 1 dx
+ I , we obtain the formula ..!!__ x• = IXX"- 1 dx
for any ,3 � - I , t h a t is, for IX � 0. However, the formula also holds trivially for IX = 0 since then x� = I and the derivative of a constant is zero. b. The Primitive Function and Its Relation to the Integral
Inverting Differen tiation
The fundamental theorem shows that the ind efin i te in te gral cf>(x), tha t is, the inte gr a l with a ·1ariable upper limit x, of a fun ct i o n f(x),
Sec. 2.9
Fundamental Theorems of the Calculus 1 87
is a solution of the following problem : Given f(x) , determine a function F(x) such that F'(x) = f(x).
This problem requires us to reverse the process of diffe ren t iation. It is typical of t h e i nverse problems that occur i n many parts of mathe matics and that we have already found to be a fruitful mathematical method for generating new concepts. (For example, the first extension of the idea of natural numbers is suggested by the desire to invert certain elementary processes of arithmetic. Again new kinds of func tions were obtained from the inverses of known functions.) Any function F(x) such that F'(x) = f(x) is called a primit ive function off(x) or simply a primitive off(x) ; thi s terminology suggests that the function f(x) is derived from F(x). This problem of the inversion of differentiation or of the finding of a primitive function at fi rst sight is of quite different character from the problem of i ntegration . The first part of the fundamental theorem asserts, however : Et•ery indefinite integral rp(x) of the function f(x) is a primitive off(x). Yet thi s result does not completely solve the problem of finding the primitive functions. For we do not yet know if we have found all the solutions of the problem . The question about the set of all primitive functions is answered by the foll owing theorem, sometimes referred to as the se c on d part of the fundamental theorem of the differential and i ntegral calculus The difference of two pr imit iv e functions F1(x) and F2(x) of the same function f(x) is always a constant Thus from any one prim itive function F(x) we can obtain all the others in the form F(x) + c by suitable choice of the constant c. Con versely , for every value of the constant c the expression F1 (x) = F(x) + c represents a p rimit ive func tion off(x).
I t is clear that for any value of the constant c the function F(x) + c i s a pri mitive function, provided that F(x) i tself is. For we have (cf. p. 1 66)
!!._ [F(x) + c )
dx
=
3._ F(x) + 3._ c dx
dx
= F'(x) = f(x).
1 88 The Fundamental Ideas of the Integrol and Differential Calculus
Ch. 2
Thus to complete the proof of our theorem it remains only to show that the difference of two primitive functions F1(x) and F2(x) is always constant. For this purpose we consider the difference £ix) - F2(x)
Clearly,
G'(x)
=
G(x).
F/(x) - F2 ' (x) = f(x) - f(x) = 0. However we had proved on p. 1 78 from the mean value theorem of differential calculus that a function whose derivative vanishes every where in an interval is a constant. Hence G(x) is a constant c, and the theorem follows. Combining the two parts just proved we can now formulate the =
FUNDAMENTAL THEOREM OF CALCULUS .
Every primitive function
F(x) of a given function f(x) continuous on an interval can be repre
sented in the form
F(x)
=
c + cp(x)
=c+
r
f (u) d u,
where c and a are constants, and conversely, for any constant values of a and c chosen arbitrarily1 this expression always represents a primitive function. Notations It m ay be surmised that the constant c can as a rule be omitted because by changing the lower limit a we change the primitive function by an additive constant ; that is, that all primitive functions are indefinite integrals. Frequently, however, we cannot obtain all the primitive functions if we omit the c, as the example J(x) = 0 shows. For this function the indefinite integral will always be zero, independ ently of the lower limit ; yet any arbitrary constant is a primitive function of f(x) = 0. A second example is the function j(x) = ,; ;, which is defined for nonnegative values of x only. The i ndefinite integral is cf>(x) = Jx% - Ja%,
and we see that no matter how we choose the lower l imit a the in definite integral cp(x) is always obtained from i(x)% by addition of a constant - Ja% which is less than or equal to zero ; yet such a function as fx'4 + I is also a primitive function for ,;; . Th us in the general expression for the primitive function we cannot dispense with the arbitrary additive constant. 1
As long as a lies in the domain off
Sec. 2.9
Fundamental Theorems of the Calculus 1 89
The relationship which we have found suggests extending the notion of. the indefinite integral so as to include all primitive functions. We shall henceforth call every expression of the form c + tP(x) = c
+
i�f(u) du
an indefinite integral of f(x), and we shall no longer
distinguish between the primitil'e function and the indefinite integral. Nevertheless, if the reader is to have a proper u nderstanding of the interrelations of these concepls, it is absolutely necessary to bear i n m i n d that in the first instance i ntegration a n d inversion o f differen tiation are two different things, and that it is only the knowledge of the relationship between them that gives us the right to apply the term "indefin ite integral" to the primitive function also. It is quite customary to use a notation which i s not perfectly clear without comment : we write
F(x)
=
Jf(x) dx,
when we mean that the function F(x) is of the form F(x)
= c
+
rf(u) du
for suitable constants c and a, that is, we omit the upper limit x, the lower limit a and the addit ive constant c and use the letter x for the variable of integration. Strictly speaking, of course, there is a slight inconsistency in using the same letter for the variable of integration and the upper limit x which is the independent variable i n F(x). In using the notation .f f( x) dx we must never lose sight of the i ndeterminacy connected with it, that is, the fact that the symbol always denotes one of the primitive functions off only. The formula F(x) = .f f(x) dx is j ust a symbolic way of writing the relation
d F(x) dx
=
f(x) .
c. The Use of the Primitive Function for Evaluation of Definite Integrals
Suppose that we know any one primitive function F(x) for the func tion f(x) and that we wish to evaluate the definite integral We know that the indefinite integral
tP(x)
=
J."'f(u) du,
f f(u) du.
1 90 The Fundamental Ideas of the Integral and Differential Calculus
Ch. 2
x
being also a primitive of f(x), can only differ from F( ) by an additive constant. Therefore cf>(x) = F(x) + c, and the additive constant c is determined at once because the indefinite
c/>(x) = J.f(u) du must take the value zero when x = a. We thus obtain 0 = cf>(a) = F(a) + fro m.which c = -F(a) and cf>(x) = F(x) - F(a). I n particular, for the v l!l'ue x = b we have the basic integral
c,
formula
ft(u) du = F(b) - F(a), F'(u ) = J(u).
if Therefore,
If F(x) is any primitive function of the continuous function f(x) what soever, the definite integral off(x) between the limits a and b is equal to the difference F(b) - F(a). If we use the relation F'(x) = f(x), this consequence of the funda mental theorem may be written in the form
(3 1 )
F(b) - F(a) =
f'
F'(x) dx
a
=
f' d ( a
F x) -- dx =
dx
f 'dF(x), a
where now F(x) can be any function with a continuous derivative F'(x), and where we use the suggestive symbolic notation dF(x) = F'(x) dx of Leibnitz. In applying our rule we often use a vertical bar to denote the difference of values at the end points, writing
f" dF(x) dx dx a
-·
=
F(b) - F(a) = F(x)
We can write (3 1 ) in the form
f'F'(x) dx.
F(b) - F(a) = _l _ b-a b-a a
(32)
I'
.
a
Recalling the definition of the average of a function in an interval from p. 1 4 1 , the rule states then that the difference quotient of the
F(x)
function formed for the points a and b is eq ual to the arithmetic mean or average of the deriwtive of F(x) in the interval with end points a and b. When we considered the motion of a particle on a straight line, we called the change in distance s divided by the change in time t
Sec. 2.9
191
Fundamental Theorems of the Calculus
the "average velocity ." We see now that i ndeed t:.s/tl.t is precis e l y the average ·of the velocities ds/dt for the given time interval i f t i s the i ndepe nd e nt variable used in fo rming t he a ve r age RELATION BETWEEN T H E MEAN V A L U E T H EOREMS
The formula F(h) - F(a)
(33)
=
.
rf(x) dx
which holds for any contin uous function ! and one of its primitives F also m akes evident the r el at i on between the mean value theorems of i nteg r a l calculus (p. 1 4 1 ) and of differential calcu l u s (p. 1 73). By the mean val ue theore m o f in te gral calcul us we concl ude from (33) that F(b) - F(a)
=
(b - a)f(�).
Si nce F i s a p r i m i ti ve off, we can replace JW by F'W a nd obtain the mean val ue theorem of d i fferential c a l cu l u s for the function F. Of course, the req u i rement that F h ave a continuous derivative i s stronger than the req u i rement of the mean val ue theorem of d i fferential calculus, that the derivative merely exist. d. Examples
I n C hapter 3 we shall m ake extensive use of the fundamental theorem in eval uating i n tegrals. For the moment we i l l ustrate the method that is based on the use of t he formula
ib dF(x ) a
dx
=
F( b ) - F(a )
by some examples. On p. 1 63 we derived the formula
dx..!!:._ xrl n xn-1 =
fo r p os i t ive in tege rs n . This formula i s rea lly a trivial consequence of the binomial theorem si nce
d.!!.x._
x"
=
lim Jt - o
=
=
lim
! h
!(x" (n xn-1 n ( n
h-0 h li m
11 - 0
[(x + h ) " -
x"]
1 n( 1 + n h x"- + n - ) h 2 x "-2 + 2 +
·
- ) h xn- 2 1 + . . . + h "1 2
·
)
·
+ h " - x" =
1 n x n- .
)
'
1 92
Ch. 2
The Fundamental Ideas of the Integral and Differential Calculus
Integrating between the limits a and b we find that
Writing m for n
-
I we obtain the formula
{"x "' dx J.
=
-1- (bmo l - a'"+') + I
m
for integers m � 0. This derivation of the expression for the integral of x "' is much si mpler than the one given on p. 1 3 1 which was based on a geometric subdivision of the interval [a, b] ; moreover, the result is now actually more general since we can dispense with the assumption that a and b are positive. The formulas
d sin x -----;;;- =
COS
X,
-( h.) d COS dx
X
•
= - sm
a:
were obtained on p. 1 65 by applying the add ition theorems for trigono sin = I . I ntegrating we im med imetric functions and using l i m ately obtain
J,, a
cos x dx = sin b
-
" •0
sin
a,
h
i"s i n
:r
dx = cos a - cos
b.
Again this derivation of the integration formulas from the fundamental theorem is si mpler than the one based on the definition of the defi n i te integral as limit of a sum.
Supp lement. The Existence of the Definite Integral of a Co n t in uo us Fun c t ion
We have yet to prove the fact that the integral of a function f(x) between the limits a and b (a < b) exists whenever f(x) is continuous in the closed interval [a, b]. The proof will be based mainly on the uniform continuity of.f (x ) (see p. 41 ) : for any given positive < the values of f at any two points � and 11 of the interval differ by less than < if � and tJ are sufficiently close to each :Jther, the degree of closeness dependent solely upon < and independent of �. 17 ; in other words, there exists a uniform modulus of continaity b(<) such that //(�) j(IJ) I < < for any values �. 'IJ in [a, b] for which / � - r1 / < b .
-
Existence of the Integral 1 93
The definition of integral as a limit of sums requires that we subdivide the interval [a, b] into n parts by successive points x0, x., . . . , xm where Xo = a , xn = b and Xo < x 1 < . . . < x,.. Let sn be a name for a particular subdivision of [a, b] of this type into n cells. The coarsen ess of the subdivision will be measured by the length of the largest of the resulting cells, that is, the largest of the quantities �x, = x, - x,_ 1 , which we shall call the "span" of S"' Because of the uniform continuity of f the values off in any two points of the same cell differ by less than E as soon as the span of S,. is less than (j = b ( E). An approximating sum based on the subdivision Sn is obtained by choosing a value � . in each cell [xH , x,] and forming n
Fn =
2 J(�,) � x, .
i=l
We have to prove that for a sequence of subdivisions S, with span tending to zero the sums F, converge toward a limit, which we shall denote by
L
b f(x) dx, and that the value of this limit does not depend on
the particular choice of subdivisions and of i ntermediate points � • . To carry out the proof we first compare the values F, and FN belonging to two subdivisions S, and S \. where the span of S, i s less than (j and where the subdivision Ss is a "refinement" of S, ; that is, all points of subdivision of S11 occur among those of SN . We have in appropriately modified n otation s F y = L f(T) ;) �Y ;· where the values Y; are the poi nts of subdivision of SN , where �Y; = Y; - Y;- 1 , and 'f); lies i n the in terval [Y ;-� > y 1] . Two successive sub division points x,_1 and X; of S, also occur among the values Y; . say xi-1 = Y r-1• X; = Y I n s;V the cell [xi-1• X;] i s broken u p into i ntervals, say [y,_ 1 , Y, ] , [y,, Yr+ 1 ] , [y_,_ 1 , Y ,, ], making the total contribution j=I
.•.
•
.
•
,
L f('YJ;)(Y; '
j=r
- Y;- 1 )
to FN . We compare this to the contribution of the cell [x; - 1 > x;] to F, given by f( � ; )(x; - x,_1 ) , which can be written as
L ·'
f(�; ) ( Y; - Y;- 1 )
(see Fig. 2 . 3 3) and find for the absolute value of the difference of the contributions
/;#.
I ;#.
[j(r/; ) - f(�;) ] (Y; - YH) S
E • (Y; - Y;- 1 ) = E ( x, - x,_1 ) .
Ch. 2
1 94 The Fundamental Ideas of the Integral and Differential Calculus
Hence, adding up the differences of the contri butions to F" and F.v for all cells [x;-�> x; ] of S", we find the estimate I F N - F ,.I :::;;
,.
I E ( x; - X;_1) = E ( b
i=l
- a),
whenever sn has span less than b(E) and SN i s a refinement of S,. If now Sn and Sm are any two subdi visions , we can consider the subd i v i sion SN formed by all the points of s ubdi v ision S, together with I a
Yr - 1
I
Xi-1
I
t; I
I
I
Y; - 1
Y.
(; I'
y.
I
X;
Figure 2.33
I
I
I
I I
I
b
SN s.
all those of Sm. Then SN will be a refinement of both S, and S"'. Assume that both S, and Sm have span less than b( •) . Choosing any i ntermed iate points 'r/; of the cells of SN to define F.v, we find l Fn - Fm l
=
I (F, - FN)
+ (Fy - Fm) l :5: l Fn - FN I + l Fm - Fs l :::;; 2 <(b - a).
We see then that any two appro x imating sums differ arbitrarily little from each other, if the spans of the corresponding subdivisions are sufficiently small. Consider now any sequence of subdivisions S, whose spans tend to zero for n -+- oo. Let F, be the corresponding approximating sums. For any • > 0 the span of S, is less than b( •) for all sufficiently large n. H ence l Fn - Fm l < 2 <(b - a)
for both n and m sufficiently large. It follows that the sequence F,. satisfies the Cauchy convergence criterion (see p. 97) ; consequently , lim Fn
=F
exists. It remains to show that the value of lim Fn does not depend on the ... particular subdivisions and intermediate points. If then S,' denotes any other sequence of subdivisions with spans tendi ng to zero, then the corresponding sum F,' has a limit F '. Since 11-·
00
as soon as the spans of Sn and S,' are less than rl(<), we find for n -+- oo that also I F - F ' l :::;; 2<(b a). Since here E is an arbitrary positive ·-
Existence of the lnte�ral ! 95
i/l>
number, it follows that F
by
=
F'. Hence the limit
(x) dx, i s uniquely determined. •
F, which we denote
The proof of the existence of the definite integral of a continuous function is thus complete. More General Approximating Sums. Our proof indicates more clearly what is essential in the approximation of an integral by a sum. I t makes evident the fact that a somewhat more general li miting process could be formulated, leading also to the i ntegral, and that the following more general form of the theorem is true : j; need not be a function value in order that the sums F, = � j; �x1 converge to the integral ; it suffices instead that l/1 - /(�1) 1 < li(E) for some point �� i n the interval [x;_1 , x,], where li( £) 0 for £ .- 0. Th is general statement is often useful. I f, for example, f(x) = (:r)!J>(x), then instead of the sum :2: f( �) �x, we may consider the more general sum .-
where � and � ," are two not necessarily coi ncident points of the cell. This sum also tends to the i n tegral ,'
JuJCx) J ' dx
=
n
( :r)lp( x) dx
as n increases, provided that the length of the longest ceJI tends to zero. A corresponding statement holds for other sums formed in an analogous way ; for example, the sum
tends to the i ntegral
i, J ( � , ' ) 2 + 1J!(� ,'')2 � x ,
V= ]
To prove these statements we only have to show that the change D i n the approx imati ng sums due to the deviation of /;," from ;; tends to zero in the limit . This is obvious in the first example where the change in the approxi mating sum is D =
2 <Pa,'l[•p(i;,") "
v= l
-
v{i;,')]D.:r,.
Si nce cp is bou nded and V' uniformly continuous, D can be made arbitrari ly small by choosing sufficiently small cells.
1 96 The Fundamental Ideas of the Integral and
Differential Calculus
The change in the second sum is represented by
D
= I< V(�. ')1 + '1'(� ;)2 n
v- 1
_
v 4>( � . ')2
+ '1'(� , ')2) t.x
Ch. 2
•.
(O,c) i n the form I v� - V�l :::; lb - c1 , we find that
Using the t ri a ngl e inequality a pp lied to the t ri a ngl e with ver ti ces (a,O), (O,b ),
1 v4>(�,')2 + '1'<;;)2
_
v 4>(� .') 2 + 'I'(�/ J2 1
:::; I 'I' (� ;>
from which follows immed ia tely that D tends to ze ro .
_ '1'(;/JI
PROBLEMS
2. 1 , Page 1 20
I. Let f be a positive monotone function defined on [a, b], where 0 < a < b. Let 4> be the i nverse off and set oc = f(a), P = f(h). U s ing the i nterpretation of integral as area show that
SECTION
SECTION 1. Prove
2.2,
J:(y)
Page
1 28
dy = hP - aoc -
for an y natural number p t ha t
("xv dx .Ia
=
f)(.r)
-1- (b�'+1 p + I
-
dx .
a ·P+i )
u si ng a subdivision of [a, b] i n t o cells of equal length. Emp l o y the techniques i n Chap ter I , m i scell an eous Problems 5 to 1 2, to evaluate the approximat ing sums Fn . 2 . Derive the formula
negative, say oc = = T , where
q-I/.•
q
- r / s,
=
for
where
· ( hfa .)
f'' "
r
.r•
d.r, a, b > 0 , when " is rational and
an d
s
are natural n u m be r s. (Hint: Set
3. By t he method used to find the integral of
l
" cos :r dx
= sin h
about J"
4. M ake a general s t ateme n t function and (b) a n even function.
S. Ca l c u l at e
Jr•/o 2
sin x dx a n d
Jro / 2c
-"
sin :r , derive the formula
- sin a .
f(x) dx when
f(x)
o s x d;r. Explain on geometrical grounds
w hy these should be the same. Furthermore, explain why
ra·+2·
for all values of a a n d
J
h.
i s (a) an odd
s i n x dx =
rb+�. b
J
cos x dx
Problems 1 97 6. (a)
Evaluate In =
..
What i s lim In ? Interpret geometrically.
r·:�·11n dx.
Jo
L �(
(b) D o the same for In =
7. Evaluate
lim
a J:n d.c.
Vn
n - "'
1
+
SECTION 2.3, Page 1 36
,, _ co
�2 \'
+..
· +
�). Vn
j(x), g(x)
* I . Cauchy's ineq uality for integrals. Prove that for all continuous functions
*2. Prove that if .f(.r) is continuous and f(x) =
then j(x) is identically zero.
f
<
f( t) dt ,
*3. Let f(.r) be Lipschitz-continuous on [0, I ] ; that is,
for all
l f(x ) - .f(y) l
:r, y
in the interval. Prove that
M I .e
-
Yl
l Joe.r(.r) d.r - � i 1(�)i < M n k�t
n
2n
SECTION 2 . 5 , Page 1 4 5 I . Prove
c
p -q p log - � --= > pq q
(q � p ) .
(Hint : Apply Ca u hy ' s inequality, Problem 2. (a) Verify that log ( I (b) Show for x > 0 that
x
+
-
.c) =
x2 2
<
*(c) More generally, show for 0 x -
x2 2
.r3 +3 -
· · · -
x2 n
ix o
1 .)
I du, where x > - I . 1 +u
--
log ( I + :r) < x.
< < x
< log (I + x)
I that
<
::r 2 .c - 2
+
x3 3
(Hint: Compare 1 /( 1 + u) with a geometric progression.)
2n
. . . +
x2n+1
2n + I ·
1 98
Ch. 2
The Fundamental ideas of the Integral and Differential Calculus
SECTION 2.6, Page 149
1. (a) Prove
i
us in g a s u b d vi sion 1 )].
lim n(\' ; (b) Find
[a, b]
of
J.'
ibex
dx = eb
-
e"
i nt o equal cells.
[Hint:
' iog ; r dx. (See S ect ion 2. 1 , Problem I .)
(c) Show fo r x � 0 that ;r,2
;.vn
;D2
xn
n!
2!
O bta i n upper and lower estimates for (xeu du Jo repeatedly.) Obtain est i mates of the same type for ex when .r < 0.
(Hint:
exxn+l
. I) ! n ! (n---
l + .r + - + · · · + - :<;; ex :<;; J + ;�: + - + · · · + - +
2!
log :x =
Apply
and
+
i ntegrate
SECTION 2 . 8c, Page 1 63
Calculate the derivatives of the fol lowing functions wherever defined d irectly as the l imits of their d i ffere n ce quotients .
1 . tan x. 2. sec 2 .c.
3. sin v:;:,
4. V sin .r. 5.
I
. -·
Slfl ;C
6. sin
7.
.r•,
!X
where "' is rational and n eg ativ e .
SECTION 2 . 8 i , Page 1 75
1. S h ow x > si n ·"' for positive x and
.r
< tan . c for
.r
in
( �). 0,
2. If f(:r) is conti nuous and differentiable for a :<;; x :<;; b, show that if :$; 0 for a :<;; x < � an d f'(x) � 0 for $ < x :<;; b, t h e function is ne v er
{'(x)
Jess than [(.;).
•3. If the c on tinuo u s function f(x) h as a derivative f'(x) at each point ;r in the neighborhood o f x = �. and if['(>:) approaches a l i m i t L as x � � . then [':)l , then f(x) = 0 i d ent ical l y .
l
SECTION 2.9, Page 1 84
• t . lf a particle traverses d istance I n time I , beginning and ending at rest, then at some point in the interval it us t have been subjected to an acceleration equal to 4 or more.
i
m
Problems 1 99
x
S U PPLE MENT, Page 1 92 . Existence of the Definite I ntegral
1. Let f.( ) be defined and bounded on [a, b]. We define the upper sum :E and lower sum a for the subdivision
to be
a =
� =
.r0
< a· 1
<
Mi :l.x i ,
11
L
:r2 · · · < a =
.
r11
11
i '--- 1
L
=
b
n1'i j,.r i ,
where M; i s the least upper bound, m, the greatest lower bound of f(.r) i n t h e cell [.r i-b a·;]. (a) Show that in any refinement of a subdivision the upper sum either decreases or remains unchanged and, simi larly, the lower sum increases or remains u nchanged . (b) Prove that each u pper sum is greater than or equal to every lower sum. (c) The upper Darboux integral F1 is defined as the greatest lower bou nd of the u pper sums and the lower Darboux integral F- as the least u pper bound of the lower sums over all subdivisions. From (h), F 1 � r-. If pr = F- we cal l the common value the Darhoux integral of ( Prove t hat the Darboux integral of .f is actually the ordinary R iemann integral ; furthermore, show that the Riemann integral exists if and only if t he u pper and lower Darboux i ntegrals exist and are equal . 2. Let f(.r) be a monotone function defined on [a, b]. (a) Show that the difference between the u pper and lower sums for a subdivision i nto n equal cells is given exactly by i=l
� -
a =
l.f(b) - /(all (b - a)jn.
and explain this result geometrical ly. (b) Use the result of (a) to prove that the Darboux i ntegral exists . ( c) Estimate � - a i n terms of f(a), f(b) and the span of the subdivision i f the cells of the subd ivision may be unequal. (d) Mostly .f(.r), i f not monotone, can be written as the sum of monotone functions,f(.r) = ¢(.r) + v{r) where ¢ is nonincreasing and v· is nondecreasing. Estimate the difference between the upper and lower sums in that case. 3. Show that if .f(.r) has a conti nuous derivative in the closed interval [a, h], then f(J:) can be written as the sum of monotone fu nctions as in Problem 2d.
M I SCELLAN EOUS PROBLE M S
f
1. Prove that (a)
l
-1
(J: 2 - 1 ) 2 d:r
=
�;
(h) ( - [ )"
fl ( �) fr'( l
2. Prove for the binomial coefficient
(� ) [ =
en + I )
_1
(.r2
that
-
-
.r ) n- r.
! )" d:r
dJ 1
=
2 2n l l ( n ! ) 2 (2n + I ) !
.
*3. If f(x) possesses a derivative {'(x) (not necessari ly continuous) at each point x of a :S: x :S: b, and if( '(.r) assumes the values m and M it also assumes every value fl between m and M.
200
The Fundamental Ideas of the Integral and Differential Calculus
2 0 for all values of in a :5: 4. on or above the tangent line at any point
Iff"(x)
x
u
Ch. 2
x :5: b , the graph of y = f(x) lies x = � . y = j(i;) of the graph. :5: x :5: b , the graph of y = j'(x) in
S. 2 0 for all values of x i n the interval l i es below the line segment joining the two points of the graph for which 6.
If f"(x) x1 :5: x :5: x2 x = x1, x = x2. If{"(x) 2 0, thenf x' ; x2 :5: f(x,) ; f(x2) .
u = u(t) f(x) *7. Let
)
(
be a function such that f"(x) 2 0 for all values of be an arbitrary continuous function. Then
� ff[u(t)] dt
2/
x and
let
G fu(t) dt) .
8. (a) Differentiate directly and write down the corresponding i ntegration formulas : (i) x 4 ; (ii) tan x. Evaluate
( b)
lim
�n
(1
+ sec2 �
4n
+ sec2
4n
�+
· · ·
+ sec2
�).
4n
9. Let f(x) have first and second derivatives for all real values of x. Prove that if (x is everywhere positive and concave, then ((x) is constant.
f
)
n � oo
3 The Techniques of Calculus
Part A 3.1
Differentiation and Integration of the Elementary Functions
The Simplest Rules for Differentiation and Their Applications
Although problems of integration are u sual ly of greater importance than those of differentiation, the latter offer less formal difficulty than the former. Therefore it is a natural procedure first to master the art of differentiating the widest possible classes of functions ; then by the fundamental theorem ( Section 2.9) the results of differentiation are available for evaluating integrals. In the following sections we shall pursue such applications of the fundamental theorem. To a certain extent we shall make a fresh start and develop techniques of integration systematically on the basis of certain general rules for differentiation. a. Rules for Differentiation
We assume that i n the interval under consideration the functions f(x) and g(x) are differentiable ; then the following rules are basic. Rule 1. Multiplication by a Constant. For any constant function f(x) = c.f(x) is differentiable, and (I)
f'(x) = cf '(x).
The obvious proof was given in Chapter 2, p. 1 6 5 . 201
c,
the
202 The Techniques of Calculus
Ch. 3
Rule 2. Derivative of a Sum. If cf>(x) differentiable and (2)
= f(x) + g(x), then
cf>(x) is
cf>'(x) = f ' (x) + g '(x) ;
that is, the operations of differentiation and addition are interchange able. The same holds for the sum of a finite number n of differentiable functions n
cf>(x) = I J. (x), for which we obtain
v=l n
cf>'(x) = I j,'(x). v= l
The proof is obvious from the definition of derivative.
Rule
3.
Derivative of a Product. If cf>(x) = f(x)g(x), then cf>(x) is
differentiable and (3)
c/>'(x) = f(x)g '(x) + g(x)f'(x) .
The proof follows from the equation
,P(x + h) - ,P( x) = f(x + h)g(x + h ) - f (x)g( x) h h + h) - g( x ) x ( x + h) - f (x) g( . + g(x/ = f( x + h ) h h Taking the limit in this expression as h -+ 0 yields Eq. (3). This formula becomes more elegant if we divide1 by cf>(x) = j(x)g(x). We then obtain
cf>'(x) f '(x) + g '(x) = . cf>(x) f (x) g(x)
Using the notation of differentials (Chapter 2, p. 1 79) we may also rewrite Eq. (3) as d(jg) = fdg + g df. By i nduction we obtain for the derivative of a product of n factors an expression consisting of n terms, each of which consists of the derivative of one factor multiplied by all the other factors of the ' We must, of course, assume that ,P(x) is nowhere equal to zero.
Sec.
3. 1
The Simplest Rules for Differentiation and Their Applications 203
original product : ' ( x) =
�
[fr ( x)j2( x ) · · · j, ( x ) ]
/,(x) + f1 ( x )N( x )fa( x ) · · · fn( x ) + · · · + /1( x )j2(x) · · · f.'( x)
= fi ' ( x)fi x ) =
· ·
·
i J: ( x) ( x) ,
v� 1
j, ( x )
or on division by (x) = f1 (x)Nx)
· · · fn(x)
j; ( x) '(x) j1'( x) + j2 ' ( x ) + . . + J,: ( x) , = . = fn (x ) v� 1 j, ( x) fi x ) ( x ) f1 ( x ) which is valid where (x) ¢ 0.
I
By repeated application of the rule for the derivative of a product we can obtain formulas for the second and higher deriuatiues as well. We h ave for the second derivative
Leibnitz's Rule. The reader should prove by induction that the nth derivative of a product m ay be found according to the following rule (Leibnitz's rule) : d" dx"
Here
( fg ) = f
(�)
efficients.
Rule 4 .
d"g + dx"
(n)
df d"- 1 g
I d x dx"-1 d n-2g + " ' + n d 2f -+ 2 2 d x dx"- 2
=
n,
() (; )
( ) -n,
= [n(n - 1 )]/2 ! , etc .
Deriuatiue of a Quotient.
n
1
d "- 1/ dg + d''f - g. dx"- 1 dx dx"
denote the binomial co
For a quotient
(x ( x ) = j ) g (x)
204 The Techniques of Calculus
Ch. 3
the following rule holds : The function cp(x) is differentiable at every point at which g(x) does not vanish, and g(x)f '(x) - g'(x)f(x) cp ' (x) = .
(4)
If cp(x) ;I:- 0, this can be written as
fg ( x) r
cp ' (x) = f '(x) _ g' (x) cp(x) g (x) f (x)
PROOF. If we assume the differentiability of c/>(x), we can apply the product rule to f(x) = cp(x)g(x) and conclude
f'(x)
=
By substituting f(x)fg(x) for '(x) , we obtain Rule 4. We can prove the differentiability of
h
h
/(x + h) - f(x) _ g(x h
g(x + h) - g(x) f (x)
g(x)g(x + h)
h
If we now let h tend to zero we arrive at the result stated ; for by hypothesis the denominator does not tend to zero but to the limit (g(x)J2, and the two terms of the numerator have limits g(x)j'(x) and g'(x)f(x), respectively. This proves both the existence of the limit on the left side and the differentiation formula. b. Differentiation of the Rational Functions
First, we derive once more the formula
!!.._ x"
dx
= nx"- 1
for every positive integer n, invoking the rule for differentiating a prod uct. We think of x" as a product of n factors, x" = x · · · x, and thus obtain
Sec.
3.1
The Simplest Rules for Differentiation and Their Applications 205
The second derivative of the function xn follows from this formula and eq. ( I ) .
d2 " x dx2
n( n
=
-
l ) x" - 2 .
Continuing, we obtain the higher derivatives
d3 -3 x" dx
=
n(n - l )(n
d" - x" dx"
=
1 ·2· ··n
2)x"-3
-
=
n !.
From the last of these formulas i t is clear that the nth derivative of
x" is a constant, whereas the (n + l)th derivative vanishes everywhere.
By using our first two rules and the rule for differentiating powers we can differentiate any polynomial y = a0 + a1x + a2x2 + · · · + a,.x", obtaining
furthermore, y " = 2a2 + 3 · 2a 3x + 4 · 3a4x2 + · · · + n (n - l )a,.x"-2, and so on. The derivative of any rational function can now be found with the help of the quotient rule. In particular, we again deduce the differen tiation formula for the function x", where n = -m is a negative integer. Application of the quotient rule, together with the fact that the derivative of a constant is equal to zero, gives the result
(dfdx)( l fx m) m or, if we take = - n ,
=
-mxm-Ijx2 m
.!!__ x" dx
=
=
- m fx m H,
n x " -1 ,
which agrees formally with the result for positive values of n and with the results given earlier (p. 1 64). c. Differentiation of the Trigonometric Functions
For the trigonometric functions sin x and cos x we h ave already obtained (p. 1 65) the differentiation formulas
d . dx
- Si n
X = COS X
and
d dx
.
- COS X = - S i n X .
206 The Techniques of Calculus
Ch. 3
The quotient rule now enables us to differentiate the functions sin x COS X and y = cot x = -.- . sm x cos X According to the rule, the derivative of the first of these functions i s , cos 2 x + sin 2 x y = ' co s 2 x cos 2 x s o that 1 - tan x = -- = sec 2 x = 1 + tan 2 x . dx cos2 x Similarly, we obtain y =
tan x
= -
·
d
1 - cot x = - -= - cosec 2 x = - ( 1 + cot 2 x). sin 2 x dx To the differentiation formulas for sin x, cos x, tan x, and cot x correspond the following integration formulas :
d
J J -1-
cos x dx = sin x,
J J
sin x dx = - cos x,
dx = tan x, -1- dx = - cot x. cos2 x sin2 x From these formulas we obtain by way of the fundamental rule of Section 2.9, p. 1 90 the value of the defin ite integral between any limits, the only restriction being that when the last two formulas are used, the i nterval of integration must not contain any point of dis continu ity of the integrand such as an odd multiple of -rr/ 2 in the first case, and an even multiple of -rr / 2 in the second. For example,
3.2
i"
cos x dx
=
sin x
1:
=
sin b - sin
a.
The Derivative of the Inverse Function
a. General Formula
We have seen on p . 45 that a continuous function y = j(x) has a continuous inverse in every interval in which it is monotonic. Precisely : If a :5: x :5: b is an interval in which the continuous function y = f(x) is monotonic, and iff(a) = oc andf(b) = {3, thenf has an im•erse function which in the inten•al between oc and f3 is continuous and monotonic.
As p o i nted out on p. 1 77, the sign of the derivative provides a simple test for seeing when a function is monotonic and therefore has an i nverse. A differentiable function is continuous, and is monotonic
Sec.
3.2
The Derivative of the Inverse Function 207
increasing in an interval wheref'(x) is greater than zero, and monotonic decreasing in an interval for which f'(x) is everywhere less than zero. We shall now characterize the derivative of the inverse function by proving the fo llow in g theorem. THEOREM. If the function y = f(x) is differentiable in the interval a < x < b, and either f'(x) > 0 or ['(x) < 0 throughout the interval, then the inverse function x = (y) also possesses a derit•ative at every interior point of its interval of definition : the derivatiues of y = f(x) and of its inverse x = <j>(y) satisfy the relation f'(x) · cp'(y) = 1 at corresponding values x, y.
This relation can also be put i n the form 1 dy -=dx dx dy This last formula again illustrates the suitability of Leibnitz's notation : the symbolic quotient dyfdx can be treated in formulas as i f it were an actual fraction. PROOF . The proof of the theorem is simple. Writing the derivative as the limit of a difference q uotient, we h ave D. y = l i m Yt - y , y ' = f'(x) = li m A x - o �x � 1 -+ a: x1 - x where x a n d y = f(x) , and x1 and y1 = f(x1), respectively denote pairs of corresponding values. By hypothesis the fi rs t of these limiting values is not equal to zero. Because of the con tin uity of y = f(x) and x = cp(y), the relations y 1 y and x1 ---'> x are equivalent. Therefore the lim i tin g value x x - x x = lim i lim i (5)
---+
x , - x yl - Y
Y t - Y Yt - Y
exists and is equal to 1 /['(x). On the other h and, the limiti ng value on the right-hand side is by de fin it i on the derivative '(y) of the inverse function <j>(y), and our formula is p ro ved . The simple geometrical meaning of the formula is clearly shown in Fig. 3. 1 . The tangent to the curve y = f(x) or x = (y) makes an angle ex with the positive x-axis, and an angle f3 with the positive y-axis ; from the geometrical interpretation of the derivative of a function as the slope of the tangent f' (x) = tan ex , c/>'(y) = tan {3.
Since the sum of the angles ex and {3 is w/2, tan ex tan f3 = 1 , and this rel ationsh ip is exactly equivalent to our differentiation formula.
208
The Techniques of Calculus
y
Ch. 3
y
"'
------��---L--�0�----�x�--� x Figure 3.1
Differentiation of the inverse function.
Critical Points
We have hitherto expressly assumed that either f'(x) > 0 or f'(x) < 0, that is, that f'(x) is never zero. What, then, happens if J'(x) = 0 ? If f'(x) = 0 everywhere in an interval, then f is constant there, and consequently has no inverse because the same value of y corresponds to all values of x i n the interval. If f'(x) = 0 only at isolated "critical" points (and ifj'(x) is assumed continuous), then we have two cases, according to whether on passing through these points j'(x) changes sign, or not. In the first case this point separates a point where the function is monotonic increasing from another where it is monotonic decreasing. In the neighborhood of such a point there can be no single-valued inverse function. In the second case the vanishing of the derivative does not contradict the monotonic character of the function y = f(x), so that a single-valued i nverse exists. How ever, the inverse function is no longer differentiable at the corre sponding point ; in fact, its derivative is infinite there. The functions 2 y = x and y = x3 at the point x = 0 offer examples of the two types. Figure 3.2 and Fig. 3.3 illustrate the behavior of the two functions upon passing through the origin and at the same time show that the function y = x3 has a single-valued i nverse, whereas the other function y = x 2 does not.
Sec. 3.2
The Derivative of the Inverse Function 209
..
,.j
]
e
.. ro..
�.rJ
u :I
"! ..., ..
.i '"'
..
210 The Techniques of Calculus
Ch. 3
b. The Inverse of the nth Power: the nth Root
The simplest example is the inverse of the function y = x" for positive integers n ; at first we assume positive values of x, hence also ' y > 0. Under these conditions y is always positive, so that for all positive values of y we can form the unique inverse function X = '\YY = ylln .
The derivative of this inverse function is immediately obtained by the above general rule as follows :
d(y11 ") dy
=
dx dy
1 _ = _1 _ = l _l__ = = _ dyjdx nx "-1 n y < n-1 ) /n
l y (lfn)-1. n
If we now change the notation and denote agai n the i ndependent variable by x, we may finally write
d\Y; � lln) l (1/nl-l = (x = X , dx dx n
w hich agrees with the result obtained on p. 1 64. For n > 1, the point x = 0 requires special consideration. If x approaches zero through positive values, d(x11")/dx will obviously increase beyond all bounds ; this corresponds to the fact that for n > 1 the derivative of the nth power f(x) = x " vanishes at the origin. Geometrically, this means that the curves y = x11" for n > 1 touch the y-axis at the origin (cf. Fig. 1 . 35, p. 48). It should be noted that for odd values of n the assumption x > 0 can be omitted and the function y = x" is monotonic and has an i nverse over the entire domain of real numbers. The formula
d( \y-y) dy
= ( 1 /n )y
still holds for negative values of y, but for x = 0, n > I , we have d(x ") f dx = 0, which corresponds to an i nfinite derivative dxjdy of the inverse function at the point y = 0. c. The Inverse Trigonometric Functions-Multivaluedness
To form the inverses of the trigonometric functions we once again consider the graphs1 of sin x, cos x, tan x, and cot x. We see at once from Figs. 1 . 37, p. 50 and 1 . 3 8 , p. 5 1 , that for each of these functions it • The graphical representation will help the reader to overcome the slight difficulties inherent in the discussion of the "multivaluedness" of the inverse functions.
Sec. 3.2
The Derivative of the Inverse Function 2 1 1
is necessary t o select a definite interval if w e are t o speak o f a unique inverse ; for the lines y = c parallel to the x-axis cut the curves in an infinite number of points, if at all.
The Inverse Sine and Cosine For the function y = sin x, for example (Fig. 3.4), the derivative y ' = cos x is positive in the i n terval - TT/2 < x < TT/2. I n this i nterval y = sin x has an i nverse function which we denote by1 x = arc sin y y
/-,
- - -/- - �...\ I
I
I
/
Figure
1
\
- - - - - - - -
3.4 Graph of
y =
sin
x
(pri ncipal value i ndicated by solid curve) .
(read arc sine y ; this means the angle whose sine has the value y). This function increases monotonically from - TT/2 to + TT/2 as y traverses the i nterval - 1 to + I . If we wish to emphasize that we are considering the i nverse function of the sine in this particular interval, we speak of the principal value of the arc sine. For some other i nterval in which sin x i s monotonic, for example, the i nterval + TT/2 < x < h/2, we obtain another inverse or "branch" of the arc sine ; without the exact statement of the i nterval in which the values of the inverse function should lie, the symbol arc sine means not one well-defined function but, in fact, denotes an infinite number of values. 2 The multivaluedness of arc sin y is described by the statement : To any one value y of the si ne there corresponds not only a specific angle x but also any angle of the form 2k1T + x or (2k + !)1r x, where k i s any i nteger (cf. Fig. 3.4). -
1 T h e symbolic notation x = sin-• y i s also used where there is no danger of con fusion with the reciprocal function 1 /sin x. ' Sometimes loosely called a multiple-val ued function .
212
The Techniques of Calculus
Ch. 3
y "'
7r
'
'l-1 '\ y "' arc sin 1 1 x = sin y
- Tr /2
-Tr
- 27r
," Figure 3.5
Graph of y
I I I I I I I I
I
I I 1
I I I ' I ' '!. I' I
"'
x
)
v
, "' I I I I
= arc sin x (principal value indicated by solid curve).
The derivative of the function x = arc sin y is obtained from Eq. (5) as follows : dx --; == - = - = - = ----= dy y' cos x ± .J l - sin 2 x
where the square root i s to be taken as positive if we confine ourselves to the first interval mentioned, that is, - 7T/2 < x < 7T/2 . l Finally, we change the name of the i ndependent variable from y to the commonly used x (Fig. 3 . 5) ; then the derivative of arc sin x i s 1 I f instead o f this w e had chosen the interval rr/2
< x < 3rr/2, corresponding t o the substitution of x + rr for x, we should have had to use the negative square root since cos x is negative in this inlerval.
The Derivative of the Inverse Function 2 1 3
Sec. 3.2
expressed by
.
- arc sm x = dx d
1 )1 -
x2
.
Here it is assumed that arc sine is the principal value which lies between - TT/2 and + TT/2, and the square root sign is chosen positive. y
-------�0�---,�--� x ," - 7f/2
,/
( \
/
Figure
/
'
/
"
"
/
- 7f '
'
3.6 Graph o f y = arc cos x (principal value indicated by solid curve).
For the inverse function of y = cos x, denoted (after again inter changing the names x and y) by arc cos x, we obtain the formula d
- arc cos x dx
= =f
1
vl
in exactly the same way. Here we take the negative sign of the root if the value of arc cos x is taken in the interval between 0 and 7T (not, as in the case of arc sin x, between - TT/2 and + TT/2) (cf. Fig. 3.6).
214 The Techniques of Calculus
Ch. 3
x =
The derivatives become infinite on approaching the end points - 1 and x = + 1, corresponding to the fact that the graphs of the inverse sine and inverse cosine have vertical tangents at these points. Inverse Tangent and Cotangent
We treat the inverse functions of the tangent and cotangent in an analogous way. The function y = tan x, h aving an everywhere positive derivative 1 /cos2 x fo r x #- 1r/2 + k1r, has a u n i que i n verse y
/
_ _ _ _ _ _ _ _ _ ......
Figure
3.7 Graph of
y
/ --
- ------ --
_,/
= arc tan x (solid curve for principal value).
i n the interval - 7T/2 < x < 7T/2. We call this inverse function (the Principal Branch of) x = arc tan y. We see at once from Fig. 3.7 that for each x we could h ave chosen instead of y a n y of the values y + k1r (where k is an i n tege r). S i mila rl y , the function y = cot x has an i nverse x = arc cot y which is uniquely determined if we req u ire that its value shall lie in the interval from 0 to 1T; otherwise the many-valuedness of arc cot x is the same as for arc tan x. The differentiation form ula s are as foll ow s : x = arc tan y, x = arc
c ot y,
dx 1 = -- = COS 2 dy dyfdx dx . = - Si n 2 dy
X
X =
1 + tan 2 x 1
= 1 + c oe
X
- ---
1
+ y2
= 1 + Y2
Sec. 3.2
The Derivative of the Inverse Function 2 1 5
or fin al ly if w e denote again the independent v a r i a ble by x, ,
d - arc tan dx
d dx
x
=
1
1 , + x2
---
- arc cot x = -
1
1 + x2 ·
---
d. The Corresponding Integral Formulas Expressed i n term s of indefinite i ntegrals, the form ulas which we have just derived are written as follows : 1
J J1 1 J -1
- x2
+ x2
dx
=
J .j J1
a rc sin x ,
dx = arc tan
1
. dx = - arc cos x, 1 - x2
1 --- dx + X2
x,
=
- arc cot x.
Alth ough the two formu las on each line express d i fferent functions by i dentical i ndefin i te i ntegrals, they do not contradict each other. In fact, they i l lustrate w h a t we l earned earlier (see Section 2 . 9 ) , that all i ndefinite i n tegral s of the same function differ only by constants ; here the constants are Tr/2 s ince arc cos x + arc sin x = Tr/2, arc tan :c + arc cot x = Tr/2. The formulas for indefinite integrals m ay i m m edi a te l y be put to use for find ing defin i te integrals, as on p. 143. ln particular,
fb a
1
dx + X2
---
=
arc tan x
lb
= arc tan b - arc tan
I f we p u t a = 0, b = 1 a n d recall t h a t tan 0 obtain the remarkable formula
(6)
7r
4
=
(I
a.
a
1
Jo 1 + x2
=
0 a n d t a n Tr/4
=
1, w e
dx.
The n u mber '" • w h ich originally arose from the consideration of the circle, is brought by this formula into a very si mple relati onship with the rational function 1 /( 1 + x2) , and represents the area indicated in Fig. 3 . 8 . This formula for '"• to wh ich we shall return later (p. 445), constitutes one of the e ar l y tri um phs of the power of calculus. M ore general ly, the i n tegral formulas of th i s sect ion perm it u s to define the trigonometric fu nctions purely analytically, without any reference to geometric objects such as triangles or circles . For example,
216 The Techniques of Calculus
Ch. 3
the relation between an angle y and its tangent x = tan pletely described by the equation x - � yJo 1 + u 2
y
is com
r
(at least for - TT/2 < y < TT/2). With this relation we may now define without appeal to intuition a numerical value for the angle y in a righ t triangle with sides a (adjacent) and b (opposite) for which b/a = x.
Figure 3.8
rr/2 ill ustrated
by an area.
Such an analytic definition in terms of numerical quantities makes the use of angles and trigonometric functions legitimate in higher analysis irrespec t i v e of a definition by geometrical construction. e. Derivative and Integral of the Exponential Function
I n Chapter 2. p. 1 50. we introd uced the exponential function as the inverse of th e l ogari thm. Precisely speaking the relations y = ex and x = log y were thus defined to be equivalent. Consequently their derivatives satisfy the relation [see (5) p . 207]
de" dy 1 1 1 - = - = - = --- = - = y = e . dx dx dx d l og y ! dy dy y •
Hence the exponential function is equal to its derit'ative :
Sec. 3.3
Differentiation of Composite Functions 2 1 7
More generally, for any positive a the function i nverse l og Y x = log. y = , log a and the derivative of a"' i s
y
= a"' has as its
da " 1 -= = (log a)y = (log a )a". dx d loga Y dy Thus for any positive constant a the derivative of the function y = --
a"
is proportional to the function itself. The factor of proportionality log a is 1 when a is the number e. On p. 223 we shall show conversely that any function which is proportional to its derivative must be of the form y = ce" , where c denotes a constant factor. By the fundamental theorem of calculus we can again translate the formulas for the derivatives of e" and of a " into formulas for indefinite i ntegrals :
3.3
fe" dx fa " dx
= e", =
1 -- a". l og a
Differentiation of Composite Functions
a. Definitions
The preceding rules allow us to find the derivatives of functions that are obtained as rational expressions in terms of functions with already known derivatives. To find explicit expressions for the derivatives of other functions occurring in analysis we must go further by deriving a general rule for the differentiation of composite or compound functions. We are confronted quite often with functions j(x) bui lt by the process of composition of simpler ones (see C hapter I , p. 52) : j(x) = g((x)), where cp(x ) is defined in a closed interval a S x S b and has there the range r; S S {3, and where g() is defined in this latter interval. I n this connection it is usefu l to remember the interpretation of functions as "operators" or mappings. As i n Chapte r I , we write the composite function simply as f = g and call g the (symbolic) "product" of the operators or mappings g and ,P.
218 The Techniques of Calculus
Ch. 3
b . The Chain Rule
For functions g and c/> which are continuous in their respective i ntervals of definition the compound function f(x) = g[cf>(x)] is con tinuous as well (see Chapter I , p. 55). The functions cf>(x) and g(c/>) are now assumed to be not only contin uous but differentiable. We then have the following fundamental theorem, the chain rule of differentiation :
The function j(x) = g[cf>(x)] is differentiable, and its derivativ e is given by the equation
(7)
J'(x) = g '(c/>) · cf>'(x),
or, in Leibnitz's notation, df df dr/> = dx de/> d x
Therefore the derivative of a compound function is the product of the derivatives of its constituentfunctions. Or : The derivative ofthe symbolic product offunctions is the actual product of their derivatives with respect to their corresponding independent variables. Intuitively, this c h a i n rule is very plausible. The quantity c/>'(x) = lim t::.. cf>(t::..x is the local ratio in which small i nte r vals are magnified by the mapping cf>. Similarly, g'(c/>) is the magnification given by the mapping g. Applying first cf> and then g results i n magnifying an x-interval first 'fold ,, and then enlarging the resulting -interval g'fold, resulting in a total magnification ratio of g'c/>' which must be the mag nification ratio for the composite mapping / = gcf>. The theorem follows very easily from the definition of the derivative. In fact, it becomes intuitively almost obvious if we assume f(x) � 0 in the closed x-interval under consideration. Then for t::..x = x2 - x1 � 0 we h ave by the mean value theorem
t::.. cf> = c/> 2 - c/> 1 = cf>(x 2) - r/>(x1) = c/>'(�)t::.x. � 0 with x1 :::;; � :::;; X2, and, with t::.g . = g(c/> 2 ) - g(c/> 1) and t::.j . = j(x2) - f(x1), we may write t::.j . t::.. g t::.. cf> = t::.. x t::.. c/> t::.. x
which is a meaningful identity because t::.. r/> � 0. Now t::.. cf> --+ 0 for t::.. x -+ 0, that is, for x2 __,. x 1 ; therefore for t::..x --+ 0 the difference quotients tend to the respoctive derivatives and the theorem is proved.
Differentiation of Composite Functions 2 1 9
Sec. 3.3
To avoid the explicit assumption c/)(x) ¥:- 0 we can dispense with the divisiop by l:l.cf> in the fol lowing slightly more subtle manner : From the assumption of differentiability of at the point cf> we know that the quantity • = as a function of l:l.cf> for fixed and ¥:- 0 has the limit zero for 0. If we define • = 0 for = 0, we have with out restriction on
g(c/>) l:l.gfl:l.cf> - g'(c/>) cf> l:l.cf> l:l.cf> -+ l:l.cf> l:l.cf> l:l.g [g'(c/>) + •] l:l.cf> . Similarly for fixed x, l:l.cf> cf>(x + l:l.x) - cf>(x) = [c/>'(x) + 1J ] l:l.x, w here lim = 0. Then for l:l.x 0 and cf> = cf>(x), l:l.g [g '(c/>) + • ] l:l. cf> [g ' (c/>) + • ] [cf>'(x) + 17 ]. l:l.x l:l.x F o r l:l.x tending t o zero through nonzero values w e have lim l:l.cf> hence lim = 0, so that l:l. l i m g = li m [ g ' ( c/>) + •] l i m [c/>' ( x ) + 17 ] = g ' ( c/>)c/>' ( x) , � =
=
'YJ
¥:-
=
=
.d x -+ O
=
0 and
t.x-o
•
X
dx - o
d x -+ 0
which proves the chain rule. By s uccessive application of our rule we i mmediately extend it to functions arising from the composition of more than t wo functions. If, for example, v = = g(u), u = cf>(v),
•p(x), y then = f(x) g [cp(1p(x))] is a compound function of x; its derivative is given by the rule dy = y ' = g '( u)c/>' ( v) 1J!' (x) dy . du . dv ; du dv dx dx y
=
=
similar relations are true for functions that are compounded of an arbitrary number of functions. Higher Derivatives of a Composite Function. y = g[ cf>(x)) can be found easily by repeated application of the chain rule and the preceding rules : dy dcf> y ' = - - = g 'cf> ' ' dcf> dx y " = g "cf> ' 2 + g 'cf> " , y"' = g"'c/>'3
+
3g "cf>'cf>"
+ g 'cf>"'.
Analogous formulas for y"" etc., can be derived successively.
220 The Techniques of Calculus
Ch. 3
Finally, let us examine the composition of two functions inverse to each other. The function g(y) is the inverse of y = cf>(x) i ff(x) = g[cf>(x)] = x. It follows that
f'(x) = g'(y)cf> ' (x) = I
which is exactly the result of Section 3.2, p. 207.
Examples. As a simple but i mportant example of an application of the chain rule we differentiate x• (x > 0) for an arbitrary real power oc. In Chapter 2, p. 1 52, we defined we also proved for cf>(x) = log x, 'l'(u) = ocu, g(y) = e• that
cf>'(x) = ! ' X
'l' ' (u) = oc,
g '(y)
=
e•.
Now x• is the compound function g{'l'[cf>(x)l}. Applying the chain rule we obtain the general formula
d - (x") = g' (y)!p ' (u )cf>'(x) dx
= e• · oc · !
X
X
x• = �- ; X
hence
.3_ (x") dx
=
ocx•->,
a result we could prove only with some difficulty had we attempted to proceed directly from the definition of x" for i rrational oc as the limit of powers with rational exponents. An immediate consequence of this d ifferentiation is, again, the integral formula (oc "*" - 1) .
As a second example, we consider
y = .J l - xz
o r y = ,J;i,,
Sec. 3.3
where
=1
-
x2
Dijjerentiation of Composite Functions 221
x < 1 . The chain rule yields = 2.J � (-2x) = - x .J 1 - x2
and - 1 <
y
'
·
Further examples are given by the following brief calculations.
= arc sin .J 1 - x2, ( - 1 � x � 1 , x ;t. 0) . dy 1 d../� dx = .j 1 - (1 - x2) dx = j";j1 .J1 -x- x2 ...; 1 - x sgn (x) . y = J� -1 - x , (-1 < x < 1). dy = -� dx 2 � dx
1.
y
1
2.
,J � .J�
-
1
2
= 2.j1 + x . ( I - x)2 = (1 + x)!-i 1( 1 - x)"' · 3 . y = log I xi - This function1 can be expressed as log x for and as log ( -x) for x < 0. For x > 0 d log JxJ = d log x = dx dx X 2
---
For
x
x
>0
--
< 0 we obtain from the chain rule that
d log lxl = d log (-x) = 1 d(-x) = 1 � dx � dx ; · Hence generally for x ¥- 0 d log Jxl = dx x 4. = By definition of (see p. 1 52) we have _
y
x a .
a
x
1 The function log x is defined only for x > except for x = 0.
0, whereas
log I xi is defined everywhere
222 The Techniques of Calculus
Ch. 3
where cf>(x) = (log a)x. Then
dax de "' de/> - = - - = e"'(log a) = (log a)ax. dx de/> dx
The same result was obtained already on p. 2 1 7 from the rule for the derivative of the inverse function. 5. y = [f(x)]"1x1 . Since [f(x)]•l x l = e <�>l• l
with cf>(x) = g(x) log [J(x)], we find
.!£ (f(x)]"1"1 dx
(
= e<�> g ' log f + g l f
=
(
r)
)
x ' (x ) . [J( x)]•l•l g '(x ) log [f (x.) ] + g ( )f f(x)
For example, when g(x) = f(x) = x we have
c. The Generalized Mean Value Theorem of the Differential Calculus As an application of the chain rule we derive the generalized mean value
theorem of d i fferential calculus. Consider two functions F(c) and G(x), c on t inuo u s on a closed interval [a, b] of the x-axis, a n d different iable on the
interior of that interval . We assume that G'(;t) is positive. The ordinary mean value theorem of differential cal c u l u s applied separately to F and G furni s h es an e x p res si on for the difference quotient F(b) - F(a)
G(b) - G(a)
F'(�)(b - a) G'(tJ)(b - a)
=
F(b) - F(a)
G(bt- Q(a)
:
F'W
G'(tJ) '
where � and 11 are suitable intermediate values in the open interval (a, b). The generalized mean value theorem states that we can write the d ifference quotient in the simpler form
F(b) - F(a) G(b) - G(a)
=
F '(O G '(O '
where F' and G' are evaluated at the same intermediate value �. For the proof we i ntroduce u = G(x) as an inde pen d en t variable in F. From the a ss u mpti o n G' > IJ we conclude that the function u = G(x) is monotonic i n the in terv a l [a, b], and hence that it has an inverse x = g(u) defined in th e interval [IX, f], where IX = G(a), {J = G(b). The compound
Sec. 3.4
Some Applications of the Exponential Function 223
function F[g(u)] = f(u ) is therefore defined for u in the interval [ex, PJ. From the ordinary mean value theorem we find that
F(b) - F(a) = f(tl)
- f(ex ) = f'(r)(tl
- ex) = f'(y)[G(b) - G(a)],
where y is a suitable value between ex and p. By the chain rule, we infer
f'(u)
=
d F[g(u)] du
= F'[g(u)]g'( u) =
F'(x)
' G (x) .
To the value u = y there corresponds a value x = g(J') = ' in the interval (a, b). Then j'(y) = F'(O/G'(O, and the generalized mean value theorem follows. 3.4
Some Applications of the Exponential Function
Some miscellaneous problems i nvolving the exponential function will illustrate the fundamental i mportance of this function i n all sorts of applications. a. Definition of the Exponential Function by Means of a Differential Equation
We can define the exponential function by a simple property, whose use obviates many detailed arguments in particular cases.
If a function y
=
f(x) satisfies an equation of the form y' = ocy,
where oc is a constant, then y has the form (8)
where c is also a constant; conversely, every function of the form ce""' satisfies the equation y ' = ocy. Since Eq. (8) expresses a relation between the function and its de rivative, it is called the differential equation of the exponential function. It is clear that y = ceax satisfies this equation for any arbitrary constant c. Conversely, no other function satisfies the differential equation y' - ocy = 0. For if y is such a function, we consider the function u = ye-•x. We then have u
'
=
y' e-•x - ocye- • x
=
e-•x(y' - ocy) .
H owever, the right-hand side vanishes, since we have assumed that y' = ocy ; h ence u ' = 0, so that by p. 1 78 u is a constant c and y = ce•x as we wished to prove. We shall now apply this theorem to a number of examples.
224
The Techniques of Calculus
Ch. 3
b. Interest Compounded Continuously. Radioactive Disintegration
A capital sum, or principal, augmented by its interest at regular periods of time, increases by j umps at these interest periods i n the following manner. If lOOoc is the percent of interest, and further more, if the interest accrued is added to the principal at the end of each year, after x years the accumulated amount of an original principal of 1 will be (1 + oc)". If, however, the principal had the interest added to it not at the end of each year, but at the end of each nth part of a year, after x years the principal would amount to
1( oc)" " +n
.
Taking x = I for the sake of simplicity, we find that the principal I has increased after one year to
(
oc)" n
I + - .
If we now let n increase beyond all bounds, that is, if we let the interest be credited at shorter and shorter intervals, the limiting case will mean in a sense that the compound interest is credited at each instant ; then the total amount after one year will be e• times the original principal (see p. 1 53). Similarly, if the interest is calculated in this manner, an original principal of I will have grown after x years to an amount e•x ; here x may be any number, integral or otherwise, The discussion in Section 3 .4a forms a framework into which examples of this type are readily fitted. We consider a quantity, given by the number y, which increases (or decreases) with time so that the rate at which this quantity increases or decreases is proportional to the total quantity . Then with time as the independent variable x, we obtain a law of the form y ' = ocy for the rate of increase, where oc, the factor of proportionality, is positive or negative depending on whether the quantity is increasing or decreasing. Then in accordance with Section 3.4a the quantity y itself is represented by a formula y
=
ce"",
where the meaning of the constant c is immediately obvious if we consider the instant x = 0. At that instant e = l , and we find that ..
Sec. 3.4
Some Applications of the Exponential Function 225
c = Yo is the quantity at the beginning of the time considered, so that we may write Y = Yoe"� . A characteristic example is that of radioactive disintegration. The rate at which the total quantity y of the radioactive substance is diminishing is proportional at any i nstant to the total quantity present at that instant ; this is a priori plausible, for each portio n of the sub stance decreases as rapidly as every other portion. Therefore the quantity y of the substance expressed as a function of time satisfies a relation of the form y' = - ky, where k is to be taken as positive since we are dealing with a diminishing quantity. The quantity of substance is thus expressed as a function of the time by y = y0e-kr, where y0 is the amount of the substance at the beginning of the time considered (ti me x = 0). After a certain time -r the radioactive substance will have diminished to half its original quantity. This so-called half-life is given by the equation
�Yo = Yoe- k' ,
from which we immediately obtain c.
-r
= (log 2)/k.
Cooling or Heating of a Body by a Surrounding Medium
Another typical example of the occurrence of the exponential function is the cooling of a body, for example, a metal plate of uniform temper ature which is i mmersed in a very large bath of lower temperature. We assume that the surr o unding bath is so large that its temperature i s unaffected b y t h e cooling process. W e further assume that a t each i nstant all parts of the i mmersed body are at the same temperature, and that the rate at which the temperature changes is proportional to the difference of the temperature of the body and that of the surrounding medium ( Newton's law of cooling). If we denote the time by x and the temperature difference between the body and the bath by y = y(x), th is law of cooling is expressed by the equation y' = - ky, where k is a positive constant (whose value is a physical characteristic of the substance of the body). From this differential equation, which expresses the effect of the cooling process at a given instant, we obtain by means of Eq. (8), p. 223, an "integral law" giving us the temperature at any arbitrary time x in the form
y = ce-k�.
226
Ch. 3
The Techniques of Calculus
This shows that the temperature decreases "exponentially" and tends to become equal to the external temperature. The rapidity with which this happens is expressed by the number k. As before, the meaning of the constant c is that of the initial temperature at the instant x = 0, Y o = c, so that our law of cooling can be written in the form y
=
Yoe-kx.
Obviously, the same discussion applies also to the heating of a body. The only difference is that the initial difference of temperature Yo is in this case negative instead of positive . d.
Variation of the A tmospheric Pressure with the Height above the Surface of the Earth
A further example of the occurrence of the exponential formula is in the variation of atmospheric pressure with height : We make use of ( I ) the physical fact that the atmospheric pressure i s equal to the weight of the column of air vertically above a surface of area one, and (2) of Boyle's law, according to which the pressure p of the air at a given constant temperature is proportional to the density a of the air. Boyle's law, expressed in symbols, is p = aa, where a is a constant depending on a specific physical property of the air. Our problem is to determine p = j(h) as a function of the height h above the surface of the earth . If by Po we denote the atmospheric pressure at the surface of the earth , that is, the total weight of the air column supported by a unit area, by g the gravitational constant, and by a(,\) the density of the air at the height ,1. above the earth , the weight1 of the column up to the height h i s given by the integral g ( h a(,\) d,\ . The pressure at height h Jo is therefore p
= f (h)
a(
= Po
-
f a(
g
A.) d ,\.
By differentiation this yields the following relation between the pressure and the density h) :
p = f(h)
ga(h)
=
-f'(h)
' = -p .
a
We now use Boyle's law to eliminate the quantity from this equation, ' thus obtaining an equation p = - (g/a)p which involves the unknown pressure function only. From Eq. (8) p . 223, it follows that p = f(h) =
1 ga(A.)
ce-••l•.
is the weight of the air per unit vol ume at the height A..
Sec.
3.4
Some Applications of the Exponential Function 227
If as above we denote the pressure f(O) at the earth's surface by p0, it follow� immediately that c = p0, and consequently p
= f(h)
= P oe-uhfa.
Taking the logarithms yields h = q_ l og £!! . g p
These two formulas are applied frequently. For example, if the constant a i s known, they enable us to find the height of a place from the baro metric pressure or to find the difference in height of two places by measuring the atmospheric pressure at each place. Again, if the atmo spheric pressure and the height h are known, we can determine the constant a, which is of great i mportance in gas theory. e. Progress of a Chemical Reaction
We n o w consider an example from chemistry, the so-called uni mo/ecular reaction. We suppose that a substance is dissolved in a
large amount of solvent, say a quantity of cane sugar in water. If a chemical reaction occurs, the chemical l aw of mass action i n this case states that the rate of reaction is proportional to the quantity of reacting substance present. We suppose that the cane sugar is being transformed by catalytic action into invert sugar, and we denote by u(x) the quantity of cane sugar which at time .r i s still unchanged. The velocity of reaction i s then - dufdx , and i n accordance with the law of mass action an equation of the form du dx
- ku
holds, where k i s a constant depending on the substance reacting. From this i nstantaneous or differential law we i mmediately obtain, as on p. 223, an integral law, which gives us the amount of cane sugar as a function of the time : This formula shows us clearly how the chemical reaction tends asymp totically to its final state u = 0, that is, complete transformation of the reacting substance. The constant a is obviously the quantity of cane sugar present at time x = 0.
228 The Techniques of Calculus
Ch. J
f. Switching an Electric Circuit On or Off
As a final example we consider the growth of a direct electric current when a circuit is completed, or its decay when the circuit is broken. If R is the resistance of the circuit and E the electromotive force (volt age), the current I gradually increases from its original value zero to the steady final value E/R. We have therefore to consider I as a function of the time x. The growth of the current depends on the self induction of the circuit ; the circuit has a characteristic constant L, the coefficient of self-induction, of such a nature that, as the current i ncreases, an electromotive force of magnitude L dlfdx, opposed to the external electromotive force E, is developed. From Ohm's law, as serting that the product of the resistance and the current is at each instant equal to the actual effective voltage, we obtain the relation dl IR = E - L - . dx E
For
f(x) = I(x) - -
R we immediately find J'(x) = - (R/L)f(x), so that by Eq. (8), p. 223, J(x) = f(O)e-RxtL. Recalling /(0) = 0, we find f(O) = -E/ R ; thus we obtain the expression
e-Rx!L) R R for the current as a function of the time. This expression shows how the current tends asymptotically to its steady value E/R when the circuit is closed. I = j (x) +
3.5
§:_ = §:_ ( I
_
The Hyperbolic Functions
a. Analytical Definition
In many applications the exponential function enters in combinations of the form It is convenient to introduce these and similar combinations as special functions ; we denote them as follows : (9a) (9b)
smh x = --- , 2
.
ex - e-x
cosh x = --- , 2
ex + e-x
Sec. 3.5
The Hyperbolic Functions 229 y
-
2
!e•
y = ---
--
-I
Figure
y = -- -
! ·-·
2
"
3.9
and we call them the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, and hyperbolic cotangent respectively. The functions sinh x , cosh x, and tanh x are defined for all values of x, whereas for coth x the point x = 0 must be excluded. The names are chosen to express a certain analogy with the trigonometric functions ; it is this analogy, which we are about to study in detail, that justifies special consideration of our new functions. In Figs. 3.9, 3 . 1 0, and 3 . 1 1 the graphs of the hyperbolic functions are shown ; the dotted lines in Fig. 3.9 are the graphs of Y = (!)ex and Y = me-x , from Which the graphs Of Sinh X and COSh X may easily be constructed. Cosh x obviously is an even function, that is, a function which remains unchanged when x is replaced by - x, w hereas sinh x is an oddfunction, that is, a function that changes sign when x is replaced by -x (cf. p. 29).
230
The Techniques of Calculus
Ch. 3 y
-I Figure 3.10 y
I
-----------
2-o�--<J---<> _-<1>------= 2 --<> -�1
Figure 3.11
"'
Sec.
3.5
The Hyperbolic Functions
By its definition, the function cosh
x
23 1
e x + e- "' = --'---
2
is positive and not less than one for all values of x. It has its least value when x = 0 : cosh 0 = 1 . The fundamental relation between cosh x and sinh x cosh 2 x
-
sinh2 x = 1 ,
fol lows immediately from the definitions. I f we now denote the inde pendent variable by t instead of x and write x
we have
= cosh t,
x2 - y2
y
= sinh t,
= I;
that is, the point with the coordinates x = cosh t, y = sinh t moves along the rectangular hyperbola x2 - y 2 = I as t ranges over the whole scale of values from w to + w. According to the defining equation, x � 1, and our formulas make i t evident that y runs through the whole scale of values w to + w as t does ; for if t tends to infinity so does e 1 , whereas e- t tends to zero. We may therefore state more exactly : As t runs from - CJJ to + w, the equations x = cosh t, y = sinh t give us one branch, namely, the right-hand one, of the rectangular hyper bola. -
-
b. Addition Theorems and Formulas for Differentiation
From their definition we obtain the addition theorems for the hyper bolic functions : ( 1 0)
cosh (a + b) = cosh a cosh b + sinh a sinh b, sinh (a + b) = sinh a cosh b + cosh a sinh b.
The proofs are obtained at once if we write
and insert i n these equations
e" = cosh a + sinh a, e b = cosh b + sinh b,
e-• = cosh a - sinh a, e-b = cosh b - sinh b.
Between these formulas and the corresponding trigonometrical formulas there i s a striking analogy. The only difference in the addition theorems is one sign in the first formula.
232
The Techniques of Calculus
Ch. 3
A corresponding analogy holds for the differentiation formulas. Remembering that d(e")fdx = ex, we readily find that d
. = smh x, d 1 - tanh x = --- , - cos h x
dx
(1 1)
cosh 2 x
dx
d
- sm h x .
dx
= cosh x,
d - coth x dx
= - sinh1 2 x --
From the first two equations it follows i mmediately that and y = sinh x are solutions of the differential eq uation
.
y
= cosh x
(12) which again only differs in sign from the analogous equation satisfied by the trigonometric functions cos x and sin x (see p. 1 7 1). c.
The InverJe Hyperbolic Functions
To the hyperbolic functions x = cosh t, inverse functions, which we denote 1 by
t = ar cosh x,
y
= sinh t, there correspond
t = ar sinh y .
Since the function sinh t is monotonic increasing2 throughout the i nterval CJJ < t < + CJJ , its inverse function is uniquely determined for all val ues of y; on the other hand, a glance at the graph (see Fig. 3.9, p. shows that t = ar cosh x is n o t uniquely determined, but has an ambiguity of sign, because to a given value of x there corresponds not only the number t but also the number - t . Since cosh t I for all values of t, its inverse ar cosh x is defined only for
-
229)
X � J.
�
We can easily express these inverse functions i n terms of the logarithm by regard ing the quantity e' u in the definitions
=
e' + e- '
X = --2- , as unknown and solving these (quadratic) equati on s for u : u=x ±
.Jx2 - I ,
u
= y + .Jy2 + I ;
= e ' can have only positive values, the square root i n the
since u second equation must be taken with the positive sign, whereas in the
1 The symbolic notation cosh-' x, etc., is also used ; cf. footnote, p . 54. 2 (d/dl) sinh 1 = cosh 1 > 0.
Sec. 3.5
The Hyperbolic Functions 233
first either sign is possible (which corresponds to the ambiguity men tioned ab,ove). In the logarithmic form, t = log u, and hence
.Jx2 - 1) ar cosh x, t = log (y + .Jy2 + I) = ar sinh y. t = log (x ±
( 1 3)
=
In the case of ar cosh x the variable x is restricted to the interval x � I , whereas ar sinh y is defined for all values of y. Equation ( 1 3) gives us two values, log (x + .Jx2 - 1 )
Jog (x - .Jx2 - I )
and
for a r cosh x , corresponding t o the two branches o f a r cosh x . Since
(x + .J x2 - ! )(x - .Jx2 - I ) = 1 , the sum of these two values of ar cosh x i s zero, which agrees with the ambiguity in the sign of t mentioned before. The inverses of the hyperbolic tangent and hyperbolic cotangent can be defined analogously, and can also be expressed in terms of loga rithms. These functions we denote by ar tanh x and ar coth x; ex pressing the i ndependent variable everywhere by x, we readily obtain ar tanh
x
( 14) ar coth x
= - log --
i n the interval - 1 <
=
i n the intervals x < - 1 , x > 1 .
1 +X
1 2
1 - X 1 X+ 1 - log -X - 1 2
x
< 1,
The differentiation of these i nverse functions may be carried out by the reader himself; he may make use of either the rule for differen tiating an inverse function or the chain rule in conjunction with these expressions for the i nverse functions in terms of logarithms. If x is the i ndependent variable, the results are
( 1 5)
1 v x2
1
d - ar cosh x dx
=
±
d - ar tanh x dx
=
1 -- , 1 - x2
- 1
,
d . hx= - ar sm
dx
d - ar coth x =
dx
1
.J x2 + 1
,
1
1 - x2
--
•
The last two formulas do not contradict each other, since the first holds only for - 1 < x < I and the second only for x < - 1 and 1 < x. The two values of the derivative d(ar cosh x)fdx, expressed by the sign ± in the first formula, correspond to the two different branches of the curve y = ar cosh x = Jog (x ± .Jx2 - 1 ) .
234
The Techniques of Calculus
Ch. 3
d. Further Analogies
The similarities between the hyperbolic and the trigonometric functions are no accident. The deeper source of these analogies becomes apparent when we consider these functions for imaginary arguments, as we shall do later in Section 7. 7a. We shall then be able to identify cosh x with cos (ix) and sinh x with ( 1 /i) sin (ix), where i = .J--:::::J . This fact makes it evident that every relation i nvolving trigonometric functions has its counterpart for hyperbolic functions. Many of those analogies have interesting geometrical or physical interpretations. (See also Chapter 4, p. 363.) In the above representation of the rectangular hyperbola by the quantity t, we did not ascribe any geometrical meaning to the "param eter" t itself. We shall now return to this subject, and encounter a further analogy between the trigonometric and the hyperbolic functions. If we represent the circle with equation x2 + y2 = I by means of a parameter t in the form x = cos t, y = sin t, we can interpret the quantity t as an angle or as a length of arc measured along the cir cumference ; we may, however, also regard t as twice the area of the circular sector corresponding to that angle, the area being reckoned positive or negative depending on whether the angle is positive or negative. We now state analogously that for the hyperbolic functions the quantity t is twice the area of the hyperbolic sector for x 2 - y 2 = 1 shown shaded in Fig. 3 . 1 2.1 It is this i nterpretation of t i n terms of areas that accounts for the names t = ar cosh x and t = ar sinh y given to the inverse hyperbolic functions.2 The proof is obtained without difficulty if we refer the hyperbola to its asymptotes as axes by means of the transformation of coordinates X - y =
or
.Jz$,
� - ($ + '1}), X= -
l "2
with these new coordinates the equation of the hyperbola is $'1} = !. Hence the two right triangles OPQ and O A B both have area i, for the lengths of OQ and QP are, respectively, 'I} and 1 /2'1}, and the area in 1 For a different proof,
t
=
2 Just
see p.
as the notation
372.
not the len gth of the hyperbolic arc.
ar cosh x refers to an
Incidentally, 1 i s
t = ar� cos x refers to an arc of the unit circle, so area connected with a rectan gular hyperbola x' - y' = I .
y
Figure 3.12
Figure 3.13
functions.
To i l l ustrate the hyperbolic
236
The Techniques of Calculus
Ch. 3
question is equal to that of the figure A B QP. Obviously the coordinates of the points A and B are and
i (z+v)/
respectively, and for double the area of our figure we thus obtain v'2
( 1 /21) ) d1J = log (x + y) = log (x ± Jx 2 - 1), lf v' 2 but by Eq. ( 1 3), p. 233, this is equal to t, proving our assertion. In conclusion, it may be pointed out that, as shown in Fig. 3 . 1 3, the hyperbolic functions can be graphically represented on the hyperbola, just as the trigonometric functions can be represented on the circle. 2
_
___
3.6 Maxima a nd Minima
As the first of a great variety of applications we consider the theory of maxima and minima of a function, in conjunction with a geometrical discussion of the second derivative. a. Convexity and Concavity of Curves
By definition the derivative f'(x) = df(x)fdx represents the slope of the curve y = f(x). The derivative of the function f'(x) or of the slope of the curve y = f(x) is given by the derivative df'(x)fdx = d2f(x)fdx2 = f"(x), the second derivative of f(x), and so on . If the second derivative j"(x) is positive at a point x-so that owing to con tinuity (which we assume) it is positive in some neighborhood1 of this point x-then throughout this neighborhood the derivative j'(x) in creases with Increasing values of x. Hence the curve y = j(x) turns its convex side downwards or is "open" upwards. We call the function f(x) or the curve y = f(x) convex. Ifj "(x) is negative, the curve and the function are concave. Therefore when f"(x) > 0, the curve in the neighborhood of the point lies above the tangent while when j"(x) < 0, ' We make use here of the intuitively obvious observation : a
continuous
function
g(x) which is positive at a point x, also is positive for all points of a sufficiently small neighborhood of x0 (as far as they belong to the domain of g). The formal proof is simple . From the continuity of g at x0 we know that for every positive • the i nequality ig(x) - g(x,)l < • holds for all x in a sufficiently small neighborhood lx - x01 < 6 of the point x0 • Since g(x0) > 0, we are free to choose for E the value lg(x0), so that jg(x) - g(x,)i < fg(x0) in some neighborhood. Since then g(xo) g(x) � i g(x) - g(x0)i < !g(x0), (t follows that g(x) > !g(x0) > 0.
Sec. 3.6 y
y
Maxima a11d Minima 237
�0�------�x--� x
(a) Figure
(b)
3.14 (a) f"(x) > 0. (b) f"(x) < 0.
it l ies below the tangent (see Figs. 3. 1 4a and 3. 1 4b) (cf. Problem 4, p. 200 and Section 5.6). Point of Inflection
Special consideration is required only in points where f"(x) = 0. On passing through such a point the second derivative j"(x) will gen eral ly change its sign. Such a point will then be a point of transition between the two cases j ust indicated ; that is, on one side the tangent is above the curve and on the other side below it, whereas at this point it crosses the curve (see Fig. 3 . 1 5) . Such a point is called a point of in flection of the curve, and the corresponding tangent is called an in flectional tangent. y
0 Figure
3.15
Point of inflection.
238
The Techniques of Calculus
Ch. 3
The simplest example is given by the function y = x3, the cubical parabola, for which the x-axis itself is an inflectional tangent at the i nflection poi n t x = 0 (see Fig. 3.3, p. 209). Another example is given by the function f(x) = sin x, for which f'(x) = d(sin x)Jdx = cos x and f"(x) = d2(sin x)jdx2 = - sin x.
Consequently,J'(O) = 1 and f"(O) = 0 ; since the sign ofj"(x) changes at x = 0, the sine curve has at the origin an inflectional tangent in clined at an angle of 45 degrees to the x-axis. It must, however, be noted that points can exist where f"(x) = 0 and the sign of{"(x) does not change with increasing x, while the tangent does not cut the curve but remains entirely on one side of it. For example, the curve y = x• lies entirely above the x-axis, although the second derivative f"(x) = 1 2x2 vanishes for x = 0. b.
Maxima and Minima-Relative Extrema. Stationary Points
A function f(x) has a maximum at a point ; if the value off at the point ; is not exceeded by the value off at any other point x of the domain off; that is, j(;) � f(x) for all x wherejis defined.1 Similarly, f has a minimum at ; if j(;) � f(x) for all x in the domain. The word extrema is used to cover both maxima and minima. The function f(x) = ._II x 2 , for example, which is defined for - 1 � x � 1 , has minima at x = ± 1 and a maximum at x = 0. It is easy to give examples of continuous functions which have no maxima or no minima. Thus the functionf(x) = 1 /( 1 + x 2) (Fig. 3.8, p. 2 1 6) in the domain oo < x < + oo has no minimum ; the function f(x) = 1 /x defined for 0 < x < oo has no extremum points at all. We recall, however, from Chapter l, p. 1 0 1 the theorem of Weierstrass, according to which a continuous function defined in a closed finite interval always has a maximum (and similarly a minimum) there. Our object is to fi nd a means of locating the extrema of a function or curve. This problem which is encountered very frequently in geometry, mechanics, physics, and other fields was one of the principal incentives for the development of the calculus in the seventeenth century. Calculus does not furnish a direct method for picking out the extrema of a function f(x), but it permits us to locate the so-called relative extremum points, among which the actual maxima and minima have to occur. The point ; is a relative maximum (minimum) off iff -
-
1 W e t a l k of a strict maximum point ; ifJW > f(x) for all x in t h e domain of / that
are different
from �.
Maxima and Minima 239
Sec. 3.6
has its greatest (least) value at � when compared not with all possible values pfj(x) but just with the values ofj(x) for x in some neighborhood of �. By a neighborhood of the point � we mean here any open interval ex < x < fJ which contains the point � but may be arbitrarily small. A relative extremum point � ofj is then a point which is an extremum point when f is restricted to all those points of its domain lying suf ficiently close to �.1 Obviously, the extrema of the function are in cluded among the relative extrema. To avoid confusion we shall use y
X
Graph of function defined on the interval [a, b] with relative minima at = a, x,, x., x,, relative maxima at x, x3, x,, b, absolute maximum at b, and absolute minimum at x, .
Figure 3.16 x
the terms absolute maxima (minima) for the maxima and minima of f in its entire domain (see Fig. 3 . 1 6). Geometrically speaking, relative maxima and minima, if not located in the end points of the i nterval of definition, are respectively the wave crests and troughs of the curve. A glance at Fig. 3 . 1 6 shows that the value of a relative maximum at one point x5 may very well be less than the value of a relative minimum at another point x2• The diagram also suggests the fact that relative maxima and minima of a continuous function alternate : Between two successive relative maxima there is always located a relative minimum. Let f(x) be a differentiable function defined in the closed interval a � x � b. We see at once that at a relative extremum point which is The formal definition of a relative maximum point � would state that there exists open interval containing � such that /(�) � .f(x) for all x of that interval for which f is defined. 1
an
240
The Techniques of Calculus
Ch. 3
located in the interior of the interval the tangent to the curve must be horizontal. (The formal proof is given below.) Hence the condition f'W = 0 is necessary for a relative extremum at the point ; with a < ; < b. If, however, j(;) is a relative extremum and ; coincides with one of the end points of the interval of definition, the equationf'W = 0 need not hold. We can only say that if the left-hand end point is a relative maximum (minimum) point, the slope j'(a) of the curve cannot be positive (negative), while if the right-hand end point b is a relative maximum (minimum) thenf'(b) cannot be negative (positive). The points at which the tangent to the curve y = f(x) is horizontal, corresponding to the roots ; of the equation f'W = 0, are called the critical points or stationary points of f All relative extrema of a differentiable function f which are i nterior points of the domain of f are stationary points. Hence : an absolute maximum or minimum of the function coincides either with a critical point of the function or with an end point of its domain. In order to locate the absolute maxima (minima) of the function we have only to compare the values of fin the critical points and in the two end points and to see which of these values are greatest (least). lf f fails at a finite n umber of points to have a derivative, we have only to add those points to the list of possible l ocations of an extremum and to check also the values of f at those points. Thus the main labor in determining the extrema of a function i s reduced to that of finding the zeros of the derivative of the function, which usually are finite in number. To take a simple example, let us determine the largest and smallest values of the function f(x) = 1\x6 - lox2 in the interval - 2 :::;; x :::;; 2 . Here t h e critical points, the roots o f the equationf'(x) = 6(x5 - x)/ 10 = 0 are located at x = 0, + I , I Computing the values of f at those points and also at the end points of the interval, we find -
.
X
-2
-I
0
f(x)
5.2
-0.2
0
2 -0.2
5.2
It i s clear that the points x = ± I represent relative minima, whereas relative maxima occur at x = 0 and x = ± 2 . The maximum value of the function, assumed in the end points of the interval, is 5.2 ; the minimum value, assumed in the points x = ± I , is -0.2 (see Fig. 3. 1 7). Without appealing to intuition we can easily prove by purely analytic methods that /'(;) = 0 whenever ; is a relative extremum point in the interior of the domain of/ provided / is differentiable at ;. (Compare
Sec. 3.6 6
Maxima and Minima 241
y
5
4 3
-1 Figure
3.17
y =
(x' - 3x2)/ I O.
the exactly analogous considerations for Rolle's theorem, p. 1 75.) If the function f(x) has a relative maximum at the point � . then for all sufficiently small values of h different from zero the expression f(� + h) - fW must be negative or zero. Therefore
for h > 0, whereas
[ /( � + h) - /( �)] <0 h [ /(� + h) - f(x)] > - 0 h
for h < 0. Thus i f h tends to zero through positive values, the limit cannot be positive, whereas i f h tends to zero through negative values, the limit cannot be negative. However, since we have assumed that the derivative at e exists, these two limits must be equal to one another, and, in fact, to the value J'(e), which therefore can only be zero ; we must h avef'(.;) = 0. A similar proof holds for a relative minimum. The proof also shows that if the left-hand end point e = a is a relative maximum (minimum) point, then at least ['(a) � 0 [f'(a) � 0] ; if the right-hand end point b is a relative maximum (minimum) point, then .f'(b) � 0 [j'(b) � 0].
242
The Techniques of Calculus
Ch. 3
The condition f'W = 0 characterizing the critical points is by no means sufficient for the occurrence of a relative extremum. There may be points at which the derivative vanishes, that is, at which the tangent is horizontal, although the curve has neither a relative maximum nor minimum there. This occurs if at the given point the curve has a horizontal inflectional tangent cutting it, as i n the example of the function y = x3 at the point x = 0. The following test gives the conditions under which a critical point is a point of relative maximum or minimum. It applies to a continuous function f, having a continuous derivative f' which vanishes at most at a.finite number of points or, more generally to differentiable functions f for which f' changes sign at most at a finite number of points :
The function f(x) has a relative extremum at an interior point � of its domain !f. and only if, the derivative J'(x) changes sign as x passes through this point; in particular, the function has a relative minimum if near � the derivative is negative to the left of � and positive to the right, whereas in the contrary case it has a maximum. We prove this rigorously by using the mean value theorem. First, we observe that to the left and right of � there exist i ntervals � 1 < x < � and � < x < �2 , in each of which f'(x) has only one sign, since .f' vanishes only at finite number of points. (Here �1 and �2 can be taken as the points nearest to � at which/' vanishes, if such points exist.) If the signs of f'(x) in these two intervals are different, then f( � + h) fW = hj'(� + (}h) has the same sign for all numerically small values of h, whether h is positive or negative, so that � is a relative extremum. Ifj'(x) has the same sign in both intervals, then hj (� + (}h) changes sign when h does, so that /(� + h) is greater than /(�) on one side and less than/(�) on the other side, and there is no extreme value. Our theorem is thus proved. At the same time we see that the value f(�) is the greatest or least value of the function, in every interval containing the point �. in which f is differentiable and in which the only change of sign off ' (x) occurs at � '
itself
The mean value theorem on which this proof is based can still be used iff(x) is not differentiable at an end point of the interval in which it is applied, provided thatf(x) is differentiable at all the other points of the interval ; hence this proof still holds if f'(x) does not exist at x = �. For example, the function y = l xl has a minimum at x = 0, since y ' > 0 for x > 0 and y ' < 0 for x < 0 (cf. Fig. 2.24, p. 1 67). The function y = tf� likewise has a minimum at the point x = 0, even though its derivative fx-"' is infinite there (cf. Fig. 2.27, p. 1 69).
Sec.
3.6
Maxima and Minima 243
The simplest method for deciding whether a critical point � is a relatiYe maximum or minimum involves the second derit·ative at that point. I t is intuitively clear that if f ' W = 0, then f has a relative maximum at � if f " (; ) < 0, and a relative minimum if f"W > o. For in the first case the curve in the neighborhood of this point lies com pletely below the tangent, and i n the second case completely above the tangent. This result follows analytically from the preced ing test, provided that f(x) and ['(x) are continuous and that j"(�) exists. For if[' W = 0 and, say, f"W > 0, we have f" W = l i m
11 � 0
f ' (� + h ) - f'W f '(� + h) = lim h
h�o
h
> 0.
It follows that f'(� + h)/h > 0 for all h rf 0 which are sufficiently small in absolute value ; hence f' ( � + h) and h have the same sign in a neighborhood of r For X near � the derivative f'(x) must be negative for x to the left of �. and positive for x to the right of � ; this implies that there is a relative minimum at .;. The situation is particularly simple in case f"(x) is of one and the same sign throughout the interval [a , b] i n which / i s defined :
A point � at which f' vanishes is a maximum point off iff"(x) < 0 throughout the interval (or if its cun:e is concat>e), and a minimum point off if throughout the interval j"(x) > 0 (that is, if the curt>e is com•ex) .
I ndeed , i f f"(x) < 0 the function f'(x) is monotonic decreasing, hence has � as its only zero. Moreover, / ' > 0 for a :::;; x < �. whereas f ' < 0 for � < x :::;; b. By the mean value theorem this i mplies again that f(x) < !W for x rf �. so that � turns out to be a strict maximum point. The minimum of f must coincide with one of the end points since there is no other critical point besides �- The same argument applies when .[" > 0 in the i nterval. Examp les
Example 1 . Of all triangles with given base and given area, to find that with the least perimeter. To solve this problem, we take the x-axis along the given base AB and the middle point of AB as the origin (Fig. 3 . 1 8) . If C is the vertex of the triangle, h its altitude (which is fixed by the area and the base), and (x, h) are the coordinates of the vertex, then the sum of the two sides A C and BC of the triangle is given by
f(x) = ..}(x + a) 2 + h 2 + ..}(x - a) 2 + h 2 ,
244
Ch. J
The Techniques of Calculus y C(x, h)
A
-a
Figure 3.18
where 2a is the length of the base. From this we obtain
,
x+a x-a + 2 2 ) J( x - a )2 + h 2 .../ ( x + a + h - ( x + a )2 1 j"(x) = , + .../ [( x + a) 2 + h 2 ] 3 ,J( x + a )2 + h 2 - (x - a ) 2 + ' + --;,====::::: v [( x - a ) 2 + h 2 ] 3 v(x - a ) " + h 2 h2 h2 + ;: +==: '11�[(= ,j [( x + a)" + h 2] 3 -; h 2;=.;]3 a ):::;2 = x=-= f'(x)
=
•
We see at once ( I ) that f'(O) vanishes, and (2) that f"(x) is always positive ; hence at x = 0 there is a least value (see p. 243). This least value is accordingly given by the isosceles triangle. Similarly, we find that of all the triangles with a given perimeter and a given base the isosceles triangle has the greatest area.
Examp le 2. To find a point on a given straight line such that the sum of its distances from two given fixed points is a minimum. Let there be given a straight line and two fixed points A and B on the same side of the line. We wish to find a point P on the straight line such that the distance PA + PB has the least possible val ue.1 We take the given line as the x-axis and use the notation of Fig. 3 . 19. Then the distance in question is given by f(x)
1
If A and B
=
Jx2 + h 2 + Jcx - a ) 2 + h12 ,
lie on opposite sides of the line,
the line with the segment AB.
P obviously is
just the intersection of
Sec. 3.6
Maxima and Minima 245
y A
Figure
3.19
Law of reflection.
and we obtain
f ' (x) = j " (x) =
x .J x 2
+
h2
+
h2 .J c x 2
+
h 2) a
x-a , .J c x - a) 2 + h1 2
+
h12 .J [(x - a) 2
The equation / '(�) = 0 means
�
or
.J e + h 2 COS
+
h / ]a
a-� .Jc � - a) 2 + h / , IX =
COS
{3 ;
hence the two lines PA and PB must form equal angles with the given line. The positive sign of f"(x) shows us that we really have a least value. The solution of this problem is closely connected with the optical law of reflection. By an important principle of optics, known as Fermat's principle of least time, the path of a light ray is determined by the property that the time the light takes to go from a point A to a point B under the given conditions must be the least possible. If the condition is imposed that a ray of light shall on its way from A to B pass through some point on a given straight line (say on a mirror), we see that the shortest time will be taken along the ray for which the "angle of incidence" i s equal to the "angle of reflection."
246
The
Techniques of Calculus
Ch. 3
Example 3. The Law of Refraction.' L e t there be g i v e n t wo poi nts A an d B on opposite sides of the x- ax i s . What is the p a t h from A to B
r e quirin g the sh ortest possible time if the velocity on one side of the i s c1 and on the other side c 2 ? Clearly, this s hortest path must consist of two porti ons of straig h t lines mee t i n g one another a t a point P o n t h e x-axis. U s i ng the notati on
x ax i s -
of Fig.
3 . 20, we obta i n the two ex p ressions J h2 + x2 and Jh1 2 + ( a - x)2 y
B
Figure 3.20
Law of refract i o n .
for the l e n g t h s P A , PB, respectively, and we find th e t i me o f passage along t h i s pa t h by d i v i d i n g the l e n gt h s of t h e two segments by the corresponding velocities and then adding ;
1 I 1 ;-.--2 f (x) = - , h- + x + - v h,- + ( a - x )-. o
c1
By differentiatio n , we obtain
f (.r) = ,
f"( x ) 1
=
1
-
c,
..!._
x
---===
.J h' + x2 hz
c , J(h2 + x ')"
0
c2
a-x 2 C2 Jh,' + ( a - x ) h/
+ ..!._ C 2 � [ h 1 2 + (a - x)2]3 •
one is not easily disposed of without calculus.
While the preced ing examples can be treated also by elementary geometry,
this
Maxima and Minima 247
Sec. 3.6
As we readily see from Fig. 3.20, the equation f'(x) equation.
0, that is, the
a-x ' C1 Jh 2 + x2 � Jh/ + (a - x)2 the condition ( l /c1) sin IX = (l/c ) sin p, or 2 X
is equivalent to
=
=
sm IX sin P
=
:1
c2
The reader should verify the fact that there is only one point which satisfies this condition and that this point actually yields the required least value. y
b
Figure 3.21
Point on ellipse having the least distance from a point on the
major axis.
The physical meaning of our example is again given by the optical principle of least time. A ray of light traveling between two poi nts describes the path of shortest time. If c1 and c are the velocities of 2 light on either side of the boundary of two optical media, the path of the light will be that given by our result, which is a form of Snell's law of refraction. Example 4. Find the point of an ellipse which is closest to a given point on its major axis (Fig. 3 . 2 1 ). Taking the ellipse i n the form
y2 x2 = + a 2 b2
1
(b
< a)
248
The Techniques of Calculus
Ch. 3
and the given point on the major axis as ( c, 0), we find for the distance of any point (x, y) on the ellipse from the point (c, 0), the expression d
=
../(x - c) 2 + b 2( l - x2fa 2) ,
where - a � x � a. The function f(x) = d2 is convex (/" > 0). It has a minimum for the same x as d itself. The only critical point of f is at x = c/( 1 - b2/a2). I f this point lies in the domain of d, it represents the minimum point ; if not, the minimum of d corresponds to the end point of the major axis closest to c . We find accordingly for the minimum distance the values
d = a - lei *3.7
The Order o f Magnitude o f Functions
Differences in the behavior of functions for large values of the argument, lead to the notion of the order of magnitude. Because of i ts great i mportance, this matter deserves a brief discussion here even though it is not directly connected with the idea of the integral or of the derivative. a.
The Concept of Order of Magnitude. The Simplest Cases
I f the variable x increases beyond all bounds, then, for oc > 0, the functions x•, log x, ex , e•x also increase beyond all bounds. They increase, however. in essentially different ways. For example, the function x3 becomes "infinite to a higher order" than x2 ; by this we mean : as x i ncreases, the quotient x3/x2 itself increases beyond all bounds. Similarly, the function x• becomes infinite to a higher order than xP i f oc > f1 > 0, etc. Quite generally, we shall say of two functions f(x) and g(x), whose absolute values increase with x beyond all bounds, that f(x) becomes infinite of a higher order than g (x) if for x __,. w the quotient lf(.r)/g(x)l increases beyond all bounds ; we shall say thatf(x) becomes infinite of a lower order than g(x) if the quotient lf(x)/g(x) l tends to zero as x increases ; and we shall say that the two functions become infinite of the same order of magnitude if as x i ncreases, the quotient lf(x)/g(x)l
Sec. 3 . 7
The Order of Magnitude of Functions 249
possesses a limit different from zero or at least remains between two fixed positive bounds. For example, the function ax3 + bx2 + c = f(x), where a � 0, will be of the same order of magnitude as the function x3 = g(x) ; for the quotient lf(x)/g(x) l = i(ax3 + bx2 + c)fx3 i has the limit Ia I as x _.. oo ; on the other hand the function x3 + x + I becomes infinite of a higher order of magnitude than the function x2 + x + I . A sum of two functions f ( x) and (x), where f (x) i s of higher order of magnitude than (x), has the same order of magnitude as f(:c). For 1 (/(x) + (x))/f(x) i = I I + (x)/f(x) l , and by hypothesis this expres sion tends to one as x i ncreases. h. The Order of Magnitude of the Exponential Function and of the Logarithm
We might be tempted to measure the order of magnitude of functions by a scale, assign ing to the quantity x the order of magnitude one and to the power x• ( IX > 0) the order of magnitude IX. A p olynomial of the nth degree then obviously would have the order of magnitude n ; a rational function, the degree of whose numerator is higher by h than that of the denominator, wou ld have the order of magnitude h. It turns out, however, that any attempt to describe the order of magnitude of arbitrary functions by the foregoing scale must fail . For there are functions that become infinite o f higher order than the power x• of x, no matter how large IX i s chosen ; again, there are functions wh ich become infinite of lower order than the power x•, no matter how small the positive number IX is chosen. These functions therefore will not fit in our scale. Without entering i nto a detailed theory we state the following theorem. THEOREM. If a is an arbitrary number greater than one, then the quotient axfx tends to infinity as x increases. PROOF. To prove this we construct the function ( x)
=
az lo g - = x log a - log x ; X
i t is obviously sufficient to show that (x) i ncreases beyond all bounds if x tends to + oo . For this p urpose we consider the derivative '(x)
=
lo g a
-! X
250 The Techniques of Calculus
Ch. 3
and notice that for x � c = 2/log a this is not less than the positive number l log a. Hence it follows that for x � c ,P(x) - (c) =
f
'(t) dt �
f!
log
a
dt � Hx - c) tog a,
,P(x) � (c) + Hx - c) log a ,
and the right-hand side becomes infinite for x --+ oo . We give a second proof of this important theorem : with ../� = b = I + h, we have b > I and h > 0. Let n be the integer such that n :5: x < n + I ; we may take x > I, so that n � I . Applying the lemma of p. 64, we have
J2 x
=
b�
.jx
=
(I
1
h nh nh +_h )" > (1 + t > + > .jx .jn + .jn + I .j2n
so that
1
=
h_ .jn , ../2
and therefore tends to infinity with x. From the fact just proved many others follow. For example : for every positive i ndex oc and every number a > I the quotient a•fx• tends to infinity as x increases ; that is, THEOREM. The exponentialfunction becomes infinite of a higher order of magnitude than any power of x.
For the proof we need show only that the octh root of the expression, that is, x
oc xfoc
oc
y
tends to i nfinity. This, however, follows immediately from the pre ceding theorem when x is replaced by y = xfoc. In a similar fashion we prove the following theorem. For every positive value of oc the quotient (log x)fx" tends to zero for x --+ oo ; that is
THEOREM. The logarithm becomes infinite of a lower order of magnitude than any arbitrarily small positive power of x. PROOF. The proof follows i mmediately if we put log x = y so that our quotient is transformed into yfe"•. We then put e• = a ; then
Sec. 3.7
The Order
of Magnitude of Functions 251
a > 1, and our quotient yfa• approaches zero as y tends to infinity. Since y approaches infinity as x does, our theorem is proved. 1 On the basis of these results we can construct functions of an order of magnitude far higher than that of the exponential function and other functions of an order of magnitude far l ower than that of the l ogarithm. For example, the function e 1 •'> is of a higher order than the exponential function, and the function log l og x is of a lower order than the loga rithm ; moreover we can i terate these processes as often as we like, piling up the symbols e or log to any extent we please. All the functions x, log x, log (log x), log [log (log x) ] , etc., eventually become arbitrarily large for sufficiently large x, but with increasing slowness. Taking, for example, for x the tremendous number x = 1 0100 we find that l og x is about 230, whereas log (log x) is only about 5.4. c.
General Remarks
These considerations show that it is not possible to assign to all functions defini te n umbers as orders of magnitude so that of two functions the one with the higher order of magnitude has a higher number . I f, for example, the function x is of the order of magnitude one and the functi on xH• of the order of magnitude I + • , then the function x log x must be of an order of magnitude that is greater than one and less than I + • no matter how small • is chosen. But there is no such number. In addition, it is easy to see that functions need not possess a clearly defined relative order of magnitude at all. For example, the function (x2 (sin x) 2 + x + l )/ [x2(cos x) 2 + x) approaches no definite l imits as x i ncreases ; on the contrary, for x = mr (where n is an integer) the value i s 1 /mr, whereas for x = (n + !}77 i t is (n + i)7T + I + l /(n + })77. Although the n umerator and denominator both become infinite, the quotient neither remains between positive bounds nor tends to zero nor tends to i nfinity. The n umerator, therefore, is neither of the same order as the denominator, nor of lower order nor of higher order. This apparently startling situation merely means that our definitions are not designed in such a way that we can compare every pair of functions. This is not a defect ; we have no desire to compare the orders of such 1 Another simple proof may be suggested : For x > I and • > 0
i"'dl; i "'
• I I; 1 if we choose < equal to "' and divide both members of this inequality by x•, then it follows that (log x)fx• ..... 0 as x ..... oo .
log x =
- <
1
l;<-1 d/; = - (x< - l ) ;
252
The Techniques of Calculus
Ch. J
functions as the numerator and denominator above ; knowledge of the value of one of them gives us no useful information about the other. d. The Order of Magnitude of a Function in the Neighborhood of an Arbitrary Point
Just as we may compare the behavior of functions for x --+ oo we may also compare functions that become infinite at the finite point X = �. We say that the function f(x) = ! f i x - �I becomes i nfinite of the first order at the point x = �, and correspondingly that the function ! f ix - ��· becomes infinite of the order t:1., provided that t:1. is positive. We recognize then that the function e 1il•- il becomes for x --.. � infinite of higher order and the function log l x - �I i nfinite of lower order than all these powers ; that is, that the limiting relations 1 lim ( l x - w e 1 1 "- '1 ) = 0
z� �
and
00
lim (lx - ��· log l x - m = 0
�4�
0
hold. To confirm this we merely put ! f i x - �I = y; our statements then reduce to the known theorem on p. 249 , since l x - W e11 1 x-- < l ·
=
!_ y•
and
l x - �� · · log l x - �I = -
log y y•
and y increases beyond all bounds as x tends to �- (The method of reducing the behavior at a point � to the behavior at infinity by the substitution ! fix :.... �I = y frequently proves useful.) e. The Order of Magnitude (or Smallness) of a Function Tending to Zero
Just as we seek to describe the approach of a function to infinity by means of the concept of order of magnitude, we may also specify the way in which a function approaches zero. We say that as x --.. oo the quantity ! fx vanishes to the first order, the quantity x-•, where t:1. is positive, to the order t:1.. We find once again that the function ! flog x vanishes to a lower order than an arbitrary power x-•, that is, for every positive t:1. the relation lim (x-• · log x) = 0 holds. In the same way we say that for x = � the quantity x � vanishes to the first order, the quantity lx ��· to the order t:1.. Wi th our results -
-
The Ordn of Magnitude of Functions 253
Sec. 3. 7
it is easy to prove the relations lim ( l x l " · log I x i ) = 0,
lim ( l xl- • . e-lll zl ) = 0,
x �o
x �o
which are usually expressed as follows :
The function 1 /log lxl vanishes as x --+ 0 to a lower order than any power of x ; the exponential function r 1 11r l vanishes to a higher order than any power of x. f. The " 0 " and "o" Notation for Orders of Magnitude
A convenient way to i ndicate that a function f(x) i s of lower order of magnitude than a function g(x) is to write f = o(g). This symbolic equation signifies only that the quotient f/g has the l imit zero, and can be used to equal advantage for functions vanishing or becoming infinite and for arguments x tending to infinity or approaching a value �.1 We can rewrite many of the results of the previous section in this notation ; for example, x•
=
log x
=
o(xP)
o (xa)
e-"' = o (x- • ) e-lf x = o (x")
for ex. < {3
as x --+ oo
for ex. > 0
as x --+
as x --+ oo through positive values
as x --->- 0
log l x l
=
o(l (x)
as x ___.. 0
x
=
o(x)
as x
1 - cos
oo
___..
0.
This notation, introduced by E . Landau, is useful for indicating the order of magnitude of the error in an approximation formula. For example,
----=='== - _.!_ +
J 1 + 4x2
2x
0
stands for the relation lim
z � ro
(�)
for x ___.. oo
x
2x 2 __ J..;:1_+ ..:. .;__4_x_ l (x
_
=
0.
1 The letter o is chosen to suggest the word "order." Observe that the relation f = o(g) for vanishing g means that f vanishes of higher order.
254 The Techniques
of Calculus
Ch. 3
Similarly, the relation between increment and differential of a function / which has a derivative at the point x can be written in the form f(x + h) - f(x) = hf'(x) + o(h)
for h
-..
0.
Equally useful is the symbolic notation f = O(g) to indicate that f(x) is at most of the order of magnitude of g (x) that is, that the quotient f(x)ig(x) is bounded for the values of x in question.1 Use of the symbol 0 is again very flexible. Thus the phrase "/ = O(g) for x oo" means that the quotient fig is bounded for all sufficiently large x as in ,
._
for x -..
oo .
Similarly, ''/ = O(g) for x -.. ; ' means that fig is bounded in a suffi ciently small neighborhood of the point x = ; as in '
e' - I = O(x)
for x - o.
More generally we can use the equation / = O(g) to indicate the bound edness offig in any domain of the x-axis without requiring x to approach a limit. Thus log x = O(x) for x > J , x
=
O(sin x)
for [ x[ < :!: . 2
Some of the earlier examples involving the symbol o can now be refined to indicate a better estimate of the error with the help of the symbol 0 . Thus we have for a function / for which /" is defined and continuous
f(x + h) - f(x)
Other examples are
=
hf'(x) + O(h')
cos x = 1 + O ( x2)
for h ._ 0 .
for all x.
The same notations can be used for sequences a n , letting the i ndex n tend to infinity. We shall meet some i nteresting examples of such "asymptotic" formulas with an error term of higher order in the sequel (cf. Stirling's formula for n ! on p. 504). A famous asymptotic law,' 1
Notice that ( = O(g) does not mean that f/g has the limit one or that the quotient
in t'tis book. See A.
necessarily has any limit at all . • The proof cannot be given
Primes,
Cambridge University Press,
1 932.
E. I n gham,
The
Distribution of
Sec. A . !
Some Special Functions 255
already mentioned in Chapter I , p. 56 states that the number 1r(n) of primes Tess than n is given approximately by n/(log n). Here the order of magnitude of the error also has been found and we have more precisely the result n 1T(n) = __ + log· n log n
o(+).
Appendix
The difficulty in appreciating a rigorous development of calculus stems from a basic dilemma : Although the fundamental concepts and procedures, such as continuity, smoothness, etc., are motivated by compelling intuitive need s, they must be made precise in order to have any l ogical meaning, and the resulting rigoro us definitions may cover phenomena beyond those of intuitive character. Thus the rigorous concept of continuity inevitably requires a degree of abstraction not completely reflected in the naive notion of a connected curve, and the concept of differentiability i s more restrictive and more abstract than the vague idea of smoothness of a curve suggests. Discrepancies of this sort are not avoidable and m ay tax the patience and understanding of a begin ner or of someone for whom l ogical finesse is not of primary interest. Nevertheless, we want to make the need for precision clearer to the reader by showing that, perhaps unexpectedly, precision and refinement are called for even by simple and intuitively comprehensible examples. A.l
Some Special Functions
As a rule such examples need not be given in terms of single analytical expressions (see Figs. 2.28, p. 38 and 1 . 30 , p. 39). Here, however, we wish to represent various typical discontinuities and "abnormal" or unexpected phenomena by very simple expressions constructed from the elementary functions. We begin with an example in which n o discontinuity is present. a . The
Function
y = e- l l x'
This function (cf. Fig. 3.22) is defined in the first instance only for values of x other than zero, and obviously has the limit zero as x 0. For by the transformatio n J fx2 = g our function becomes y = e-< and __..
256 The Techniques of Calculus
Ch. 3
lim e-< = 0. Hence it is natural to extend our function so that it is
continuous for x = 0 by defining the value of the function at the point x = 0 as y(O) = 0. By the chain rule the derivative of our function for x ;e 0 is y' = - (2/x3)r1 1'' = 2�'l->e-<. If x tend s to zero, this derivative also has the limit zero, as we find immediately from p. 250. At the point x = 0 itself the de r i vati ve ' ' y( h) y(O) ' y (0) = I 1m = I tm h� o h�o h h ,_ .,
e-tth'
-
--
can also be continu ou s l y defined as zero. y
Figure
3.22
For the higher derivatives when x ;tf 0, we obviously always obtain the product of the function e-11•' and a polynomial in I jx, and the passage to the limit x ---+ 0 always yields the l i mit zero. Hence all the h i gher derivatives vanish, l i ke y', at the point x = 0. Thus our function is continuous everywhere and differentiable as many ti mes as we please, and yet at the point x = 0 it vanishes with all its derivatives and yet does not vanish identically. We shall later realize (Appendix T . l in Chapter 5) how remarkable or "abnormal " this behavior is. b. The Function y
=
e- 1 /x
As easily seen, for positive values of x this function behaves in the same way as the functi on just dealt with ; if x tends to zero from one side, through positive values, the function tends to zero, and the same i s true of all its derivatives. I f w e define the value o f the functio n a t x = 0 as y(O) = 0 , a l l the right-hand derivatives a t the point x = 0 h ave the value zero. It is quite ar.other matter when x tends to zero thro u gh negative values ; for then the function and all its derivatives become infinite, and left-hand derivatives at the point x = 0 do not exist. At
Sec. A. I
Some Special Functions 257
the point x = 0, therefore, the function has a remarkable sort of discontinuity, quite unlike the infinite discontinuities of a rational functions considered on pp. 36, 37 (cf. Fig. 3.23). y
0 Figure
c.
.
The Functwn
y =
I
3.23
tanh -
.r
As already seen on p. 65, functions with "ju mp" discontinuities can be obtai ned from simple functions by a passage to the limit. The exponential function defined on p. 1 5 1 together with the principle of compounding of functions give us another method for constructing functions with such disconti nuities from elementary fu nctions, without any further limiting process. An example of this is the function y =
1 e lf x _ e-l/ x tanh - = ---x elfx + e-l/x
and its behavior at the point x = 0. The function is in the first instance not defined at this point. If we approach the point x = 0 through positive values of x, we obviously obtain the limit I ; if, on the other hand , v.e approach the point x = 0 through negative values, we obtain the limit - I . This point x = 0 is thus a point of j ump dis continuity ; as x increases through 0 the value of the function j umps by 2 (cf. Fig. 3 .24). On the other hand, the derivative ' y = -
1 1 cosh 2 ( 1 /x) x2 4 1 x2 ( elfx + e- l/ xl
258
The Techniques of Calculus
Ch. J y
--------0�-- x
Figure 3.24
approaches the limit zero from both sides, as follows readily from 1 Section 3.76, p. 249. d. The Function y = x tanh l jx
I n the case of the function
e- 1 /X e1/ X 1 y = x tanh - = x -:-:----..,..,lfx e x + e-1/x _
the preceding discontinuity is removed by the factor x. This function has the limit zero as x -.. 0 from either side, so that we can again appropriately define y(O) as equal to zero. Our function is then con tinuous at x = 0, but its first derivative
1 y ' = tan h x
-
1 - ---,---x cosh2 (1 /x)
has just the same kind of discontinuity as the preceding example. The graph of the function is a curve with a corner (cf. Fig. 3.25) ; at t he point x = 0 the function has no actual derivative but a right-hand derivative with the value + I and a left-hand derivative with the value -I. 1 Another example of the occurrence of a "jump" discontinuity is given by the func
tion y = arc tan
I fx as
x -+
0.
Sec. A .l
Remarks on Dif!erentiabiliry 259
y
-1
0
Figure
e. The Function
y =
3.25
x sin ! Jx, y(O)
=
0
We have already seen that this function is not composed of a finite number of monotonic pieces-as we may say, it is n ot "sectionally" or "piecewise" monotonic-but that it is nevertheless continuous (p. 40 and Fig. 1 .3 1 ). Its first derivative y
,
=
. I 1 I sm - - - cos X
X
X
(x
¢ 0),
on the contrary, has a disconti nuity at x = 0; for as x tends to zero this derivative oscillates continually between bounding curves, one positive and one negative, which themselves tend to + oo and - oo respectively. At the actual point x = 0 the difference quotient i s [y(h) - y(O) ]/h = sin ( 1 /h ) ; since this expression swings backward and forward between I and I an infinite number of times as h 0, the function possesses neither a right-hand deri vative nor a left-hand derivative at x = 0. -
A.2
_.,
Remarks on the Differentiability of Functions
The derivative of a function which is continuous and has a derivative at every point of an i n terval need not be continuous. As a simple example we consider the function given by y
= f(x) = x2 sin
and j(O) = 0.
! X
for
x ¢
0
Ch. J
The Techniques of Calculus
260
This function is defined and continuous everywhere. For all values of x different from zero the derivative is given by the expression
!'( X)
(
2 = - X COS
-X.) -X2 + 2 1
1
.
X Sin
1
-X
= - CO S
1
-X + 2
1
.
X Sin - .
X
When x tends to zero, f'(x) has no limit. If, on the other hand, we form the difference quotient [f(h) - f(O)]/h = (h 2 sin 1 /h)/h = h sin l jh, y
'
'
/
'
/
'
/
'
/
'
/
'
/
'
/
'
/
'
/
'
/
'
/
'
/
Y = x2 // / '
/
/
/
/
/
/
/
/
.....
/
��
'
'
'
y=
'
'
'
-x2 ',
'
'
'
'
'
'
'
Figure 3.26
we see at once that this tends to zero as h does. The derivative therefore exists for x = 0 and has the value 0. To grasp i ntuitively the reason for this paradoxical behavior we represent the function graphically (cf. Fig. 3. 2 6). It oscillates between the curves y = x2 and y = - x2 , which it touches alternately. Thus the ratio of the heights of the wavecrests of our curve and their distances from the origin steadily becomes smaller. Yet these waves do not become flatter, for their slope, given by the derivative f'(x)
=
2x sin 1 /x - cos 1 /x,
Part B
Techniques of Jnt�gration 261
is equal to - 1 at the points x = l f2mr where cos l fx = l, and to + I at the points x = l /(2n + l )1r where cos l fx = - I . I n contrast to the possi bility illustrated here (that a derivative exist everywhere and yet not be continuous) we state the following simple theorem , which throws light on a whole series of earlier examples and discussions.
THEOREM. If we know that in a neighborhood of a point x = a, the function f(:-c) is continuous, and that for x ¢ a it also has a derivative .f'(:r) and if in addition the equation lim f'(�:) = b holds, then the derivative ['(;r) exists at the x� a
point a also, andj'(a) = b. PROOF. The proof follows immediately from the mean value theorem. For we have [/(a + h ) - f(a)]/h = {'(�), where � is a value intermediate between a and a + h . If h now tends to zero by hypothesisf'W tends to b, and our statement follows at once. A companion theorem may be proved in a similar way : lf the function ((�") is continuous in a s ;r s b and for a < x < b possesses a derivative which increases beyond all bounds as x tends to a, the right-hand difference quotient [/(a + h) - f(a)]/h also increases beyond all bounds as h tends to zero, so that no finite right-hand derivati ve exists at x = a. The geometrical meaning of this statement is that at the point with the (finite) coordinates [a,f(a)] the curve has a vertical tangent.
Part B Techniques of Integration Explicit Functions
A wide class of functions can be constructed from the elementary functions1 by repeated rational operations, that is, addition, m ulti plication, d ivision, and furthermore by the operations of forming inverse functions and of compounding functions. The functions thus descri bed form the class of "explicit"functions or "closed expressions."2
As a result of Part A of this chapter we state the rather general fact : E1•ery explicit function can be differentiated and its deriuative is again an explicit function.
Thus we have attained a fairly complete mastery of the operation or the "algorithm" of differentiation. Yet, the inverse process, that of
I t should be emphasized that the distinction between "elementary " and "explicit" functions and others is in itself somewhat arbitrary. For us the term "elementary " function includes just the rational functions, the trigonometric and exponential functions, and their inverses. • This name indicates that we shall encounter many other functions which cannot be represented in this fashion but which can be constructed by means of limiting processes such as infinite series. 1
262
The Techniques of Calculus
Ch. 3
integration, is generally speaking more important and presents the major challenge. To a certain extent the challenge is met by the fundamental theorem of calculus : To every formula of differentiation F '(x) = f(x) there corresponds an equivalent formula for the primitive functions F(x) to f(x) or the integral :
(
J
f(x) dx = F(x).
More precisely we have F(x)
= J.t(u) du
)
+ constant .
Thus as
more explicit formulas of differentiation are derived, addi tional explicit functions can be integrated in terms of explicit functions. A first table of integrals is listed on p. 264 ; in principle, it would not be difficult, although i mpractical and confusing, to extend such a table very much . In the early phases of the development of calculus many mathema ticians tried to find, i n explicit or closed form, the integral or primitive function for every explicitly given function. I t took some time before i t was realized that i n principle this problem cannot be solved ; on the contrary, for some quite elementary inte grands the integral j ust cannot be expressed i n terms of elementary functions (see p . 298). Thus the need for studying new types of functions generated by integration processes from elementary functions became an i mportant stimulus for the development of analysis. Nevertheless the desire to integrate-when feasible-given explicit functions explicitly without getting hopelessly entangled in tedious consultation of tables or numerical computations has led to some simple devices which provide a certain flexibility for transforming given integrals ; in fact, these devices permit us to carry out the integration by reduction to one of the elementary integrals in the Table of Integrals. Section 3.9 will be devoted to the development of such useful devices. I n this connection the beginner should be cautioned against merely memorizing the many formulas obtained by using these technical devices. The student should instead direct his efforts toward gaining a clear understanding of the methods of integration and learning how to apply them. M oreover, he should remember that even when inte gration by these devices is impossible, the integral does exist (at least for all continuous functions), and can actually be calculated to as high a degree of accuracy as is desired by means of numerical methods which will be further developed later (Section 6 . 1). I n Part C of this chapter we shall endeavor to extend our conceptions of i ntegration and integral, quite apart from the problem of the tech nique of integration.
Table of Integrals 263
Sec. 3.8
F'(.r) I.
xa
= j(x)
Table of Elementary Integrals
=
F(x) a + I
(a � - 1 )
I 2. X
fJ(x) dx
l xl
log
3 . e" 4 . ax (a � I )
log
5 . sin x
6. COS X
I ( = cosec2 x) 7. sin z ,, 1 8 · -- ( = sec2 :r ) cos2 x
9. sinh x 1 0 . cosh x I 11. ( = cosech 2 x) sinh 2 .., I ( = sech2 a:) 1 2 · -cosh2 x
1 3. 1 4.
1
(l.rl < I )
--=== vi a·z
-
I I +
x2
1 1 5. --= v i + :r2 1 6.
(lxl
± v x2 - 1
1 1 7. 1 - x2
a
-cos x
{ l:rl
< 1
lxl > I
> I)
sm x -cot x tan
x
cosh
.r
sinh :r
- coth x
tanh x
Jarc sin x l - arc cos .r
{
arc tan .r - arc cot .r
log (x + v 1 + x2)
ar sinh x
=
ar cosh .1:
= l og (x ± Y :t:2 - I )
I 1 +X ar tanh x = 2 log 1 :c ar coth
x
= 2 log
1
_
X + I x
_
1
264 The Techniques of Calculus
3.8
Ch. 3
Table of Elementary Integrals
To each of the differentiation formulas proved earlier there corre sponds an equivalent integration formula. Since these elementary i ntegrals are used time and again as materials for the art of integration, we collect them in a Table. The right-hand column con tains a number of elementary functions and the left-hand column the corresponding derivatives. If we read the table from left to right, we obtain in the right-hand column an indefinite integral of the function in the left-hand column. We also remind the reader of the fundamental theorems of the differential and integral calculus, proved in Section 2.9, i n particular, of the fact that any definite integral is obtained from the indefinite integral by the formula1
F(x)
ff(x) dx
= F(x)
1:
= F(b)
-
F(a).
In the following sections we shall attempt to reduce the calculation of integrals of given functions in some way or other to the elementary integrals collected in our Table. Apart from special artifices which are learned only from experience, this reduction is based essentially on two useful methods : "substitution" and "integration by parts." Each of these meth ods enables us to transform a given integral in many ways ; the object of such transformations mostly is to reduce the given integral, in one step or in a sequence of steps, to one or more of the elementary integration formulas given above. 3.9
The Method of Substitution
Integrating Compound Functions The first of these methods is the introduction of a new variable (that is, the method of substitution or transformation). It aims at reducing the integration of composite functions-such as functions of c or of ax + b to that of simpler functions.
x
-
-
a. The Substitution Formula. Integral of a Composite Function
The rule for integrating composite functions follows from the corre sponding chain rule for differentiation. For a composite function
1 We shall not discuss in this chapter the somewhat different problem of calculating
special definite integrals witho\.lt first finding a general primitive function.
Sec. 3.9
G(u)
=
( 1 6)
The Method of Substitution 265
F
[c/>(u) ] we have (see p. 2 1 8) dG(u) dF[c/>( u)] F '[ c/>( u) ] c/>' ( u). du du =
=
cf>( ) u,
It is sufficient for the validity of this formula that the functions x = u and are continuously differentiable i n their arguments x respectively, and that F(x) is defined for the values x assumed by the function = u (that is, the range of the function cf> m ust belong to the domain of F). I n tegrating the formula between the limits u = <1. and u = {i, we find
F(x) x c/>( )
G ({i) - G(<1.) F[ c/>(/3)] - F [c/>(<1.) ] rF '[cf>(u ) ]c/>'(u) du. c/>(fi) b , c/>( rJ.) we have F[c/>(/3) ] - F[c/>(<1.) ] F(b ) - F(a ) ),.f!' F'(:c) dx. Setting F'(x) f(x) we obtain the basic substitution formula ' ) dx (P. f[c/> ( u ) ] c/>' (u) du, {'J(x ( 1 8) x cf>( u ) J J =
=
( 1 7)
I f here
=
=
= a,
=
=
=
=
or, written suggestively in Leibnitz's notation with the differen tial de/> = c/>'(x) dx,
J
( 1 8 a)
f(x) dx
=
Jf(c/>)
dcf>.
Here = c/>(u ) may be any function which is defined and has a con tinuous derivative i n the interval J with end points <1. and /3; it maps those end points i nto = a and = b respectively ; the function f (x) is assumed to be continuous in an interval / containing the images of all poi nts of J under the mapping For F(x) we can take any primitive fu nction off(x). As should be noticed the substitution rule ( 1 8) does not require that the mapping = cf>(u) map points between <1. and fi only on points between a and b or that different values u are mapped into different all that matters i s that <1. and fi are mapped into a and b and that f(x) is defined for the val ues taken by cf>(u) for u between <1. and fl. I n terms of i ndefinite integrals the substitution rule takes the form
x
x
x cf>.
x
x;
x
( 1 9)
G( u ) Jf[cf>(u)]c/>'( u ) du Jf(x) dx F( x) F[c/>( u )] . =
=
=
=
The Techniques of Calculus
266
Ch. 3
The differential symbols
cf/(u) du =
dx du du
dx
and
become identical if we formally cancel the symbols du in the numerator and denominator.
Examples. We apply formula ( 1 8) to the integrand f(x) = 1 /x and make the substitution x = c/>(u), assuming c/>(u) yt:. 0 in the interval considered ; then =
- =
J c/>cf>'((u)u) du J dxx -
log lxl = log l c/>(u)l
or c h a n gi n g the name of the variable
J c/>c/>(x)'(x) dx
(20)
u
aga i n into x,
= log l cf>(x) l .
If in th i s i mportant formula we su b sti t u te particular functions, such as cf>(x) = log x, cf>(x) = s i n x, or cf>(x) = cos x, we obtain1
(2 1 )
J
J
dx
X log X
-- =
cot x dx = log I sin
xl ,
Fur/her Examples.
J
¢(u)¢ ' (u) du
where f(.t")
= x.
We finally consider
.r
J
x
This yields for ¢(u)
(22)
Here
=
J
log u
11-
J
du
log I log
J
d.r
=
=
tan
=
xl,
x dx
! ,�2
=
= - log I eos
xl.
t[¢(u)]2,
log u i (log u)2•
sin" u cos u du.
= sin u = ¢ (u), and hence
J
sin" u cos u du =
J
x" dx
;cn+l sin"+I u = -- = -- . n + i n + i
1 These and the following formulas are easily verified by showing that differentiation of the result gives us back the integrand .
Sec. 3.9
The Method of Substitution 267
The same substitution .r = sin u gives for any function j(x) continuous in the interval - I ::;; ·" ::;; I 11 sin � j(l') d.r. j(sin u) cos 11 du =
J
i
Taking here :x = 0 and {J = 2, gives us an example for applying the substitu tion formula to a case where the mapping function x = rf>(u) = sin u = x is not monotonic throughout the interval :x ::;; u ::;; {J. We find �in
:x
i�w
j(si n u) cos u du
Other Forms of the Rule
=
io
:x
j(.t) dJ·
= 0.
I n many applications the integral to be eval uated is given i n the form
f
F(u) = h (c/>(u ) ] du
in which the integrand appears as the composite function h [c/>(u) ) without the additional factor c/> (u) We can apply the substitution rule ( 1 8) if we succeed i n writing the i ntegrand h (c/>(u) ] in the formJ(c/>(u)]c/>'(u). This can always be achieved under the assumption that the function x = cf>(u) has a continuous derivative cf>'(u) which does not vanish. For then there exists an inverse function u = 1p(x) with a continuous derivative dufdx = 1p'(x) = 1 /c/>'(u). Taking for f(x) the function h(x)1p'(x) we have indeed h [ cf>(u)] = J [c/>(u)J/v/(x) = f (c/>(u) ]c/> '(u) and we obtain from the substitution rule '
(23)
f
h [cf>(u) ] du
=
f f
.
f
f (cf>( u)]c/>' (u ) du = j(x) dx
f
= h(x)!p'(x) dx = h(x) '!!!. dx. dx The assumption cf>'(u) � 0 has been introduced in order to prevent the expression dx/du in formula (23) from becoming i nfinite. The beginner must never forget that in substituting u for 1p(x) i n an i ntegral one must not merely express the old variable x i n terms of the new one, u , and then integrate with respect to this new variable ; instead, before integrating one must multiply by the derivative of the original variable x with respect to the new variable u. This, of course, i s dx suggested by Leibnitz' notation h dx = h d du. I n the definite integral
r
h [ 1p(x)] d x =
r
u
h (u) c/> (u ) du '
we must not forget to change the limits a, b for x into the corresponding limits (1. = 1p(a) and fJ = 1p(b) for the variable u.
268 The Techniques of Calculus
Examples.
In order to calculate
h(u) = sin u. We have
Ch. J
J sin 2x dx we
du dx dx = IJ!'(x) = 2• du
=
2I
take u = 'P(x) = 2x and ·
If we now introduce u = 2x i nto the integral as the new variable, then it is transformed, not into J sin u du but into
�2
J
sin u du = - � cos u = - � cos 2x '· 2 2
this may, of course, be verified at once by differentiating the right-hand side. If we integrate with respect to x between the limits zero and 7T/ 4 , the cor responding limits for u = 2x are zero and 1rj2 and we obtain
i
•/4
0
sin 2x dx = I sin u du = 2 0 -
i"'" i� 2
Another simple example is the integral
v;, from which x = cf>(u) = u
r· dx_
VX
J,
=
•
-
I 2
- cos
u
1 " '2 0
=
I 2
- .
dx ---:. . Here we take u Vx
Since cf>'(u) = 2u, we have 1
I" 2 " du = z I" du U
J,
)1
=
=
v{x)
=
2.
As another example w e consider the integral o f s i n 1 /x for the interval = 1 /x or x = 1 /u, dx = - duf u2, and hence
i :::; "' :::; I . We have for u I
1'
A
sin - d.r .c
= -
l' 2
sin u -2-
u
du =
i2 I
sin u
-2-
u
du.
*b. An Alternative Derivation of the Substitution Formula
Our i ntegration formula ( 1 7) with a s lig h t change of notation can also be i n terpre t ed i n a direct manner, based on the meaning of the definite i ntegral as a l imit of a sum instead of b e ing deduced from the chain rule of differentiation.' To calculate the integral
fh [!p(x)] dx
(for the case a < b), we begin with an a rbi trary subdivision of the interval a � x � b, and then make the su bdi v is i o n finer and finer. We choose these subdivisions in the fol l ow i ng way. If the function u = 1p(x) is assumed to be monotonic i ncreasing, there is a one-to-one correspondence between the interval a � x � b on the x-axis and an 1 The
g gene ral than formula ( 1 8) furnished by the chain rule (on p. 265).
result obtained i n this w�y is a ain restricted to monoton ic subst itutions and
thus is less
Sec. 3.9
The Method of Substitution 269
interval rx � u � f3 of the values of u = 1p(x), where rx = 1p(a) and f3 = 1p(b). We divide this x-interval i nto n parts of length' �x ; there is a corresponding subdivision of the u-interval into subintervals which, i n general, are not all of the same length . We denote the points of division of the x-interval by and the lengths of the corresponding u-cells by The
integral we are considering is then the limit of the sum n
L h { 1p(�,)} �x.
v- 1
where the val ue �. is arbitrari ly selected from the vth subinterval of the
� h(v,) ::, �u,,
x-subdivision. This sum we now write in the form '
'
where t', = 1p(� , ) . By the mean value theorem of the differential calculus �xf�u, = cp'( rJ,), where fJ, is a suitably chosen intermediate value of the variable u in the vth subinterval of the u-subdivision and x = cp(u) denotes the i nverse function of u = 1p(x). If we now select the value �. i n such a way that v , and f/ , coincide, that is, f/ , = 1p(�,) �. = cp( rJ,) , then our sum takes the form "
L h(rJ,) f(rJ,) � u , .
\' = ]
I f we here ma ke t h e passage t o the limit letting n -+ 00 ,2 we obtain the expression dx h( u) du du
fp •
as the limiting value, that is, as the value of the integ r al we are con sidering , in agreement with formula (23) given before. Thus we arrive at the following result. THEOREM.
Let h(u) be a continuous function of u in the interval = 1p(x) is continuous and monotonic and has a continuous nonvanishing derit•ative dufdx in a � x � b, and oc
� u � {3. Then if the function u
1 The assumption that the lengths of these subintervals are all equal is by no means essential for the proof. 2 This limit exists (for t.x -+ 0) and is the integral, since on account of the uniform continuity of u = 'l'(x) the greatest of the lengths t.u, tends to zero with t.x.
270
The Techniques of Calculus
lb
Ch. 3
tp(a) = rL and tp(b) = fJ, then h{ tp{x)} dx
=
lb
h(u) dx =
J
P
dx h(u) - du .
du This derivation exhibits the suggestive merit of Leibnitz's notation. I n order to carry out the substitution u = tp(x), we need only write (dxfdu) du in place of dx, changing the limits from the original values ofx to the corresponding values of u . a
c.
a
•
Examples. Integration Formulas
With the help of the substitution rule we can in many cases evaluate a given integral S f(x) dx if we reduce it by means of a suitable sub stitution x = cf>(u) to one of the elementary integrals in our Table. Whether such substitutions exist and how to find them are questions to which no general answer can be given ; this is rather a matter in h which practice and ingenuity i n contrast to systematic met ods , come into their own. dx As an example, we evaluate the integral 1 by means of the
f
,
,
'l a2 - x2 substitution 1 x = cf>(u) = au, u = tp(x) = xfa, dx = a du, by which, using No. 13 of our Table we obtain
f
f
(24)
. x . dx a du = arc sm = u = arc sm a ...;1 a 2 - x 2 2 J a 1 u
f 2 dx 2 = f 2( a1 du 2) = u x a a a f dx = xa )a2 x2 f --:::d=x== = a
,
for l xl < I a ! .
By the same substitution we similarly obtain (25)
(26) (27)
---
.
h
x 1 arc tan u = - arc tan a
- ,
a
ar sm - ,
+
J x2 - a'
ar cosh � ,
{l
(28)
1
-
+
+
for l x l > l a l ,
ar tanh �
for l xl < l a l ,
- ar coth
for l xl > la l ,
f a 2 � x2 ; =
a
:
a
For the sake of brevity we a;ain take the liberty of writing the symbols dx and du se parately, that is, dx = cp'(u) du instead of dx/du '(u) (cf. p . 1 80). 1
=
Sec. 3.10
Further Examples of the Substitution Methods 271
formulas which occur very frequently and which can easily be verified by differentiating the right-hand side. 3.10
Further Examples of the Substitution Method
In this section we collect a number of examples which the reader may consider carefully for practice. By the substitution u = I ± �'2, du = ± 2�, d.c, we deduce that
J
(29)
(30)
x d�· ---= = ± V I ± x2,
VI
J
±
_
.r2
-- = ± � log I I ±
x d.c
I ± ,.2
x21 .
In these formulas we must take either the plus sign in all three places or the minus sign in all three places. By the substitution u = + b, d.u = d.r (a "" 0), we obtain
(3 1 ) (32) (33 )
Jax
ax
d.c
J
J(a>'
+ b
l ax -1�
I = � log
+ b)• d.c = -
( + I)
a o::
sin (a.r + b) d.,· = -
a
+ bl ,
(al' + b)• f 1
( o::
"" - 1 ),
cos (a.r + b) ;
similarly, by means of the substitution u = cos J ' , du = -sin x d:r, we obtain
(34)
J
tan .r d.r = - log l cos :r l ,
a n d b y means of the substitution u = sin J', du = cos x d.r ,
J
cot .c dx = log I sin xl
(35)
[cf. (2 1 ) p. 266]. Using the analogous substitutions u = cosh '"· du = sinh x dx and u = sinh x, du = cosh .1' d.r, we obtain the formulas
(36) (37)
J J
tanh x dx = log cosh x, coth x d.r = log I sinh x l .
272
The Techniques of Calculus
By virtue of the substitution at the two form u la s (3 8 )
J
11
Ch. J
= (a/b) ta n x, d11 = (a/b) sec2 x dx, we arrive
J
dx I dx I = a2 sin 2 x + b2 cos2 x b2 (a2fb2) tan 2 x + I cos2 x
(� (�
an d
( 39)
J
b2 cos 2 :r
-
-;;b arc cot -
dx a2 sin 2 x
arc tan
I
=
We evaluate the integral
-
tan x
.
ta n x
ar tanh
'
tan x
·a
1
-
(� ) (� ) (� ) ( )
- - ar coth - tan :r ab b
f.!!:_
·
Sin .r
by writing sin x = 2 sin (:r / 2) cos (.r/2) = 2 tan (x / 2) cos2 (x/ 2), and putting 11 = tan (.r / 2), so that du = ! sec2 (.r / 2) d.r ; the integral then becomes (40) If we replace
(4 1 )
.r
J
-.- = Sin X d.t:
J
du U
- =
I
by .r + rr/2, this formula becomes
J
-- = log tan
d.r
COS X
.r l
log tan -
.
2
- +4 2
I ( ") I "'
.
The substitution u = 2x yields, if we also apply the known trigonometrical formulas 2 cos 2 x = I + cos 2.r and 2 sin 2 :r. = 1 - cos 2.r, the freq uently used formulas
J
(4 2)
and (4 3)
cos 2 :r. d.t· = � (x + s i n x cos :r)
J
sin2 ,. d.r = i(x - s i n x cos x).
By the substitution = cos 11, equivalent to generally, = a cos u (a ..= 0), we can reduce
x
x
f.Jt - x2 dx
and
u
=
arc cos
f.Ja2 - x2 dx
x, or, more
Furrlrer Examples of the Substitution Method 273
Sec. 3./0
respecti.vely to these formulas. We thus obtain
(44)
fJ
•
-2--2 a - x dx =
X ;---. X a2 - arc cos - + - ...; a 2 - x·. 2 a 2
-
Similarly, by the substitution x = a cosh u we obtain the formula
(45)
J
/-2--2
...; x
2 X X (-.-. a a dx = - - ar cosh - + - " x2 - a2 2 a 2
-
and by the substitution x = a sinh u
(46)
, --. f ; -.1 '-1 a · + x- dx
•
=
, --. • x x -.ar Sinh - + - y; a · + x2• 2 a 2
-
a
2
The substitution u = afx , dx = - ( aj u 2 ) du leads to the formulas
(47) (48) (49)
J J J
I . a = - - arc S in - , a x x J x 2 - a"
dx
dx --=== = xJ x2 + a 2
dx --=== = -x ,l a 2 - x2
-
a . 1 - ar stnh - , a x
I a - ar cosh - . x a
Finally, we consider the three integrals
J
sin mx sin nx dx,
Jsin mx cos nx dx, Jcos mx cos nx dx,
where m and n are positive integers. By well-known trigonometrical formulas we can divide each of these i ntegrals into two parts, writing sin mx sin nx = Hcos (m - n)x - cos (m + n)x], sin mx cos nx = ! [sin (m + n)x + sin (m - n)x],
cos mx cos nx = ! [cos (m + n)x + cos (m - n)x]. If we now make use of the substitutions u = (m + n)x and u = (m - n)x respectively, we obtain directly the following system of
274 The Techniques of Calculus
Ch. 3
formulas :
(50)
(! {si n (m - n)x _ sin (m + n)x} if m � n, 2 m-n ) m+n fsin mx sin nx dx = ! (x sin 2mx if m = n ; 2 2m (5 1 ) {- !{cos(m + n)x + cos(m - n)x} if m n 2 m + n) m-n sin mx cos nx dx f _ ! (cos 2mx if m = n; 2 2m _
�
=
(5 2)
{( !
sin (m + n)x sin (m - n)x} 2 m + n + m - n if m � n, cos mx cos nx dx = f ) sin 2mx -21 (--if m n. x + 2m If, in particular, we integrate from t o +1r, w e obtain from these formulas the extremely important relations if m n, 0 sin mx si n nx dx f+• { if m n, (5 3) J::si n mx cos nx dx = 0, +J •cos mx cos nx dx = {0 if m = n.n, =
- 7T
=
-·
�
7T
=
if m �
7T
-·
These are the orthogonality relations of the trigonometric functions, which we shall encounter again in Section 8.4e. 3. 1 1
Integration by Parts
a. General Formula
The second widely used method for dealing with integration problems expresses in integral form the rule for differentiating a product :
(/g) '
= f'g + fg ' .
Integration by Parts 275
Sec. 3.11
The corresponding integral formula is (cf. p. 1 89) .
or
(54)
J
J
f(x) g (x) = g(x)f ' (2:) dx + f ( x) g ' ( x) d x
J
J
f(x)g'(x) d:r = f(;r)g (x) - g(.r)f'(x) dx.
Using Leibnitz's differential notation, th is becomes
ff dg - fg df .
(54a)
= fg
This formula will b e referred t o a s the formula for integration by parts. lt reduces the calculation of one integral to the calculation of another integral . Since a given integrand can be regarded as a prod uctf(x)g'(x) in a great many different ways, this formula provides us with an effective tool for the transformation of i ntegrals. Written as a formula for definite integration, the formula for inte gration by parts is
(54 b )
rf(x)g'( x) dx = f( x)g(x) 1: -rg(x)f'(x) dx = f (b )g ( b ) - f( a )g ( a )
-f
g ( x)f'( x) d x .
This follows either directly by i n tegrating the formula for the derivative of a prod uct between the limits a and b or by formi ng the difference at the points b and a in formula ( 54). We c a n give a simple geometrical interpretation of formula (54b) : Let us suppose that y = f(x) and z = g(x) are monotonic, and that f(a) = A , f(b) = B, g(a) = r:t. , g( b) = fJ; we can then form the i nverse of the first function and substitute in the second equation, thus obtaining z as a function of y. We assume that this function is monotonic increas ing. Since dy = J'(x) dx and dz = g'(x) the formula for integration by parts can be written [cf. the substitution rule ( 1 8), p. 265] .
dx
f
y dz +
J�
z dy = B{J - Ar:t.,
i n agreement with the relation made clear by Fig. 3.27, area NQLK + area PML Q = area OMLK - area OPQN.
276
The Techniques of Calculus
a
Ch. 3
Q
N
p M ��------�--------------------��__. y
0
A _J
f------ B -----� Figure 3.27
The following example may serve as a first illustration :
J
J
log x dx = log x · I d.r .
We write the integrand in this way in order to indicate that we put f(x) = log x and g'(x) = I , so that we havef'(x) = l fx and g(x) = x. Our formula then becomes
(55)
J
log x dx = x log x
-J;,
dx = x log x -
x.
This last expression is therefore the i ndefinite integral of the logarithm, as may be verified at once by differentiation. b. Further Examples of Integration by Parts
With f(x) = x, g'(x) = e"', we have f'(x) = I , g(x) = ex, and
( 5 6) In a similar way we obtain
(57) and (58)
J J
x
J
x e"' dx = e"'(x
-
1 ).
sin x dx = -x cos x + sin x
X COS
X dx
=
X
Sin
X
+ COS X.
Sec. 3.JJ
For
J.(x)
= log
x, g'(x)
fx"
(59)
=
Integration by Parts 217 x",
we have the relation
log x dx =
a +
x<> + I
I
( - -)I log x
1 -
a +
Here we must assume a "F - I . For a = - I we obtain
I
I
- log x dx = (log x)2
X
-f
.
log x · - ; dx X
transferring the integral on the right-hand side over to the left, we have [cf. (22), p . 266]
f�
(60)
log x dx = ! (log :r)2
.
We calculate the integral f arc sin x d.r by takingf(.r) = arc sin Hence x dx are sin '" dJ: = .c arc sin x v' I J'2 .
-f -
f
:r,
g'(x) = I .
The i ntegration on the right-hand side can be performed as in (2 9), p. 27 1 ; we thus find (6 1 )
f
arc sin x d:x: =
x
In the same way we find
f
(62)
arc tan x dJ· =
x
-
arc sin ,, + V I
arc tan .r
-
x2.
� log (I + x2)
and many others of a similar type . The following examples are of a somewhat different nature ; here repeated integration by parts brings us back to the original integral, for which we thus obtain an equation. In this way we obtain
f
eax sin bx dx
=
=
-� -
eax cos bx
+
b ea• cos bx
+
I
� feax cos
bx dx
a
Solving this equation for the integral f eax sin b:r dx, (6 3 )
f
eax sin
bx dx =
a2
:
b2
In a similar way it follows that
(64)
fe"x
cos bx dx =
-1+ b2
a2
e'""(a sin bx
- f a2
b2 e'"" sin hx
b2 eax sin bx dx ;
-
ea•(a cos bx
b cos bx) .
+
b sin bx).
278
The Techniques of Calculus
Ch. 3
c. Integral Formula for f(b) + f (a) As a last example we derive a remarkable formula expressing the sum j(b) + [(a) as a definite integral (instead of the difference [(b) - [(a) given by the fundamental formula). Integration by parts will be applied by introducing I = g' (x), whereg(x) = x - m with a constant m at our disposal. Then we have for the i ndefinite integrals
Jj(x) J
dx + j'(x)(x - m) dx = j(x)(x - m)
and for the integral between
ib '
J(x) dx +
f'' a
a and b
{'(x) (.r - m) dx = j(b)(b - m) - f(a)(a - m).
If for arbitrary a and b we choose for m the mean value m between a and b, we obtain, as the reader will easily verify b -a [ + f(b )] = -2 f(a)
d. Recursive Formulas
i' a
j(x) d.r +
i" a
= (a
+ b) / 2 ,
(.r - m )/'(x) d.r .
I n many cases the integrand is not only a function of the i ndependent variable but also depends on an integer index n ; on integrating by parts we sometimes obtain, instead of the value of the i ntegral , another similar expression in which the i ndex n has a smaller value. We thus might arrive after a number of steps at an integral which we can deal with by means of the Table of Integrals, p. 263 . Such a process i s called recursive. The following examples are ill ustrations : By repeated integration by parts we can calculate the trigonometrical integrals
J
sin" x dx,
J
sin"' x cos" x dx,
provided that m and n are positive integers. For using [(x) = cos"- 1 x,
g(x) = sin x we find for the first integral that
J
cos" x dx
=
J
cos"-- 1 x sin x + (n - I) cos"- 2 x sin2 x dx ;
Sec. 3.11
Integration by Parts 279
the rig� t-hand side can be written in the form cos"- 1
x sin x + (n - 1) Jcos n- 2 x dx - (n - 1 ) Jcosn dx; x
thus a recursive relation is obtained :
(65)
Jcos" x dx = � cos"-1 x sin x + n � 1 Jcos"-2 x dx.
This formula enables us to diminish the i ndex i n the integrand step by step until we finally arrive at the i ntegral
Jcos x dx = sin x
or
depending on whether n i s odd or even. I n a similar way we obtain the analogous recursive formulas · n 1 sin"-2 sin" - sin"- 1 + ( 66) and
J x dx = �
(67)
J x x dx = sin"'
cos"
sin"'+ 1
�
x cos x
J
x dx
x cos"-1 x + -- I Jsin"' x cosn- 2 x dx.
m + n
n
m + n
I n particular, we calculate the integrals
and
Jsin 2 x dx = � (x - sin x cos x)
Jcos2 x dx = �(x + sin x cos x),
as we h ave already done by the method of substitution [Eqs. (42), (43), p. 272]. It need hardly be mentioned that the corresponding integrals for the hyperbolic functions can be calculated in exactly the same way :
(68) (69)
Jsinh2 x dx = � ( -x + sinh x cosh ) Jcosh 2 x dx = � (x + sinh x cosh x).
x ,
280 The Techniques of Calculus
Ch. 3
Further recursive formu las are given by the following transformations :
I I fr"'
(log x)'" dx = .r(log x)m -
(70) (7 1 )
x'"e" d:c = x"'e" - m
(7 2 ) (73)
f
:v'"
m
I
I
(log x)m-t dx,
xm-le• dx,
sin :r d.r = _ ,.m cos x +
cos x dx = ,, m sin :r -
m
(74)
m
I
I
xm-l
x m-t
cos x dx,
sin :r d.r,
- __!!!_ a + I
J
x•(log x)m-t dx (a '# - 1 )
.
e. Wallis's Infinite Product for 1T
The recursive formula for the i n tegral .f sin" x dx w i t h n > I leads to a fasci nating expression for the n umber 1r as an "infini te product." In the formula fsin" x dx
= -
�
we insert the limits 0 and (75 )
i
•/ 2
o
SID" X .
dx
s i n - 1 x cos x + "
1rj2,
1 fsin"- 2 x dx
thus obtaining
n - 1
= --
n
�
n
•i /2 . o
S!D"- 2
X
for n > l .
dx
I f we repeatedly apply the recursive form ula, we obta i n , d i st in g u ishi n g between the cases n = 2m and n = 2m + l , 12 i ' sin 2 '" x dx
(76)
o
=
---
2m - 1 2m - 3 2m 2m - 2
12 2m (76a) i " sin 2 '"+1 x dx = --whence (77)
( 7? a )
--- ·
2m - 2 2m + 1 2m - 1
o
"12 • 2 m X dx = i0 SID
•1o 12
•
.
sm· "'+l x d x
· --- ·
·
·
·
-1
2
2 3
'/2
o
i
·
· -
i
·
''2
o
dx,
sin x dx,
1 1T 2m - 1 2m - 3 2;;:;-- . 2m - 2 2 . 2 ' 2m . 2m - 2 2 = 2m + 1 2m - 1 .
.
.
.
.
·3 ·
Sec. 3.11
lnlei;ratioll by Parts 281
By divjsion this yields
(78)
�
2
=
2_·_ 2 . 4_·_4 . 6_·6_. . .
1 ·3 3·5 5·7
i_::_:o: 2s i •/2 . 2 nH 1 1
2m · 2m ( 2 m - 1 ) . (2m + 1 )
•
in 2 m x d x
_____
Sin
0
X
dX
The quotient of the two integrals on the right-hand side converges to I as m i ncreases, as we recogn ize from the following considerations. In the interval 0 < x < 1rj2, where 0 < sin x < I , we have 0 < s i n 2m + l x sin 2m x sin - l x ; � consequently, �
2m
l'12si n2m+l x dx l' 2s n 2m x dx J: \i n 2 1 dx. � � 2 If we here d ivide each term by i' ' sin 2 m x dx and notice that by o formula ( 75) 2sin2m-l x d x i'-"--o•/2m + = = + -' i"1 2sin2'--2m "H x dx 2m 0 <
1
i
m·
'
:r
H
l
0
1
J
•2
we have
i sin2m x dx J 1 +' • / 2 2m i � sin 2 m+! x dx � 0 /
0
fro m which the above statement follows. Consequently, the relation
(79)
7T -
2
=
2 -2 h• m -
4466
m -"' 1 3 3 5 5 7 -
- - -
·
·
·
2m
2m
2m - 1 2m + l
--- ---
holds. This product formula (due to Wallis), with its si mple Jaw of formation, gives a most remarkable relation between the number 7T and the integers. Product for
.J;
As an easy consequence we can derive an equally remarkable ex pression for .J;. If we observe . Itm
2m
m -oo 2m + 1
--- =
1,
282
The Techniques of Calculus
we can write
Ch. 3
. 22 · 42 · · · (2m - 2)2 1T hm 2m = - ; 3 2 5 2 • • • (2m - I ) 2 2
m � oo
·
taking the square root and then multiplying the numerator and denominator by 2 · 4 · · · (2m - 2), we find .
·
-
-
(2m 2) ,j 2m A2 = m � oo 23 · 45 ·· ·· ·· (2m - l) hm
=
. 2 2 4 2 · ( 2m - 2 )2 ,j 2m m � oo (2m - 1 ) ! lim
•
•
•
. 22 · 42 · · · (2m) 2 -,J2m = hm 2m . (2m) ! m � oo . (22 . 1 2)( 2 2 . 2 2) (22 . 32 ) . . . ( 2 2 . m2 ) = h m '-----'-'----'"'-:::=l::: - ...o.:---'.c:__:_ :. (2m) !,j2m
From this we finally obtain (80)
.
l1m
m � oo
( m !?22 m = ,j1r , (2m) !,jm
a form of Wallis's product which will be of use to us later (cf. Chapter 6, Appendix) . *3.12
Integration of Rational Functions
During the seventeenth and eighteenth centuries mathematicians were preoccupied with discovering classes of elementary explicit functions which could be i ntegrated explicitly. A wealth of i ngenious devices was invented and at the same time the basis for deeper under standing created. When one later realized that achieving integration of all explicit functions in closed form was neither an attainable nor really an important goal, the tedious technicalities which had been developed in connection with such problems were gradually deempha sized. Yet, a significant general result remained :
All rational functions R(x) of a variable x can be integrated explicitly in terms of the elementary integrals listed in Table 3. 1 .
This general result can be obtained much more easily in the context of the more advanced theory of functions of a complex variable. Yet, it is still worthwhile to sketch an elementary derivation employing only real variables.
Sec. 3.12
Integration of Rational Functions 283
The rational functions are th ose of the form R(.r)
(81)
=
f ( .r ) .
g(�·) wh e re .f( :r ) and g (:r) are poly nomi als :
g(x) = b , x" + h, _1 x " - 1 + · · · + b0
(b , ¥- 0) .
As we recal l , every p o ly n o m ial can be in te g ra ted at once and it s integral is i tself a poly n o mial . We therefore need consider only those rational functions for which the denom i nator g(x) i s not a con s ta n t . Moreover, we can always assume that the degree of the n u m erato r is less than the degree n of the denominator. For otherwise, dividing the polynomi a l .f(x) by the polynomial g(.r), we obtain a remainder of d egree less than n ; i n other words, we can write .f(.r) = q(x)g(x) + r(x), w here q(x) and r(x) are also polynomials and r(x) is of lower degree than n . The i n teg r a t i on of.f(x)/g(.r) is then red uced to the integration of the polynomial q(x) and of the "proper" fraction r(x)/g(x). We notice fu rther that the functio n f(x)/g(x) can be represented as the sum of the functions a,x'/g(x) , so that we need only consider integrands of the fo r m x'/g(x) . a. The Fundamental Types
We proceed in s tep s to the integration of the most general rational fu nction of t he type (8 1 ) , s tud y i ng fir s t o nly those functions with de nomi n at o r g(.r) of th e particu l arly s i m ple type or
g(:c) = x",
g(x) = ( I + x2)" ,
where n i s a n y positive i n te ge r . To this case we can then red uce the somewhat more ge n era l case in which g(x) = (C<x + �)" , a power of a l i near expression C<X + � (IX ¥- 0), or g(x) = (ax2 + 2bx + c)", a power of a definite1 quadratic
1 A quadratic expression Q(.r) = ax ' + 2b.r + c is s a i d to be definite i f f o r a l l real val ues of x i t takes val ues having one and the same sign , that is, if the equation Q(x) = 0 has no real roots . For this it is necessary and sufficient that the "dis criminant" ac - b' is positive. This folloY.s, of cou rse, from the expl icit formula ( -b ± V b' - ac)/a for the roots. Equivalently, a definite quadratic expression is one that cannot be factored into two rea/ l inear factors.
284 The Techniques of Calculus
Ch. 3
expression. If g(x) = (cxx + {3)" we introduce � = cxx + f3 as a new variable. Then d�fdx = <X, and x = (� - {3)/<X is also a linear function of �. Each numerator j(x) becomes a polynomial ,P(;) of the same degree, and consequently,
f
x f( )
( <X X + (/)" In the second case, we write
dx = !
IX
a•
1
f
(�) �n
d�.
ax2 + 2bx + c = - (ax + b) 2 + a a
since we have assumed our expression to be quadratic and definite, ac - b 2 must be positive and a # 0. By introducing the new variable
�=
ax + b d
we arrive at an integral with the denominator [(d2/a)( l + �2)]". Hence in order to i ntegrate rational functions whose denominators are powers of a linear expression or of a definite quadratic expression it is sufficient to be able to integrate the following types of functions : X
n '
We shall, in fact, see that even these types need not be treated in general, for we can reduce the integration of every rational function to the integration of the very special forms of these three functions obtained by taking v = 0. Accordingly, we now consider the integration of the three expressions X
(x2
+ lt '
b. Integration of the Fundamental Types
Integration of the first type of function, 1 /x", i mmediately yields the ex pression log lxl if n = I , and the expression - 1 /(n - l )xn- 1 if n > l, so that in both cases the integral is again an elementary function. Functions of the third type can be integrated immediately by introducing the new variable ; = x2 + I , from which we obtain 2x dx = d� and
f
x =�f� =
_ xd (x" + l )n
2
;n
(t
log (x" + I ) I 2(n - l )(x2 + on-1
if n = I ,
if n > I .
Sec. 3. 12
Integration of Rational Functions 285
Finally in order to calculate the integral In = J I )" where If we puthas any value exceeding one, we make use of a recursive method: •.
d:< (a:2 +
'
n
so that
(x2 + 1 )" 1
(.�:2 + 1 )"
- (:1'2 + 1
)
"
•
f<x2 � I )" = f(x2 :�)" 1 - f(x:2:� )" ' we can transform the right-hand side by integrating by parts, using formula (54) on p. with = g'(x) Then, as we have just found, = and consequently, we obtain = dx = - l (x2 1 )"- 1 - ) J(x2 +dxI )" 1n J( 2 + I )" The calculation of the integral /" is thus reduced to thatI of the integral /._1• If I > we apply the same process to the latter integral, and conti n ue until we finally arrive at the expression J dx 1 = arc tan We thus see that the integraP can be explicitly expressed terms of rational functions and the function arc tan >: . Incidentally, we could also have integrated the function directly, using the substituti o n = tan we should then have obtained = sec2 t and I /( I = cos2 so that J dx = fcos and we have already learned [Eq. (65) p. how to evaluate this integral. 275
f(x)
X
= J: ( 2 + I )" .
x,
g(x)
n -
)
2(n
x
x
+
+
2n - 3
2(n
I
x.
x2 +
In
+ x 2)
t;
t,
x
(x2 + I )"
in
l /(.r2 + I )" dx
dt
2n-2t dt '
279]
The integral of the function 1 /(x' - I )n can be calculated in the same way ; by t'he corres ponding recurrence method we reduce it to the integral 1
I_!:__ = I - x'
ar tanh x
(or ar coth x).
286
The Techniques of Calculus
Ch. 3
c. Partial Fractions
We are now in a position to integrate the most general rational functions. We make use of the fact that every such function can be represented as the sum of so-called partialfractions, that is, as the sum of a polynomial and a finite number of rational functi ons, each one of which has either a power of a linear expression for its denominator and a constant for its numerator, or else a power of a definite quadratic expression for its denominator and a linear function for its numerator. If the degree of the numerator f(x) is less than that of the denominator g (x), the polynomial does not occur. We know already how to inte grate each partial fraction. For according to p. 284 the denominator can be reduced to one of the special forms x n and (x2 + It, and the fraction is then a combination of the fundamental types integrated on p. 284. We shall not give the general proof of the possibility of this resolution into partial fractions. We shall merely confine ourselves to making the statement of the theorem intelligible to the reader and to showing by examples how the resolution into partial fractions can be carried out in typical cases. In practice only comparatively simple functions are dealt with, for otherwise the computations become too cumbersome. As we know from elementary algebra, every real polynomial g(x) can be written in the form1 r 1 g(x) = a(x - a:1) 1(x - a: 2) , · · · · · · (x 2 + 2b1x + c1)'•(x 2 + 2b x + c 2) '' 2
•
•
• •
Here the distinct numbers a:1, a:2, are the real roots of the equation indicate the multiplicity g (x) = 0, and the positive integers /1 , /2 , of these roots ; the factors x2 + 2b ,x + c, indicate definite quadratic .
.
•
•
•
•
expressions, of which no two are the same, with conjugate complex roots, and the positive integers r1, r2 , give the multiplicity of these roots. We assume that the denominator is either given to us in this form or that we have brought it to this form by calculating the real and imaginary roots. Let us further suppose that the numerator f(x) is of lower degree than the denominator (cf. p. 283). Then the theorem on resolution into partial fractions can be stated as follows : For each •
1
•
•
The actual proof of this so-called fundamental theorem of algebra does not belong
to algebra. It is achieved most easily by methods belonging to the theory of functions
of a
complex variable.
Sec. 3.12
Integration of Rational Functions
287
factor {x - 1X) 1 , where IX is any one of the real roots of multiplicity one can determine an expression of the form
/,
A, A ., A1 + -" -.. + · · · + -. X - IX ( :r - IX)" (.T - 1X)1 --
x2
and for each quadratic factor Q(:r) = + 2hx + c i n our product which is raised to the power r we can determine an expression of the form
in such a way that the function j(:r)/g (x) is the sum of all these ex pres· sions ( A v, Bv, Cv, are constants). In other words, the quotient j(x)fg(x) can be represented as a sum of fractions, each of which belongs to one of the types integrated above.'
J'2 -
I n particular cases the decomposition into partial fractions can be done I , we see at once that easily by i nspection. l f, for example, g(.r) =
so that
J
-- = l !o
dx
x2 - 1
2
l :c - 1 1
-g :r + l ·
1 We give a brief sketch of a method by which the possibility of this decomposition
into partial fractions can be proved without using the theory of functions of complex variables, once g(x) can be factored completely into linear factors. If g(x) = (x - ex)kh(x) and h(ex) ;C 0, then on the right-hand side of the equation f(x) g(x)
_
{(ex) h(ex)(x - cx )'
=
_1_
h(ex)
f(x)h(ex) - j(ex)h(x) ( .c - ex)'h(.c)
the numerator obviousl y vanishes for x = ex ; it is therefore of the fo rm h(ex)(x - cx)"'_(,(x), where.j;(x) is also a poly nomial, the integer m � 1 , and /1 (ex) ;C 0. Writingf(ex)/h(ex) = (J, this gives us j(x)
g(x)
(x
fi_ = ex)' (x
-
_ _
-
j;(x)
ex)k mh(x)
Continuing the process, we can keep on diminishing the degree of the power o f (x - ex) occurring in the denominator until finall y no such factor is left. On the remaining fraction we repeat the process for some other root of g(x), and do this as many times as g(x) has distinct factors. By doing this not only for the real but also for the complex roots, and by combining conjugate complex fractions we eventually arrive at the complete decomposition into partial fractions.
288
The Techniques of Calculus
Ch. 3
More generally, if g(x) = (.r - ex)(x - {3), that is, if g(.r) is a nondefinite quadratic expression with two real zeros ex and {3, we have (x - ex)(x - {J)
so that
J
= (ex
dJ: (>: - ex)(x -
(ex - {J) (x - {J)
- {J) (x - ex)
-
{3)
g x - {J •
=
1
ex
- {J
lo
I
I
/" -"I
d. Examples of Resolution into Partial Fractions. Method of Undetermined Coefficients
If g(x) = (x - Ot:1)(x - Ot:2) • • • (x - Ot:,), where Ot:; """ Ot:k if i """ k, that is, if the equation g(x) = 0 has only simple real roots, and if f( x) is any polynomial of degree < n , the expression in terms of partial fractions has the simple form
x) f( -
g( x)
a 1 + -a, . a 2 + " ' · + -= -X - 0(1
X - Ot: ,
X - 0(2
We obtain explicit expressions for the coefficients a�o a2, • • • if we multiply both sides of this equation by (x - Ot: 1 ) , cancel the common factor (x - Ot: 1 ) i n the numerator and denomi nator on the left and i n the first term on the right, and then put x = Ot: 1 • This gives a
/(Ot:l) ------- (�----��--�-10(1 - 0(. )( 0(1 - 0(3) • • • ( 0(1 - Ot: ,)
The reader will observe from the rule for the derivative of a product that the denominator on the right is g '( Ot: 1), that is, the derivative of the function g(x) at the point x = Ot:1 • Similar formulas for a2 , a3 , • • • , obtained in this way, lead to the explicit partial fraction expan sion
f(x)
g(x)
=
f( Ot:l)
g'( Ot:1)( x - Ot:1)
+
+ . . . f(Ot:.) g'( Ot:2)(x - Ot:2)
+
f(Ot: ,) g ' ( :x,)( x - Ot:.,. )
As a typical example of a denominator g(:r) with mul tiple roots, we consider the function 1 /[x2(x - 1 )] . It has a representation
I x2(x - 1)
=
i n accordance with p. 287. If we - 1 ), we obtain the equation
x2(x
I
=
a
multiply
(a + b).r2
b
c
x - I + ; + ;2
both
sides
- (b - c).r - c,
o f this equation by
Sec. 3.12
Integration of Rational Functions 289
true for all values of from which we have to determine the coefficients b, This condition cannot hold unl ess all the coefficients of the polynomial (a + b)x2 - (b - c)x - - I are zero: that is, we must have a + b c + I 0 or c -I, b -I, a We thus obtain the resolution .T ,
c.
h
c =
a,
c
=
=
=
=
=
I.
and consequently,
log \.r - I I - log l·rl + ;:I . + I )] whose denomi n ator has Next we decompose the function complex zeros in accordance with the equation
I d.c :r ,,.2(
_
I)
=
l /[x(x2
I
J:(.r2 + I )
=
a
:-;:
For the coefficients we obtain a + b = :t·(x2 + I )
and therefore
I
+
h.,. + c .r2 + I .
=a - I
c
=
0
so that
:r
:;: - x2 + I '
log lxl - ! log (x2 + I ). As a third example we consider the function l /(:c.4 + whose integration was a challenge even in Leibnitz' time. We can represent the denominator the product of two quadratic factors :1 as
x4 + I
=
dx = ( 2 + I)
X X
I ),
(J:2 + 1 )2 - 2.c2
=
(x2 + I + v'2x)(;:2 + I - v'Zx).
We know therefore that the resolution into partial fractions will have the form -- =
,. 4 + 1
ax + b
+
"'2 + v2x + I
To determine the coefficients
a,
b, c, d,
ex + d "'2 - v'zx + I
.
we use the equation
(a + c).c3 + (h + d - a v'z + c v'2);·2
+ (a + c - b v'Z + dvZ)x + (b + d - I )
=
0,
The factorization of x4 + I into real quadratic factors corresponds to the factori zation into conjugate complex linear factors 1
where
x' +
I
=
< =
[(x -
e)(x
- c')][(:r - •')(x - c')],
cos 4' + i sin 4 rr
rr
=
I
Z
r v 2(1
+ i)
is one of t he eighth roots of + I , and a fourth root of - I (see
p . I 05).
290 The Techniques of Calculus
Ch. 3
which i s satisfied by the values 1
a = --,
b
2 V2
We therefore have
x• + I
=
=
c =
!.
x + v2 . z v2 x2 + V 2x + I
. z v2 :c2 -
x -
-
v2
V 2x
+ I
'
and, applying the method given on p. 284, we obtain
J
dx --
.c•
+ I
I I = --= log lx2 + V 2x + I I - -- log l:c2 - V 2x + I I 4V2
4 V2
I I + --- arc tan ( V2x + I ) + --- arc tan ( V2x - I ), -
2 V2
2 V2
which may easily be verified by differentiation. The preceding examples illustrate a general method of integrating a rational function f(x)(g(x). We first divide and are reduced to the case where the degree of [ is less than that of g. We factor g(x) i nto linear and definite quadratic factors, grouping the product into powers of such factors. We write down the app rop riate partial fraction representation for fig with indeterminate coefficients a, b, c, . . . M ulti plying through with g(x) and comparing CO(!fficients of equal powers in the resulting polynomial identity, we obtain a system of l i near equations for the unknown coefficients that should j ust be adequate for determining those coefficients, if we really have the correct form for the partial fraction expansion. We are then ready to i ntegrate any of the resulting partial fractions by the rules discussed before. .
3.13
Integration of Some Other Classes of Functions
a. Preliminary Remarks on the Rational Representation' of the Circle and the Hyperbola
The integration of some other general classes of functions can be reduced to the i ntegration of rational functions. We shall be able to better understand this reduction by first stating certain elementary facts about the trigonometric and hyperbolic functions . If we put t = tan (x/2), elementary trigonometry yields the simple formulas .
21 I + t2
s r n x = -- ,
1
Sometimes cal led "uniformi.zation . "
cos x =
12 · 1 + 12 ,
1
-
--
Integration of Some Other Classes of Functions 291
Sec. 3.13 indeed, from
1
1+
---
t2
X
= cos - 9
2
and
(2
+ 12
--
1
. X = sm 2 - ,
2
and from the elementary formulas
sin x = 2 cos2 � tan � 2
2
and
cos
x
= cos 2 � 2
-
sin 2 � , 2
we obtain these equations. They show that sin x and cos x can both be expressed rationally in terms of the quantity 1 = tan xj2. By differen tiation we have dt dx
so that
----
2 cos 2 xj2
1 + 12 = -2
(82) hence the derivative dxjdt i s also a rational expression in
t.
*The geometrical representation of our formulas and their geometrical meaning are given in Fig. 3-28 . Here the circle u2 + 1' 2 = I in a u,v-plane is shown. If :r denotes t he angle TOP in the figure, then u = cos "' and v = sin x. The angle OSP with its vertex at the point u = - I , v = 0 is equal to xf2, by
Figure 3.28
Parametric representation of the trigonometric functions.
292
The Techniques of Calculus
Ch. J
a theorem in elementary geometry, and we can read off the geometrical meaning of the parameter 1 from the figure ; r = tan !J• = OR where R is the "projection" from S of the point P of the circle onto the v-axis. If the point P starts from S and describes once the circle in the positive direction, that is, if x runs through the interval from - to +rr, the quantity r will run through the whole range of values from + oo to + oo exactly once. (Notice that the point S itself corresponds to 1 = ± oo ) . We have here a representation of the general point (u, v) of the circle u2 + v2 = I in terms of the rational functions u = (I - r2)/( l + 12), and v = 2t/( l + 12) of the parameter t. These formulas then define a rational mapping of the r-line onto the circle in the u,v-plane (which incidentally is the two-di mensional analogue of the stereographic projection of a sphere mentioned on p. 2 1 ). At the basis of this rational representation of the ci rcle l ies obviously the identity rr
u• - 1 J2 + (2rJ• = u• + o• .
Curiously enough, this formula is of interest also i n number theory since it generates for each integer r Pythagorean integers a = 12 - I, b = 21, and c = r • + I which satisfy the identity a 2 + b2 = c2 , that is, determine a right triangle with commensurable sides. Thus r = 2 gives rise to the well-known triple a = 3, b = 4 , c = 5; for 1 = 4 we obtain a = 15, b = 8 , c = 1 7, etc. It is remarkable, and, of course, no accident, that the same algebraic identity is of significance in such diverse contexts as integration i n closed form, geometry, and number theory. Linking different fields in such a manner is the typical trend i n modern mathematics, although our particular example goes back to antiquity.
Similarly we may express the hyperbolic functions cosh x = i( e ' + e-') and sinh x = He' - e-') as rational functions of a third quantity. The most obvious way is to put e' = .,. , so that we have
(
(
!). ,
!)' ,
sinh x = ! .,. .,. 2. which are rational expressions for sinh x and cosh x. H ere again dxfd.,. = 1 /.,. is rational in -r . However, we obtain a closer analogy cosh x = ! .,. +
2
.,.
with the trigonometric functions by introd ucing the q uantity t tanh (x/2) = (T - 1 )/(-r + I ) ; we then arrive at the formulas
=
-2 1 1 + 12 - . sinh x = cosh x = --2 ' 1 - 1 1 - 12 By differentiating t = ta'lh (x/2) we obtain, as in Eq. (82) on p. 29 1 ; the rational expression
(8 3 )
--
2 dx = dt I - t2
Integration of Some Other Classes of Functions 293
Sec. 3.13
v
Figure
3.29
Parametric re p resentation
of the hyperbolic
functions.
for the derivative dxjdt. H ere again the quantity t has a geometrical mea n i ng similar to that which it has for the trigonometric functions, as we see at once from Fig. 3.29. We have here a rational representation of the hyperbola u 2 - v 2 = 1 in the u,l'-plane by means of the equations u = ( I + !2)/( 1 - 1 2 ) t• = 2tj( l - 1 2 ). The points on the right-hand branch of the curve are of the form u = cosh x, v = sinh x and correspond to values of t with l t l < I . The other branch is obtained for i t i > I . We now proceed to our integration problems. *h. Integration of R (cos x, sin x)
Let R(cos x, sin x) denote an expression which is rational in the two functions sin x and cos x, that is, an expression which is formed rationally from these two functions and constants, such as 3 sin 2 x + cos J' 3 cos 2 x + sin x
294
The Techniques of Calculus
I f we apply the substitution
J
Ch. J
I =
tan x/2, the integral
R(cos x, sin x) dx
is transformed into the integral
J
R ( 1 - 12 ' _2_1_2·) -2- d t '
I + 12 1 + 1 1 + t 2 and under the integral sign we now have a rational function of I . Thus we have i n principle obtained the integral of our expression, since we can now perform the integration by the methods of the pre ceding secti on. c.
Integration of R(cosh x, sinh x)
In the same way, if R(cosh x, sinh x) i s an expression which i s rational in terms of the hyperbolic functions cosh x and sinh x, we can effect its integration by means of the substitution 1 = tanh xj2. Recalling Eq . (83), we have
JR(cosh
x,
sinh x) d x
( 122 , �)· -2- dt . . I - 1 1 - I - r2
= R 1
J
+
r'
(According to a previous remark we could also h ave introduced T = ez as a new variable and expressed cosh x and sinh x in terms of T.) The i ntegration is once again reduced to that of a rational function. *d.
Integration of R (.r, J 1
- x 2)
J 1 - x2
The integral J R (x , J l x2 ) dx can be red uced to the type treated in Section 3 . 1 3b by using the substitution
X = COS U, = sin u, dx = - sin u du ; from this stage the transformation 1 = tan u/2 brings us to the inte gration of a rational function. Incidentally, we could have carried out the reduction i n one step instead of two by using the substitution
-)]-=-; 1 + -. 2t 1 --, I - x· = --2 ' 1 + 1 t=
X
;
·
X=
2 I - 1 ; -2 1 + 1
dx = -41 dt (1 +
1 2) 2
;
that is, we could have i ntrod uced t = tan u/2 directly as the new variable and thereby obtained a rati onal integrand.
Integration of Some Other Classes of Functions 295
Sec. 3./3
x2 x2 - dx
*e. Integration of R(x,
.J
I)
The i ntegral J R(x, ·/ I) i s transformed by the substitution cosh u into the type treated in Section 3 . 1 3c. Here again we can arrive at our goal directly by introducing
x
=
t =
*f. Integration of R(x,
Jx -+ 1
1
x
=
u tanh 2
.
.J x2 + I )
The integral J R(x, .J(x2 + I )) dx is reduced by the transformation x = sinh u to the type considered i n Section 3 . 1 3c (p. 294) and can therefore be i ntegrated in terms of elementary functions. I nstead of the further red uction to the integral of a rational function by the sub stitution e" = r or tanh u/2 = r , we could have reached the integral of a rational function i n a single step by either of the substitutions r =
:r
l =
+ .J x2 + 1 ,
- I + .Jx2 + 1 X
0
*g. Integration of R(.r, .j ax 2 + 2 b:r + c )
T h e i ntegral J R(x, ·/ ax 2 + 2bx + c) dx of an expression which is rational i n terms of : r and the square root of an arbitrary polynomial of the second degree in . r can i m mediately be reduced to one of the types just treated. We write (cf. p. 2 84) "
a :c- + 2 b .r +
c
=
1 ., ac - b 2 - (a x + b t + --- . a a
I f ac - b2 > 0 we i ntrod uce a new variable !; by means of the transformation !; = (ax + b)/.J ac - b 2 , whereupon the surd takes the form .J(ac - b2)(/;2 + I )/a. Hence our integral when expressed in terms of !; is of the type of Section 3 . 1 3f. The constant a m ust here be positive in order that the square root m ay h ave real values. If ac - b 2 = 0, and a > 0, then by way of the formula "I a x.,- + 2 b x +
c
=
( b).
v; a . x + . a
we see that the integrand was rational in x to begin with.
296
The Techniques of Calculus
Ch. 3
I f, finally, ac - b2 < 0, we put � = (ax + b)JJ b2 - ac and obtain for the surd the expression J(ac - b2)(�2 - I )Ja. If a is positive, our integral is thus reduced to the type of Section 3. 1 3e ; if, on the other hand, a is negative, we write the surd in the form
J(b 2 - ac)( l - �2)/( - a) and see that the integral is thus reduced to the type of Section 3. 1 3d . * h . Further Examples of Reduction t o Integrals of Rational Functions
Of other types of functions wh ich can be integrated by reduction to rational functions we shall briefly mention two : ( I ) rational expressions i nvolving two different square roots oflinear expressions, R(x, J�, J ocx + {J) ; (2) expressions of the form R(x, '(:1(ax + b)/(ocx + {3)), where a, b, oc, fJ are constants. In the first type we i ntroduce the new variable � = J + {J, so that ocx + fJ = �2, and consequently
ocx
e - tJ
and
X = -OC
dx d�
=
2�
then
fR(x, Ja.r +
b, Jocx + {J) dx =
fR \w oc--{3 , Jt� [a� - (a{J 2
- boc)], �
)�
u
d�,
which is of the type discussed in Section 3. 1 3g. I f i n the second type we i ntrod uce the new variable � we have �
n
=
=
Jax +
b,
o:x + fJ
ax + b , ocx + fJ
and we i m mediately arrive at the formula
which is the integral of a rational function.
dx d�
a{3 - boc (oc�" -
al
n,
,,_1
,
Sec. 3.13
Integration of Some Other Classes of Functions
i. Remarks
on
297
the Examples
The preceding discussions are chiefly of theoretical interest. I n complicated expressions the actual calculations would be far too i nvolved . It is therefore exped ient to take advantage, when possible, of the special form of the integrand to simplify the work. For example, to i ntegrate l /(a2 sin2 x + b2 cos2 x) it is better to use the substitution t = tan x instead of that given on p. 294 ; for sin2 x and cos2 x can be expressed rationally in terms of tan x , and it is therefore unnecessary to go back to t = tan x/2. The same is true for every expression formed rationally from' sin2 x, cos2 x , and sin x cos x. M oreover, for the calculation of many i n tegrals a trigonometrical form is to be preferred to a rational one, provided that the trigonometrical form can be evaluated by some simple recurrence method . For example, although the i n tegrand in .f x"( J I - x2) "' dx can be red uced to a rational form, it is better to write x = sin u and bring it to the form J sin" u cos "' t l u du, since this can easily be treated by the recurrence method on p. 279 (or by using the addition theorems to reduce the powers of the sine and cosine to sines and cosines of m ultiple angles). For the evaluation of the integral
f
d.l'
a cos .r + b sin :r
i nstead of referring to the general theory we write sin 0 = A , (/
cos 0
b
A
= - .
The integral then takes the form
I fsin
A
(:r· d.t·+ 0) '
and on i ntroducing the new variable :r: + 0 we find [(cf. Eq. (40), p. 27 2)] that the value of the i ntegral is
I I + (j I
A For sin tan x .
1
x
cos x
=
tan
x
log tan ---z- . .r
cos' x can, of course, be expressed rationally in terms of
298
The Techniques of Calculus
Part C Further Steps
3.14
Ch. 3
in
the Theory of Integral Calculus
Integrals of Elementary Functions
a. Definition of Functions by Integrals. Elliptic Integrals and Functions
With the examples already given of types of functions which can be integrated by reduction to rational functions, we have practically exhausted the list of functions which are i ntegrable in terms of ele mentary functions. Attempts to express indefinite integrals such as (for n > 2) dx ·•
J .J a0 + a 1 x +
or
f.Jao
+ a1x +
J
· · ·
· · ·
+
G 71 X " '
+ a 71 x" dx,
e'
-;;: dx
in terms of elementary functions have failed ; i n the nineteenth century it was finally proved that it is actually impossible to carry out these integrations in terms of elementary functions. If therefore the object of the integral calculus were to integrate functions explicitly, we should have come to a definite halt. However, such a restricted objective has no intrinsic j ustification ; it is of an artificial nature. We know that the i ntegral of every continuous function exists as a limit and is itself a continuous function of the upper limit whether or not the integral can be expressed i n terms of elementary functions. The distinguishing features of the elementary functions are based on the fact that their properties are easily recog nized, that their application to numerical problems is facilitated by convenient tables, or that they can easily be calculated with as great a degree of accuracy as we please. Whenever the integral of a function cannot be expressed by means of functions with which we are already acquainted, there i s no objection to i ntroducing this integral as a new "higher" function, which really means no more than g1ving the integral a name. Whether the intro d uction of such a new function is convenient depends on the properties which it possesses, the frequency with which it occurs, and the ease with which it can be manipulated in theory and in practice. In this
Sec. 3. 14
Functions not Integrable 299
sense the process of integration is a general principle for the generation of new functions. We are already acquainted with this principle from our dealings with the elementary functions. Thus we were forced (p. 1 45) to introduce the i ntegral of l fx as a new function, which we called the logarithm and whose properties we could easily derive. We could have introduced the trigonometric functions in a similar way, making use only of the rational functions, the process of integration, and the process of i nversion. For this purpose we need only take one or other of the equations dt a rc tan x = --.
.
or
arc stn
x =
ix o 1 ixo
+ r dt
- 12
--=
v' I
as the definition of the function arc tan x or arc sin x respectively, and then obtain the trigonometric functions by i nversion. By this process the definition of these functions is divorced from intuitive geometry, (in particular, from the intuitive notion of "angle"), but we are left with the task of devel oping their properties, independently of geom etry.' (Later, in Sectio n 3 . 1 6 we shall give another purely analytic discussion of the trigonometric functions.) *Elliptic Integrals The first i mportant example which leads beyond the set of elementary functions is given by the elliptic integrals. These are integrals in which the integran d depends rationally on the square root of a polynomial of third or fourth degree. Among these integrals the function u (s)
=
"
( Jo -/ ( 1
dx
- x2 ) ( 1 - k 2 x 2 )
has become particularly important. Its i nverse function s(u) similarly plays an i mportant role. 2 This function s( u) has been as thoroughly examined and tabulated as the elementary functions.3 • We shall not go into the development of these ideas here. The essential step is to prove the addition theorems for the inverse functions, that is, for the sine and the tangent . 2 For t h e special value k = 0 we obtain u(s) = arc sin x a n d s(u) = sin u respectively. 3 The function s(u), one of the so-cal led Jacobian elliptic functions, is usually denoted by the symbol sn u to indicate that it is a generalization of the ordinary sine-function.
300
The Techniques of Calculus
Ch. 3
It is the prototype of the so-called elliptic functions which occupy a central position in the theory of functions of a complex variable and occur in many physical applications (for example, in connection with the motion of a simple pendulum ; see p. 4 1 0) . The name "elliptic integral" arises from the fact that such integrals enter into the problem of determining the length of an arc of an ellipse (cf. Chapter 4, p. 37 8 ). We point o u t further that integrals which a t first glance have quite a dif ferent appearance turn out to be elliptic integrals after a simple substitution. As an example, the integral dx
I
v'cos "' - cos x
is transformed by means of the substitution
u
= cos xf2 into the integral k =
the integral
by means of the substitution
and finally the integral
u
I
I
1 --
cos (<X/2)
· •
dx
v' cos 2J:
= sin
x
becomes
dx
v' ! - k2 sin2 .c
is transformed by the substitution u = sin x into
b. On Differentiation and Integration
Another remark on the relation between differentiation and inte gration should be inserted. Differentiation may be considered a more elementary process than integration, because it does not lead us out of the domain of "known" functions. On the other hand, we must remember that the differentiability of an arbitrary continuous function is by no means a foregone conclusion but a stringent assumption. In fact, as we have seen, there are continuous functions which are not differentiable at certain isolated points, whereas since Weierstrass'
Sec. 3.15
Extension of the Concept of Integral 301
time many examples of continuous functions have been constructed which do not possess a derivative anywhere. 1 In contrast, even though integration in terms of elementary functions is generally not possible, we are certain at least that the integral of a continuous function exists. Taken all in all, i ntegration and differentiation cannot be contrasted simply as more elementary and less elementary operations ; from some points of view the former and from other points of view the latter could be thought of as more elementary. l nsofar as the concept of integral is concerned , we shall free ourselves in the next section from the assumption that the integrand is everywhere continuous ; we shall see that it may be extended to wide classes of functions which have discontinuities. 3.15
Extension of the Concept of Integral
a. Introduction. Definition of " Improper " Integrals
il•
In Ch apter 2, p. 1 28, we defined j(x) dx by form ing the " Riemann a sums" F,.
=
2 /(0 llx; "
i=l
based on a subdivision of the interval [a, b ] into n subintervals of length s ll:r; and a ch oice of intermediate points $, in those subintervals. If the sequences F,. tend to the same limit Fab • for any sequence of sub divisions and intermediate points, as long as the largest value llx, tends to zero, we define
fJ (x)
d:r to be that limit F. b . This limit was shown to
exist when f(x) i s continuous i n [a, b]. However, we are often con fronted with the need for defining an integral when f(x) is not defined, or not contin uous, in all points of the closed i nterval I or when the i nterval of integration extends to i nfinity. We would wish, for example, to attach an appropriate meaning to expressions such as
r .jx�
dx
or
f''e-x dx
or
o
r oo i o
1
sin - dx, XI sin x dx ' X2
etc.
' Compare Titchmarsh, The Theory of Functions, Oxford, 1 932, 1 1 .23, pp. 3 50-354.
Sections
1 1 . 2 1 to
302 The Techniques of Calculus
Ch. 3
We first of all extend the concept of the integral to functions that are continuous in the open interval (a, b) but are not necessarily defined or continuous at the endpoints a, b. For any numbers IX, b with a < IX < {I < b the ordinary ("proper") integral f(x) dx is then defined. If F
= f p, lim
£-0
f(x) dx
12 £
exists when a < IX, < {J, < b and lim IX,
=
a, lim {J,
=
b, and if F is
independent of the particular choice of IX, and {J, we say that the im-
proper integral
J:
c--o
E-0
f(x) dx converges and has the value
F.
Sectionally Con tinuous Integrand. I f, more generally, f(x) is defined and continuous in (a, b) w ith the possible exception of a finite number of intermediate points c1, c2 c, and f is continuous in each of the •
open i ntervals (a, c1), ( cb c2),
•
•
•
•
•
(c, , b) we define
,
i"
j(x) dx as the
sum of the improper integrals over the subintervals. provided each of those converges. , The improper integral j(.r) dx always converges when f is contin-
f.
uous and bounded in the open interval (a. b). For example, the integral
J0l sm. X dx = hm i l sm. I
-
.
c
..
o
l
x
I d
-
X
converges. To prove this general statement we may assume, for brevity, that f is continuous at b, but not necessarily at a. Then by defin ition
fJ(x) dx = :� b fx F(IX),
where F(1X) for a < IX <
is defined as
j( ) dx. If
M
i s an upper
bound for lf l and IX, a sequence tending to a, we have by the mean value theorem of integral calculus I F( IX,) - F(1Xm ) l � M I IX , - IXml ; hence, by Cauchy's convergence test l im F(1X,) exists . Since this is the
case for any sequence IX, convergi ng to a, it follows that lim F(1X) exists.
f
As a m atter of fact, when f is continuous and bounded i n (a, b) we can assign to f any values at the endpoints a, b and also obtain
J(x) dx
directly as a "proper" integral defined as the limit of Riemann sums. It is easily seen that fer continuous bounded f both definitions apply and lead to the same value, independently of the choice off(a) an dj(b) .
Extension of the Concept of Integral 303
Sec. 3. 15
The same is true more generally for bounded functions that are defined and continuous in (a, b) with the possible exception of a finite number of points. I n particular,
f
j(x) dx always exists when f is continuous
except for a finite number of jump discontinuities. Altogether conf(x)
Figure
3.30
In tegral of a function with discontinui ties.
vergence of the i mproper integral of a function over a finite interval demands attention only when f becomes infinite. We note that the geometrical interpretation of the integral as the area under the curve is unchanged from the i nterpretation for a con tinuous f (Fig. 3 . 30) . b. Functions with Infinite Discontinuities We begin with the i ntegral J
=
fl
where a. is a positive number. The integrand 1 /xa becomes infinite for :r 0. We therefore must define J by taking the integral J, from the positive limit £ �
304
The Techniques of Calculus
Ch. 3
to the limit I , and final ly letting • tend to zero. According to the elementary rules of integration, we obtain, provided o: ¢. I ,
We i mmediately recognize the following possibilities : ( I ) o: i s greater than I ; then for • � 0 the right-hand side tends to i nfinity. (2) o: is less than I ; then the right-hand side tends to the limit 1 /( 1 - ex ) In the second case, therefore, we shall simply have to take this limiting value as the integral d.c � . In the first case we shall say that the integral from 0 to I does J =
ll 0
.
X
not exist or diverges. (3) In the third case, where o: = I , the integral is equal to - log < and therefore for < 0 does not approach a limit, but tends to i nfinity ; that is, the integral
11 o
- = J does not exist or is divergent. �
d"'
.r
Another example for an i n tegrand with an i nfinite discontinuity is given by J(;c) = 1 / V l - :r2 . We find
•
(1 I o,
•
. 2 V I - .c dx
= arc sin (I - <) .
For • 0, the right-hand s i de converges to the limit, 7T/2 ; this therefore is the value of the i ntegral ( 1 d.r 7T = 2 Jo V I - :c2 ' �
although the integrand becomes i nfinite at the poi nt c.
.r
=
I.
Interpretation as A reas
Improper i n tegrals can be interpreted as areas of regions extending to infinity defined by means of a passage to the limit from bounded regions. For example, the preceding results for the function 1 /x" assert that the area bounded by the x-axis, the line x = I , the l i ne x = < , and the curve y = 1 /.c" tends to a finite limit as • � 0, provided that o: < I , and that it tends to infinity if ex � I . This fact may be simply expressed as follows : The area between the .r-axis, the y-axis, the curve y = l f.r", and the line x = I is finite or infinite according as o: < I or o: � I . I ntuition can, of course, give us no reliable information about the finiteness or infiniteness of the area of a region stretching to i nfinity. Figure 3 . 3 1 illustrates the fact that for o: < I the area under our curve remains finite, whereas for o: � I it is infinite, a fact which i s certainly not suggested by geometrical intuition.
Sec. 3.15
Extension of the Concept of Integral 305
y
a< l
a=l
0�----- x
Figure 3.31
To i l lustrate the convergence or divergence of i mproper integrals.
d. Tests for Convergence To check the convergence of an integral of a function [("") with an infinite discontinuity at the point ;: = b we can often use the following cri terion. Let the function J(�·) be continuous in the interval a ::=;:: .r < b, and let lim f(.>·) =
w.
Then the integral
C
... a
f(x) d.r converges if there exist both a
positive number I' less than I and a fixed number M independent of :r, such that everywhere in the interval a ::=;:: .r < b the inequality l.f(.r)j ::=;:: M/(b - x) 1' is true ; in other words, if at the point x = b the function.f{x) becomes infinite lif a lower order than the first : .f(.r) = 0[1 /(b - J.') '') for some I' < I . On the other hand, the integral diverges if there exist both a number v ::2: I and a fixed number N, such that everywhere in the interval a ::=;:: x < b the i nequality fCr) ::2: Nj(b - .r )' is true ; in other words, if at the point ,. = b the positive .fimction f(x) becomes infinite of the first order at least. The proof follows al most i mmediately by comparison with the very simple special case j ust discussed. In order to prove the first part of the theorem we observe that for 0 < < b - a we have :r -h
<
and hence also
0 ::=;::
::=;:: (b -
x ) 1'
M
0
r c -T
b
+ j(
x) ::=;:: (b - :J:j"
�:r)'' + j(x)J dx
2M
::=;::
f
/:!x)'' dx .
.(
As < � 0 the integral on the right, which is obtained from the integral f da;fx'' by a si mple substitution of b - x for .r, has a limit and therefore s tay s
306
The Techniques of Calculus
Ch. 3
-.
bounded. Moreover, the values of the integral in the middle increase monotonically as £ 0; since they are also bounded, they must possess a limit and the integral
r(
(b
f dx.
':!x) '' dx + f(x)) dx -E E dx + r f(x) dx) l' ':!x ) (b !�( f
-
=
converges. The convergence of the integral of M/(b that of
f(x)
x) 1' then also i mplies
The proof of the second part of the theorem is left as an exercise for the reader. We li kewise see at once that exactly analogous theorems hold where the lower boundary of the integral is a point of i nfinite discontinuity. If a point of infinite discontinuity lies in the i nterior of the interval of integration, we merely separate the i nterval i nto two subintervals by this point and then apply these considerations to each of these. As an example we consider the elliptic integral
e J o V( l
- x2)(1 - k2x2)x x) ) dx
(k" < 1 ).
x -. I
From the identity I x2 = ( I - ( l + we see at once that as the i ntegrand becomes infinite only of order !, from which it follows that the improper integral converges. (For k = I the i ntegral diverges.)
e. Infinite Interval of Integration
Another i mportant extension of the concept of integral concerns an i nfinite i nterval of integration. For a precise formulation, we i ntroduce the following notation : If the i ntegral
lAf(x) dx,
of f(x) over the i nfi nite i nterval x :;:::
with
a
fixed, tends to a definite limit for A a,
as
--+
oo,
we define the integral
lim [Af(x) dx = [ f ( x) dx. Ja A - 00 Ja 00
Again, such an integral is called convergent.
Examples. Si mple examples of the various possibilities are again given by the functionsj(x) = lfx",
fA dx --I 1 x•
-
=
1 - "'
(A1-• - 1 ).
Sec. 3.15
=
Extension of the Concept of Integral 307
Here we see that, if we again exclude the case "' exist�; for the case "' > I, and, i n fact,
f oo � = -1 . 1
.rll
0:
I , the integral to infinity
- 1 '
when "' < I , the integral no longer exists. For the case "' = I the i ntegral again clearly fails to exist since log x tends to infinity as x does. We see therefore that with regard to integration over an infinite interval the functions I fx• do not behave in the same way as for integration up to the origin. This statement also is made plausible by a glance at Fig. 3 .3 1 . For obviously, the larger "' is, the more closely do the curves draw towards the x-axis for x oo ; thus it is plausible that the area under consideration tends to a definite limit for sufficiently large values of "' The following criterion for the existence of an integral with an infinite limit is often useful . (We agai n assume that for sufficiently large val ues of x, say for x � a, the integrand is continuous.) �
·
Criterion of Convergence The integral
i
00
j'(;r) d.r converges if the function f(a:) vanishes at infinity to
a higher order than the first, that is, if there is a number v > I such that for all values of :r, that are sufficiently large, the relation 1/(>:)1 � Mfx' is true,
where M is a fixed number i ndependent of
.r .
In symbols : f(.r) =
o(�).
Again, the integral diverges if the function remains positive and vanishes at infinity to an order not higher than the .first, that is, if there is a fixed number N > 0 such that "f(:c) � N.
J 1
The proof of these criteria is exactly parallel to the previous argument and can be left to the reader. oo A very simple example is the in tegral 2 dx (a > 0). The integrand
A
n
X
vanishes at i nfinity to the second order. We see at once that the integral I I 1 2 d.t = - - - , and therefore converges, for A a , .r
J
i oo
Another equally simple example is 0
I
--2 d.r I +X
Then obviously also
=
lim (arc tan A - arc tan 0) A -· 00
+oo
I -d.r = " ' _ , l + x2
J
=
2.
"
308
The Techniques of Calculus
Ch.
3
since the integrand is an even function. It is curious that the area between the curve and y = 1 /( 1 + x2) and the x-axis (see Fig. 3.8, p. 2 1 6) that extends to infinity turns out to be the same as that of a circle of radius on e .
f. The Gamma Function
A further example of particular importance m analysis is that of the so-called gamma function
(n > 0).
x e-'x"-1 1/x"
Splitting up the interval of integration into one part from = 0 to � . we see that the i ntegral over the first part clearly converges, since 0 < < with flo = 1 - n < I . For the integral over the second, infinite part, the criterion of convergence i s also satisfied ; for example, for v = 2, we have lim = 0, since the exponential function tends to zero
x = 1 and another one from x = 1 to x = x2e-'x"-1
e-•
to a higher order than any power 1 /xm (m > 0) (see p. 25 3). This gamma function which we consider as a function of the number n (not necessarily an integer) satisfies a remarkable relation obtained by integration by parts as follows. First, we have (with f(x) = g'(x) e-')
I
=
e-'x"- 1
dx = - e -•x•-1 + ( n - 1 ) Ie-•x•-2 dx.
x•<- �,
If we take this integral relation between 0 and A and then let A i ncrease beyond all bounds, we immediately obtain
f(n) = (n - 1 )
L"e-•x•-2 dx = (n - 1 )f(n - I )
for n > 1 ,
and by this recurrence formula, provided flo is an integer and 0 < flo < n, it follows that
f(n) = (n - l )(n - 2) ·
· ·
( n - flo)
L"e-•x•-H dx.
I n particular, if n is a positive integer, we have for flo = n - I
f(n) and since we have a
=
(n - l )(n
- 2)
· · ·
3·2
·
l l"'e-• dx,
f(n ) = ( n - l )(n - 2) · · · 2 · I = (n - 1 ) ! ,
most useful expression of a factorial by an integral.
Sec. 3.15
Extension of the Concept of Integral 309
Other Examples. The i ntegrals "' ' e-x d.r'
i
also converge, as we may easily deduce from our criterion. The first one is i dentical with ! f(!l. the second one with ! J'[(n + 1 )/2] for n > - l . as is seen by the substitution .r2 = U, d.r = ( 1 /2 V;;) du. g. The Dirichlet Integral
In many applications we encounter integrals whose convergence does not follow directly from our criterion . An important example is furnished by the integral Sin X I =
i oo -0
.r
dJ·
i nvestigated by Dirichlet . If the upper l imit is not infinite but finite, the i ntegral is convergent since the function (sin x)Jx is continuous for all finite x ; sin x for x = 0 it is given by lim -- = I for J' 0 The convergence of the ,. i ntegral / is due to the periodic change in sign of the integrand, which causes contributions to the i ntegral from neighboring i ntervals of length almost to cancel one another (Fig. 3 . 32). Thus the sum of the infinitely many areas
-> ) .
(,
y
Figure 3.32
G raph of y
sin x
= ---;- .
1r
310
The Techniques of Calculus
Ch. 3
sin x between the x-axis and the curve y = -- converges, if we count areas above x the x-axis as positive and below the x-axis as negative. (On the other hand. the sum of the numerical values of all areas, that is, the integral,
l"" 0
\ sin x i
X
--
dx'
can easily be shown to diverge.) The alternating character of the function sin "' accounts for the fact that its indefinite integral
is bounded for all
.r.
J
sin >, d:r = I - cos x
We make use of this fact in estimating the expression
JA R =
J
-d.r "'
R sin x
t
I ntegration by parts shows that
- I - cos B /_j 11 B
l""
Hence
o
-. d.r =
sin J" .r
-
l
R � d( i - cos x) . d". d:�· x
A
I - cos A A
hm •
A -o JJ -.. 00
JA B =
+
l"' o
lB _1
I - cos X d J. ..r 2 •
I - cos 1· d --.2-- ;·, J
where the integral on the right-hand side clearly is convergent. In other words, the integral I exists. In Section 8.4c we shall establish further the remarkable fact that I has the value -rr/ 2.
h. Substitution. Fresnel Integrals
Obviously, all rules for the substitution of new variables, etc., remain valid for convergent improper integrals. Often such trans formations can lead to different, more tractable expressions for the integral . As an example, to calculate
l
l
we introd uce the new variable 0
oo
xe
x
'
dx
1
= -
2
o
oo
e
u ·-
=
" du
:r2 and obtain = hm •
A-,
1 - (I
2
- ·I - e · ) =
1
-
2
.
Extension of the Concept of Integral 3 1 1
Sec. 3. 15
Another example i n the i nvestigation of i mproper integrals i s given by the Fresnel integrals, which occur in the theory of diffraction of light : "' F1 sin
= l (x2) dx, The substitution x2 = u yields � l sin;-u d 12 u u, F _
"'
0
'I
Integrating by parts, we find
J
H
A
sin� du
.Ju
= 1 -.jBcos B
_
1 - �os A + I 'l A
2
J
H1
-
3CO S
u •
.t
u du .
As A and B tend to zero and infinity respectively, we see by the same argument as for the Dirichlet integral that the integral F1 converges. The convergence of the integral is proved in exactly the same way. These Fresnel integrals show that an improper integral may exist even if the integrand does not tend to zero as oo . I n fact, an i mproper integral can exist even when the i ntegrand is unbounded, as is shown by the example
F2
x -..
L""2u cos ( u 4 ) du .
u4 =
u=
=
When m r , that is, when {/m r , n 0, I , 2, . the i ntegrand becomes 2{/ mr cos 111r ± 2 {/n1r, so that the integrand is unbounded . By the substitution however, the integral is red uced to
= u2 = x,
.
.
f'' (x2) dx, c os
which we have j ust shown to be convergent. By means of a substitution an improper integral may often be transformed i nto a proper one. For example, the transformation sin gives
x= u
2 dx Jo .J 1 - x2 = J(•1o du = "::2 . [1
On the other hand, integrals of continuous functions may be trans formed into improper integrals ; this occurs if the transformation i s such that at the end of the i nterval of i ntegration the derivative vanishes, so that is infinite .
u = (x)
'(x)
dxfdu
3 1 2 The Techniques of Calculus
3.16
Ch.
3
The Differential Equations of the Trigonometric Functions
a. Introductory Remarks on Differential Equations
Integration is merely the first step into a much more extensive field : Instead of inverting differentiation by integration, that is of solving the equation y ' = f(x) with given f(x) for y = F(x), we might aim at finding functions y = F(x) which satisfy more general relation ships between y and derivatives of y. Such "differential equations" occur everywhere in applications as well as in strictly theoretical contexts. Penetrating studies far beyond the framework of this book are made of these equations : we shall return to some elementary aspects of the theory of differential equations later in this and the following vol ume. At this stage we confine ourselves to a quite simple, yet significant, example. We shall discuss the differential equations of the functions sin x and cos x, which we have already mentioned on p. 1 7 1 . Although i n elementary trigonometry these functions and thei r properties were taken from a geometric standpoint, we now discard the reliance on geometric i ntuition and put the trigonometric functions in a simple way on a precise, analytical basis, in accordance with the general trend of development mentioned before. b. Sin x and cos x Defined by a Differential Equation and Initial Conditions
We consider the differential equation u" +
u
=
0
w ith the aim of characterizing solutions u(x) which we shall identify with the sine and cosine functions. Any function u = F(x) satisfying the equation, that i s for which F"(x) + F(x) = 0, is called a solution.1 At once we realize that together with a solution u = F(x) the function u = F(x + h) for arbitrary constant h i s also a solution, as i mmediately verified by differentiating F(x + h) twice with respect to x. Similarly, it is immediately seen that with F(x) the derivative F'(x) = u is also a solution, as is of course, cF(x) with a constant factor c. I n addition, together with F1(x) and F.(x) any linear combination c1F1 (x) + c2F2(x) = F(x) with constants c1 and c2 is a solution. Of course, it is always understood that the functions under consideration are sufficiently differentiable.
1
Differential Equations of Trigonmetric Functions 3 1 3
Sec. 3. 16
T o single out from the multitude o f solutions o f the differential equatidn a specific one, we impose "initial conditions" stipulating that for x = 0 the values of u = F(O) and u' = F'(O) be prescribed as a and b respectively. We state first : The solution is uniquely determined by these initial l'alues. For the proof we start with a general remark valid for any solution u. By multiplying the differential equation with 2u' we find because of 2u"u' = (u'2) ' and 2u'u = (u2)' the equation 0 = 2u"u' + 2u'u = [(u')2 + u2 ) ' ,
which can be integrated at once and impl ies u'2 + u2 =
c,
where c is a constant, that is, does not depend on x; therefore c must have the same value as the left-hand side for x = 0. Thus we have for any solution u u'2(0) + u2(0) = c .
Now, s uppose we have two solutions u1 and u 2 with the same initial conditions : Then the difference z = u1 - u 2 is a solution with z'(O) = z(O) = 0. Hence we have c = 0 and for all x z'2 + z2 = 0 ; this means that z = 0 and z ' = 0 which obviously proves our statement. We now define the functions sin x and cos x as those solutions of the differential equation u"(x) + u(x) = 0 for which the initial conditions are, respectively, for u = sin x, u'(O) = h = I , u(O) = a = 0, and for u = cos x, u(O) = a = I ,
u'(O) = b = 0.
We take for granted here the fact that such sol utions exist and are arbitrari ly often differentiable, since its proof will be given later anyway in a more general context ( see Section 9.2). 1 The only solution u of u" + u = 0 for which u = a, u ' = b for x = 0 is then the function u = a cos x + b sin x. This proves that every sol ution of the differential equation is a linear combination of cos x and sin x. N ow we obtain the basic properties of the trigonometric functions from our differential equation u" + u = 0 applied, for example, to the 1
u'' + u' = I , x a s well as for cos x and from whose equi valent form dxfdu = i n verse functions of sin x and cos x are immediately obtained by
Incidentally, we can infer these facts i m mediately from the equation
1/VI
which i s valid for s i n -
u'
the
i n tegrations.
314
The Techniques of Calculus
Ch. 3
function u = sin x. Obviously, with u also v = u is a solution : v " + v = 0. Because of u" + u = v' + u = 0 we have v'(O) = - u(O) = 0 whereas v(O) = u'(O) = I . Hence v(x) = cos x =
.!!._ sin x. dx
Similarly, we derive (dfdx) cos x = - sin x. The central theorem of trigonometry is the addition theorem cos (.r + y) = cos x cos y - sin x sin y.
I t now follows immediately from our approach : First, the function cos (x + y) as a function of x, with y remaining constant for the moment, is a solution u(x) of the differential equation u" + u = 0 satisfying for x = 0 the initial conditions u(O) = cos y = a and u'(O) = - sin y = b. Now, as verified immediately the solution-according to the preceding statement, the only one-for which u(O) = a and u'(O) = b is a cos x + b sin x. Hence we have at once for our solution cos ( x + y) the expression cos (x + y)
=
cos x cos y - sin x sin y,
as we wanted to prove. The remarks in this section should suffice to indicate how trigono metric functions can be introduced in an entirely analytical manner without any reference to geometry. Without going into further details we mention the following. The number !7T could now be defined as the smallest positive value of x for which cos x = 0. The periodicity of the trigonometric functions likewise follows easily from the analytic approach. We shall return to the analytical construction of the trigonometric functions by infinite power series (see Section 5 . 56).
SECTION 3 . 1 , page 201
PROBLEMS
l. Let P(x) = a0 + a1x + ar:2 + · · · + anxn. (a) Calculate the polynomial F(x) from the equation
F(x) - F'(x)
=
P(x).
*(b) Calculate F(x) from t he equation c0F(x) + c1F'(x) + c2F"(x) = P(x).
Problems 3 1 5 2. Find the limit a s n --+-
1 /x at the point x n
=
2.
3. Prove if fl"1(x)
=
-
oo
of the absolute value of t h e nth derivative o f
0 for all
I , and conversely.
x,
then f is a polynomial of degree at most
4. Determine the form of a rational fu nction
lim xr (.t r(x) '
)
=
for which
r
0.
x- oo
5. Prove by i nduction that the nth derivative of a product may be found according to the following rule (Lei bnitz's rule) :
d" g d:r" (f )
Here
=
(n) d 2J' d"- 2g ( n ) df d"-1g . . . + + 2 d.r2 dxn-2 1 J:r d.,- "-1
f dd>·g" + "
(� ) ( ; ) = n,
6. Prove that
=
n(n
I i.r•-l
n-1
.
•- 1
=
S ECTION 3 . 2, page 206
(
+
1
n ) d"-1f dg d"f n - 1 dx"··1 ;h + dx" g .
� ) , etc. ; denote binomial coefficients. 2 (11
- l )x" - n:t·n -1 + I 2 (.<. . - I )
1 . Let 11 = ex( a sin ,. + b cos :r ) . Show that y " can be expressed as a linear combination of and y ' , that is,
y
y" =
p !! ' + qy,
where p and q are constants. Express all higher derivatives as linear com binations of y' and y. •2. Find the nth derivative of arc sin x at ,,. = 0, and then of (arc sin x)2 at x = 0. SECTION 3 . 3 , page 2 1 7
I . Find the secon d derivative o f j[g{h(�:)}].
2. Differentiate the following function : log,.< X I u(x), [that is, the logarithm of u(x) to the base v(:c) ; v(x) > 0]. 3. What conditions must the coefficients lXX
Y(a.,-2
+ {I
"'•
{1,
a, b, c satisfy in order that
+ 2bx + c)
shall everywhere have a finite derivative that is never zero ? 4. Show that d" (e"' 'l 2)/dx" = u,(x)e"''i 2 , where un(•·) is a polynomial of degree n. Establish the recurrence relation •s.
By applying Leibnitz's rule to
� (e"'l2)
=
xex'l2,
3 1 6 The Techniques of Calculus
Ch. 3
obtain the recurrence relation U n +l
= .TUn
+ nUn-1•
*6. By combining the recurrence relations of Problems 4 and 5, obtain the differential equation u/' + X Un ' - nun = 0 satisfied by u,(�·). 7. Find the polynomial solution Un (x)
= X"
+ a1xn�l +
of the differential eq ua t ion u." + xu. · - u,
a)
p>�+ I ,
+
· · · =
0.
On
Z"n ! dx" (x2 - ! )", prove the relations ;r2 - I p (n + 2)x , n + 2 2(n + I ) + n+T P,. + -2- Pn -
•s. If P,.(;r) =
(
n
I
d"
_
"
"
(b) P�+l = "'Pn ' + (n + I )Pn .
(c) dx [(x2 - I )P.;] d
n(n + J )P.
9. Find the polynomial solution p
"
=
(2n) !
2"(n !)2
of the differential equation
dc [(x2 d
.-c"
! )P, ' ]
=
0.
+ a1xn�I + . . . + a n
- n(n + J )P, I
=
0.
. 10. Determme t he po I ynom1al P,(x) = 2!'1 --;:;; (.r - I )" by using the n d . bmom1al theorem . *ll.
Let A,,.(.r) =
= .! ' "
=
,,
n
(�):r•(l
1 A,.,.(x), �o
, ((�) )
1
n�k
P
-
SECTION 3 .4, page
.
- ,,y•- n ,
n =
0,
d"
2
·
I , 2,
.
.
.
, p. Show that
).,. , .(:r) ,
,k
223
1. The fu nction f(x) satisfies the equation
j (x + y) = j (x)j( y) . (a) I f ( (.r) is differentiable, either f(x) = 0 or f (x) = e"'. *(b) If j(x) is continuous, e\ther f(x) = 0 or j(.r) = e"'.
2. If
a differentiable function f(x) satisfies the equation
Problems 3 1 7
f(xy ) = f(.-r) + f (y), then f(:r) = "' log x . 3. Prove that if f(x) is continuous and
/(>') = then f(x) is identical ly zero . SECTION 3.5, page 228
1. Prove the formula
. h a + sm . hb = sm
2
lz
[(t) dt,
. ( ) (-) a a
a +b a b smh -2- cosh -2- . .
Obtai n similar formulas for s i n h sinh b , cosh + cosh b , cosh a cosh b. 2. Express tanh (a + b) i n terms of tanh a and tanh b. Express coth (a ± b) in terms of coth a and coth b. Express sinh !a and cosh !a in terms of cosh a. 3. Differentiate (a) cosh x + sinh x ; ( b) e '••• ' � coth (c) l og sinh (x + cosh 2 x) ; (d) ar cosh :r + ar sinh :r (e) ar sinh ( Hosh x) ; ([) ar tanh (2x/(I + x 2)). 4. Calculate the area bounded by the catenary y = cosh x, the ordinates x = a and x = b, and the x-axis. •,
x
SECTION 3.6, page 236
1. Determine the maxima, minima, and points of inflection of :r3 + 3p +
q. Discuss the nature of the roots of x3 + 3px + q = 0. 2 2. Given t he parabola y 2 = px, p > 0. and a point P(x = �. y = "'I) within it ( r/ 2 < 2pl;), find the shortest path (consisting of two line segments)
=
leading from P to a point Q on the parabola and then to the focus F(x �p. y = 0) of the parabola . Show t hat the angle FQP is bisected by the normal to the parabola, and that QP is parallel to the axis of the parabola (pri nciple of the parabolic mirror) . 3. Among all triangles with given base and given vertical angle, the isosceles triangle has the maximum area. 4. Among all triangles with given base and given area, the isosceles tri angle has the maximum vertical angle. • s. Among all triangles with given area, the equilateral triangle has the least perimeter. *6. Among all triangles with given perimeter the equi lateral triangle has the maximum area. *7. Among all triangles i nscribed i n a circle the equilateral triangle has the maximum area. 8. Prove that i f p > I and x > 0, xP I ;z. p(x I).
-
-
318
The Techniques of Calculus
Prove the i neq uality I > (sin x)fx � 2/.,, 0 � x � .,;2. Prove that (a) tan :r � x , 0 � .,;2. (b) cos ;; � I - 1·2j2. * 1 1 . G iven a1 > 0, a > 0, . . . , an > 0, determi ne the minimum of 2
Ch. 3
9.
10.
a l + . . . + an-1 + x
n
for x > 0. Use the result to prove by mathematical induction that (cf. Problem 1 3, p. I 09)
(a) G iven is a minimum . 12.
n
fixed numbers a 1 , . . . , a,, determi ne
*(b) Minimize L I a , - �· 1 . "
i=t
* (c) M inimize L "
i=l
A, I a,
-
x l , where
A;
.r
so that
L (a, "
- x)2
i=t
> 0.
13. Sketch the graph o f the function !J =
(x2)", y(O)
Show that the function is continuous at minima, or points of inflection ? * 14. Find the least value "' such that
(1
)
I Xf •
+ :;;
"'
=
I.
= 0. Has the function maxima,
> e
for all pos1t1ve x. (Hint: It is known that [I + ( 1 f:�:)Y i 1 decreases mono ton ically and [1 + ( l fx)]" i ncreases monotonically to the l imit e at infinity.) * 1 5. (a) Find the point such that the sum of the distances to the three sides of a triangle is a minimum. (b) Find the point for which the sum of the distances to the vertices is a minimum. 1 6. Prove the fol lowing inequalities : (a) e" > 1 /( 1 + x), x > 0. (b) e" > 1 + log (I + x), x > 0. (c) e" > 1 + ( I + x) log ( I + x), x > 0. 17. Supposef"(x) < 0 on (a, b). Prove : (a) Every arc of the graph within the interval l ies above the chord joining its endpoints. (b) The graph lies below the tangent at any point within (a, b). * 1 8. Let f be a function possessing a second derivative on (a, b). (a) Show that either condition a or b of Problem 22 is sufficient for ( "(x) � 0.
Problems 3 1 9
t(x ;y) � j(x) ;j(y)
(b) Show that the condition
for all :r and y in (a, b) is sufficient for ("(x) � 0. * 19. Let a, b be two positive numbers, p and q any nonzero numbers p < q. Prove that ( lia '' + (I - O)b PJ11 J• [ lia" +
- li)b"]1i• � 1
(I
for al l values of li in the i nterval 0 < li < I . (This i s Jensen's inequality, which states that the pth power mean [ lia 1' + (I li)b"]1 1 " of two positive qualities a, b is an i ncreasing function of p.) 20. Show that the equality sign i n the above i nequality holds if, and only i f, a = b. 21. Prove that lim ( lia 1' + ( I li)b"Jl l '' = a 8b1-e. -
-
22. Defining the zeroth power mean of a, b as a8b'-8, show that Jensen's i nequality applies to this case, and becomes (a -F b),
a8b1 -8
[ Oa• +
(I
- li)b"] l l"
a 8b 1 - 8 For q = I , 23. Prove the i nequality
a8b1-8
according to whether q � 0
+ ( I - O)b.
�
lia
�
Oa + ( I
-
O)b,
> 0, 0 < () < I , without reference to Jensen's i nequality, and show that equality holds only if a = b. (This inequality states that the li, 1 - () geo metric mean is less than the corresponding arithmetic mean .) *24. Let f be continuous and positive on (a, b] and let M denote its maxi mum value. Prove
a, b
M
= 2 i�
71
f[j(x)]" dx.
SECTION 3.7, page 248 1 . Let j(x) be a continuous function vanishing, together with its first derivative, for x = 0 . Show that j( x) vanishes to a higher order than x as - o.
x
a0xn + a , n -1 + · · · + Gn 2. Show that j( x ) = b "' oX + b1 x 1 + . . . + b , when a0 , b0 -F 0, is of the same order of magnitude as xn -m, when x x
m
m
�
oo .
*4. Prove that e' cannot satisfy an algebraic equation with polynomials i n x as coefficients. 5. If the order of magnitude of the positive function j(x) as x - oo is
*3. Prove that e" i s not a rational function.
higher, t he same, or lower than that of xm, prove that corresponding order of magnitude relative to x"'+l.
i"j(i;) a
d!; has the
320
The Technique., of Calculus
6. Compare the order of magnitude as X [(x) for the following func,tOns [(x) : a ( )
e v;;
v;:
(b) e'.
(c)
.
-
(d)
SECTION 3.8, page 263 ' ' as n 1. Fm d th e l 1m1t •2.
Find the limit of bn
03. If
=
a.
oc
xe>'.
Ch. 3
- i' d� rm
of
oc;
relative to
log x.
=
o f a,
I
+
n + 1
I
n +2
+
···
.
+
I
2n .
1 I I + . . + --:;"";;===;' + + = ===;;:;; v n2 - 0 v n2 - 1 v n2 - (n 1 )2 v n2 - 4
---=
--=
--=
is any real number greater than - 1 , evaluate
.
n ----. ro
hm
I " + z• + 3" + . . . + n•
-
n• ti
SECTION 3 . 1 1 , page 274 1. Show that for all odd positive values of n the integral I e�x'x" dx can be evaluated in terms of elementary functions. 2. Show that if n is even, the integral I e-•'xn dx can be evaluated in terms of elementary functions and the integral I e-x' d"· fo which tables have been constructed). 3. Prove that
f [ff( tl dt] du = fr
(r
- u) du.
04. Problem 3 gives a formula for the second iterated integral. Prove that the nth iterated integral of f(x) is given by
--- lx (n J 1 ) !
5.
0
G) fx''( l '- x)n-k rl {(u)(o·
Prove for the binomial coefficient
(: ) [ =
en + 1 )
u) n -l du.
·
that
dx
6 . Obtain a recursive formula for
J
x"(ax" + b)•
dx
and use this relation to integrate r
J
x3(x1 + 1 )4 dx.
*7. (a � Let P (x) = ..
l J i J
(b) Prove that
I d" ( r2 - ! )" . Show that 2 nn ! dx" .
f/
-1
.. (x)P,.(x) dx = 0,
Pn2(x) dx = 2
Problems 321
if m '#- n .
2 n + I
-- ·
-1 (c) Prove that x"'P,.(x) d:t· = 0, if m < n.
(d) Evaluate
1
x"P,.(x) da:.
1
SECTION 3 . 1 2, page 282
* I . Integrate
2. Use the partial fraction expansion to prove Newton's formulas
for k = 0, I , 2, . . . , n - 2 for k = n - I , where g(x) is a polynomial of t he form ,,., + <X 1 :r" -1 + · · · with distinct roots ll.h . . . , cxn · SECTION 3 . 1 4, page 298
* 1 . Prove that the substitution x = (<Xt + fJ)/()'t + a) with <XO - yfJ '#- 0, transforms the integral
f
dx
Vax4 + bx3 + cx2 + dx +
e
into an integral of simi lar type, and that if the biquadratic a>·4 + b x3 + cx2 + dx +
e
has no repeated factors, neither has the new biquadratic in t which takes its place . Prove that the same is true for
J R(x, V
a:r4 + bx3 + cx2 + dx + e) d:r,
where R is a rational function . 2. The function
cp(x) =
rx
J0
du
\I I - k 2 sin2 u
is k nown as the elliptic integral of the first kind. (a) Show that ¢ is continuous and i ncreasing and hence has a continuous inverse. (b) Let am(x) denote the inverse of cp(x). Prove sn(x ) = sin [am(x)], where sn(x ) is defined on p. 299, footnote 3 .
322
The Techniques of Calculus
Ch. J
SECTION 3 . 1 5, page 301 •t. Prove that
i [ ( �) J i "' = i "' "'
sin 2 " x +
•2 . Prove that hm
k-"' .
0
dx
I + kX
10 -
3. For what values of s is (a) 04. Does
i oo
dx does not eKist.
0
sin t -- dt converge ? 1 +t
0.
x•-1 dx, (b) I +X
--
i "' 0
sin x -,- dx convergent ? X
•s. (a) If a is a fiKed positive number, prove that 0
h-- ...0+ lim
fa
-a
dx = "· 2 + X2h h
-
(b) Ifj(x) is continuous in the interval - 1 � x � I , prove that
lim
h-0
06. Prove that lim e-x'
J1
-1
h (x) dx = "f(O) . +X j
h·�
= 0.
{ "e'' dt
Jo
Assuming that I ocl ,t. I ill. prove that T f sin oc x sin px dx = 0. lim o J 7'---.. oo " j (x) dx converges for any positive value of a, and if (x) tends to a •s. If j x - / ( /lx) dx converges for "' and !i hm1t L as x � 0, show that x-co
7.
.
�
f
.
i "' /(=)
a
0
positive and has the value L log � .
X
"'
9. By reference to the Problem 8, show that
i"' e-•x - e-�x (b)l "' f
(a)
0
0
X
dJ: = log -fl . OC
cos ocx - cos {3x d {3 x = I og - . X
<X
j (x) dx converges for any positive values of a and b, and if f(x) X tends to a limit M as � ro and a limit L as x � 0, show that "
• t o. If
a
,
{ "' j( rxx) - j( {3x) dx x
Jo
= (L - M) log � . "
Problems 323
11. Obtain the fol lowing expressions for the gamma function :
i"" dx, f ( �r-1d.r.
r(n)
=2
rcnl
=
SECTION 3 . 1 6, page 3 1 2
x2"-l e-�·
1og
1 . Obtain the addition formula for sin (x +
y).
2. Without using the addition formulas prove that cos x is an even fu nction and sin x, odd. 3. (a) * Prove for some positive h that cos "' < for 0 < .1· < h. (b) Prove if cos z > 0 for 0 s z s 2nx that
1
cos (2" 1 1x) < 2" (cos "' - I ) + I .
(c) Combining the resu lts (a) and (b) prove that cos :r has a zero.
4. Let a be the smallest positive zero of cos "' · Prove that
sin (:r + 4a) = sin ""·
cos (J' + 4a) = cos :r.
has no zeros, then sin ,,. is monotonical ly i ncreasing for x 2 0.
5. Fill in the steps of the fol lowing i ndirect proof that cos x has a zero : (a) If cos
x
(b) The functions sin x and cos .r are bounded from above and below.
(c) The limit of sin J' as (d) The equation
x
tends to i nfinity exists and is positive.
cos J.· = I -
stands in contradiction to (b).
i
' sin t dt
M ISC E L LANEOUS P R O B L E M S 1 . Prove
!!:.__ d.r"
when
1 =
{(log x) ·
log
.r.
=
(!!... - 1) (!!...dt - 2). ·
r" !!... dt dt ,
Here, we employ
(!!... k)
-
=
'!f. dt
-
k,P
· ·
•
(!!... - 1). dt
n
+
f(t) ·
where q, is any function of t and k is a constant. 2. A smooth closed curve C is said to be convex if it l ies wholly to one side of each tangent. Show that for the triangle of minimum area circumscribed about Z that each side is tangent to C at its midpoint.
4 Applications in Physics and Geometry
4.1
Theory of Plane Curves
a. Parametric Representation
Definition The representation of a curve by an equation y = f(x) imposes a serious geometrical restriction : A curve so represented must not be intersected at more than one point by any parallel to the y-axis. Usually, this restriction can be overcome by decom posing the curve into portions each representable in the form y = f(x). Thus a circle of radius a about the origi n is given by the two functions y = )a 2 - x 2 and y = J a• x2 defined for - a s x s a. H ow ever, for as simple a curve as a parallel to the y-axis this device does not work . More flexibility is obtained by an implicit representation through an equation �(x, y) = 0 which involves a function � of two independent variables. For example, the circle of radius a about the origin i s completely described by �(x, y ) = x2 + y2 - a2 = 0. A n y straight line i n the plane has an implicit equation of the form ax + by + c = 0, where a, b, c are constants and a and b do not both vanish ; for b = 0 we obtain a parallel to the y-axis. The implicit description of a curve has the disadvantage that to find points (x, y) of the curve at all, say for a given x, we must solve the equation �(x, y) = 0. This problem we shall discuss in detail in Volume I I . -
-
324
Sec. 4. 1
Theory of Plane Curves 325
The most direct and most flexible description of a curve is a para metric r-epresentation. Instead of considering one of the rectangular coordinates y or x as a function of the other we think of both coordi nates x and y as functions of a third independent variable t, a so-called parameter ; 1 the point with coordinates x and y then describes the curve as t traverses a corresponding interval. Such parametric repre sentations have already been encountered ; for example, the circle x2 + y2 = a2 has the parametric representation x = a cos t, y = a s i n t. Here t den o te s the angle at the center of the circle. y
Figure 4.1
For the ellipse x2/a2 + y2/b2 = 1 we have the similar p ara me tr i c representation x = a co s t , y = b s i n t, where t is the so-called eccentric angle, that is, the angle at the center corresponding to the point of the circumscribed circle lying vertically above or below the point P = (a cos t, b sin t) of the e llipse We assume here that b < a (see Fi g . 4 . 1 ). In both cases the point with the coordinates x, y describes the complete circle or ellipse as the parameter t traverses t h e interval 0 � t < 27T. In general, curves C are p ara metrically represented by two fu nct i o n s of a parameter t, .
x
=
cp(t)
=
x(t),
y
=
lj!(t)
=
y(t) ;
1 This word denotes an auxiliary variable which we do not want to emphasize
primarily.
326
Applications in Physics and Geometry
Ch. 4
the shorter notation x(t) and y(t) will be used when there is no danger of confusion. 1 We assume throughout that cp and !JI possess continuous derivatives unless the contrary is said. Mapping of Parameter Interval on Curve-Sense of Direction For a given curve these two functions cp(t) and !JI(t) must be deter mined in such a way that the set of pairs of functional values x( t ) and y(t) corresponding to a certain interval of values t defines all the points on the curve and no other points. We have then a correspondence between the points of the curve and the values of t in an interval of the t-axis. The parameter representation defines a mapping of the t-axis onto the curve, the original point t on the t-axis being mapped onto the point x = cp(t), y = 1p(t) of C. Since x(t) and y(t) are assumed continuous, neighboring points on the t-axis correspond to neighboring points on the curve. Since the points of the t-axis are ordered, we may in an obvious manner assign an order or "sense" to the points of C by saying that the point onto which the number 11 i s mapped precedes the point onto which 1 2 is mapped if 1 1 < t 2 (see p. 334). The parametric representation thus gives precise mean ing to the vague intuitive notion of a curve as a set of points i n which the points are arranged in the same order as on a straight line. b. Change of Parameters
The values of the parameter t serve to distinguish the different points on the curve C; they play the role of "names" for the individual points of the curve. The same curve C admits of many different parameter representations. Any quantity that varies continuously along the curve and has different values in different points of the curve can serve as parameter. If, say, the curve originally is given by an equation y = f(x), we can choose for the parameter t the variable x and describe the curve by the functions x = t, y = f(t). Similarly, for a curve described by giving x as a function of y, say x = g (y) , we can use y as parameter t and write X = g(t), y = t. • The notation x = cp(t), etc., puts emphasis on the specific functional connection
between the dependent and independent variable ; the notation x(t), etc., just means that t is to be considered as the independent variable which determines the function value of x in some prescribed way .
Sec. 4.1
Theory of Plane Curves 327
For a curve given by an equation r = h(O) in polar coordinates r, (} (see Chapter I , p. 1 0 1 ) we can choose 0 as parameter t and obtain the parametric representation X = r COS (} = h(t) COS I
cp(t),
y = r sin 0 = h(t) sin t = tp(t). =
From a given parametric representation x = cp(t), y = tp(t) of a curve C we can always derive many other parameter representations. For that purpose we take an arbitrary function 7' = x(t) which is monotonic a n d continuous in that t-interval corresponding to the points of C; the function x has then a monotone and continuous inverse I = a( 7') in a corresponding 7'-interval . The coordinates of the points (x, y) of C can then be represented in the form The functions ot( 7') and �( 7' ) are again continuous ; moreover, different points of C correspond to different values of t and hence, because of the monotone character of the function a, to different val ues of 7' . The total effect of the change of parameter from t to 7' is that of "renaming" the points on C. Thus the line y = :r has the parameter representation r = t, y = t, where - c:tJ < 1 < c:tJ. Substituting 7' = 13 gives rise to the parameter representation x = 7' 1 , y = 7' \ :i for the same line. Similarly, the elli pse 2 fa 2 + y2fh2 = I ad mits of the parameter representation x = a cos I , y = h sin I, where 0 � t < 27T. Defining t = c� + d, for c , d real numbers (c ;1:. 0) yields another representation x W = a cos (c� + d), YW = b sin (c� + d ) for the same ellipse, with � varying in the interval - dfc � � < (27T - d)/c for c > 0, and (2rr - d)/c < � � - d/ c , for c < 0. The substitution 7' = tan (t/2) leads to the "rational" parameter representation (see p . 292)
' x
x=
---
a( l - 7'2) 1 + 7'2 '
for the ellipse ; [.S 7' runs through all real values we obtain all points of the ellipse with the exception of the point S = ( -a 0). Singularities in ordinary representation may disappear i f a suitable parameter is used . For example, we can represent the curve y = � by the smooth functions x = t3, y = t2• The point with coordinates x, y then descri bes the whole curve (semicubical parabola) as t varies from - c:tJ to + c:tJ . ,
328
Applications in Physics and Geometry
Ch. 4
This flexibility in the choice of the parameter often permits us to simplify the study of geometrical properties which, of course, do not depend on specific representations. I n particular, we may sometimes find i t convenient to use a repre sentation y = f(x) for C or part of C. Such a representation is always possible for a portion of the curve 10 � t � 11 in which one of the functions , IJ!, say x = c/>(t), is monotonic. Indeed for this portion we have a unique inverse function t = y(x) and thus y = !J![y(x)].l
c.
Motion along a Curve. Time as the Parameter. Example of the Cycloid
Motion along a Curve Very often the parameter t has the natural physical meaning of time. Any motion of a point in the plane may be expressed by representing its coordinates x and y as functions of the time such that at the time t, the point (x, y) is at (x(t), y(t)). These two functions therefore deter mine the motion along a path or trajectory C in parametric form ; they constitute a mapping of the time scale onto the trajectory. 2
The Cycloids and Trochoids An example i s furnished by the cycloids, the paths of points on a circle rolling uniformly without slipping along a straight line or another circle. In the si mplest case a circle of radius a rolls along the x-axis ; the path of a point P on its circumference is a "common" cycloid. We choose the origin of the coord i nate system and the initial time in s uch a way that for time t = 0 the point P is at the origi n and that at the time t the circle has turned from its original orientation by the angle t. This means that the circle turns clockwise with "angular velocity" one. The circle is assumed to roll uniformly along the x-axis without sliding so that at the time t the distance of the point of contact from the origin is exactly equal to the length of the arc from the point of contact to P. Thus at the time t the center M of the rolling circle must be at the point (at, a) ; the center moves with constant velocity a to the right. For the
1
Th is is, of cou rse, merely a statement about a property "in the small" of a curve, meaning a statement made only for a su itably sma l l portion. Usually (for example, in the case of a circle), the va r i a b l e x cannot be used as a parameter throughout the whole curve but only on a portion. 2 To a c hange of the parameter 1 there would correspond then a change in the time scale acco rding to which the �urve C is described by the moving point.
Sec. 4. 1
Theory of Plane
Curves 329
coordinates of P at the time I we find then (see Fig. 4.2) the parametric represerrtation (I)
x =
a(t
-
y = a( I
sin 1),
-
cos 1).
By eliminating the parameter r we can obtain the equation of the curve in nonparametric form, at the cost, however, of neatness of y
t
Figure
-
4.2
-
expression. We h ave cos t = a
a
y ,
t = arc cos
and hence ( Ia) thus obtaining
J' ;r
= a arc cos
Cycloid.
a
a
y ,
a - y
· --
a
sin t = ±
1 =F y y(2a
J�
_
(a
� cr
- y),
as a function of y.
Epicycloid Our next example is that of an epicycloid, defined as the pat h of a point P fixed on the circumference of a circle of radius c, as it rolls at a uniform speed along the circu mference and outside of a secon d circle of radius Let the fixed circle be centered at the origin of the ":,y-plane. Suppose the moving circle is rolling along the fixed one i n such a way that its center has rotated about the origin to an angle r at time t ( Fig. 4 . 3 ) Then we find for the position at the time t of the point P = (x(f), y(t)), which at the time t = 0 is the poi nt of contact 0) , the parametric equations
(a,
( 2)
:r(r) y(f)
(a + = (a =
c) cos
+ c) sin
r 1
. - (a + ) - (a + ) c
c
cos --
c sin
c
c
-- I c
I
.
.
.
a.
,
330
Ch. 4
Applications in Physics and Geometry
y
Figure 4.3
Epicycloid.
y
Figure 4.4
Cardiod.
Theory of Plane Curves 3 3 1
Sec. 4.1
When a = c, t he curve formed is called a cardiod, (Fig. 4.4) and is given by the parametric equations x(t) = 2a cos t a cos (2t), ( 3) y(t) = 2 sin t - a sin (21 ) .
-
a
A third variety of cycloids is obtained as the locus of a point attached to the circ•1mference of one circle rolling along the circumference of another fixed circle, but interior to it. To find the parametric equations for this "hypocycloid," let be the radius of the fixed circle and c that of the rolling circle. Let the point P on the circumference of the moving circle be located
a
y
Figure 4.5
Hy pocycloid.
at (a, 0) at time 1 = 0. Suppose that the rolling circle is moving along the fixed one in such a way that at time 1 its center has rotated about the origin through an angle 1 (Fig. 4 . 5 ). Then we find the parametric equations for the hypocycloid to be (4)
:r(l) = (a
y(l) =
(a
- r) cos
I
+ c cos
- c) sin t -
c sin
(a--.- )' (a---.- ) -c (
(
c
t
I
,
.
In the special case when the fixed circle has twice the radius of the moving one, c = � a. we find x(l) = a cos I,
y(l) = 0,
332
Applications in
Physics and Geometry
Ch. 4
and the hypocycloid degenerates into the diameter of the fixed circle, de scribed back and forth. The interesting feature of this example is that it provides a mechanical solution of the problem of drawing a straight line by using merely circular motions ( Fig. 4.6).
y
Figure 4.6
A point
P on the rim of a circle rolling inside a ci rcle of twice the radi u s
describes a straight l i n e segment.
I f the radius of the fixed circle is three times that of the moving one, then c = a/3, and x(t) = fa cos I + �a cos (21),
y(t)
= !a sin
t
-
�a sin (21).
By an elementary computation we find
x2 + y2
=
g a2 + : a 2 cos (31),
so that the hypocycloid meets the fixed circle at exactly three points and the curve appears as shown in Fig. 4 . 5 .
Trochoids M ore general curves called trochoids (epitrochoids, hypotrochoids) are obtained if we consider the motion of a point P attached to a circle (but not necessarily on its rim) when t'<�at circle rolls along a straight l i ne or along the outside or inside of another circle (see Fig. 4. 7). The same type of curve arises as the path of a point moving uniformly on a circle while the center of the circle itself moves uniformly along a line or circle. These curves play a
Sec. 4. 1
Theory of Plane Curves 333
y
X
Figure 4.7
Trochoid.
central role in the Ptolomaic description of the apparent motion of the planets. Some of the remarkable properties of cycloids will be discussed later on in this chapter (p . 428). d. Classifications of Curves. Orientation
Definitions Among the m ost obvious features of a curve are the number of separate pieces or branches and the number of loops which it has. A hyperbola is an example of a curve consisting of two disjoint branches ; another such example is the curve y2 = (4 - x 2 )(x2 - I ) which consists of two separate ovals. We shall be concerned mainly with curves consisting of one piece, the connected curves. A con nected curve can intersect itself, l i ke the trochoid (Fig. 4.7) or the "lemniscate" of Fig. 1 53 , p. 1 03 . A connected curve without self-intersections i s called simple. Among the simple curves we can still distinguish the closed curves, such as circles or ellipses, from the ones that are not closed, such as parabolas or straight-line segments. We shall not attempt here to give either a rigorous or a complete classification of curves, but only point out certain " topological" features of a curve relevant for parameter representation.
Simple Arcs A parameter representation of a curve C by two contin uous functions ,P(t), y = -rp(l) defines a mapping of the t-axis or of a portion of it
x =
3 34
Applications in Physics and Geometry
Ch. 4
onto C. We call C a simple arc if it can be represented in such a way that the parameter t describes a closed interval [a , b] on the t-axis, forming the domain of the functions cp(t), tp(l), and if in addition different 1 in the interval correspond to different points P on C. An example is the parabolic arc x = t, y = 1 2 for 0 :::;; t :::;; I . The same arc C (that is, the same points i n the plane) can be repre sented parametrically in many ways. Any monotone continuous function -r = x( t ) for a S t :::;; b defines a parameter -r such that x and y are continuous functions of -r in a suitable closed interval [o:, p], different values of -r corresponding to different P. As a matter of fact, as is easily seen the continuous monotone substitutions -r = x(t) provide the m os t general continuous parameter representations of a s imple arc that assign different points of the arc to d i fferent parameter values. (See the remarks on p. 55 about one-to-one continuous mappings.) To a special parameter representation x = x( t ), y = y(t) of a simple arc C belongs a definite sense on C corresponding to the direction of increasing t. Given any two distinct points P0, P1 we say that P, follows P0 if P1 belongs to the larger value of the parameter t. If we i ntroduce a new parameter -r by a continuous increasing function -r = x(t) the order of the pairs of points with respect to -r is the same ; the parameter -r defines the same sense on C. I f x(t) is decreasing, the sense is reversed. Direction or Orientation of Arcs
A directed or oriented simple arc is one on which a definite sense has been selected (for example, that sense corresponding to an increase in a particular choice of the parameter 1) ; that sense is then called the positil•e sense on the arc. The positive sense i s completely specified . if we know wh ich of the two end points of the arc follows the other one. We call the end point that follows, the final point of the arc, and the other one the initial point. Given any parameter representation x = x(-r), y = y(-r) of the oriented arc, where a S -r :::;; b, the positive sense will be that of increasing -r if the parameter value -r = a corre sponds to the initial point and -r = b to the final point ; otherwise the sense of increasing -r will be the negative sense on the arc (Fig. 4.8) . Any t w o distinct points P0, P1 o n a simple arc C define a sub-arc with end points P0, P1 , which consists of the points with parameter values between those of P0 and P1 • If C is a directed arc and P, follows P0 i n the positive sense on C, we obtain a directed sub-arc with initial point P0 and final point P1 • A finite number of points of subdivision on a directed simple arc C breaks up that arc i nto a sequence of directed
Sec. 4 . 1
Theory of Pla11e Curves 335 0 0
a
(j
Figure 4.8
to
b
t,
\
Po
T!
ro
a
T
!A
Sense and parameter representat ion.
sub-arcs, the initial point of one sub-arc being the final point of the preceding one. Often i t is impractical to restrict oneself to simple arcs and to insist that different parameter values t shall belong to different points of the curve. If, for example, the equations x = x(t), y = y(t) give the posi tion of a moving particle P at the time t, there is no reason why the particle should not stand still for a while or why its path should not be allowed to cross itself so that the particle returns to the same position at a l ater time. y
Figure 4.9
A curve with a loop : x = t' - I , y =
13 -
t with
sense of i ncreasing I .
336 Applications in Physics and Geometry
Ch. 4
An example is the curve x = 1 2 - I , y = t3 - t [which also could be described completely by the cubic equation y2 - x2( 1 + x) = 0]. As t varies from - oo to + oo the curve crosses the origin twice, for t = - I and t = + I (Fig. 4.9). We verify easily that all other points of the curve belong to a unique value of 1. Geometrically, the interval - I < t < + I corresponds to a loop of the curve. Here again the sense of increasing t defines a certain order among the points of the curve, at least if we visualize in some way the points corresponding to t = - 1 and t = + I as distinct, one lying "on top" of the other one. The whole oriented cubic curve can be decomposed i nto directed simple arcs, for example, i nto the arcs corresponding to n � t � n + 1 , where n runs over all integers. Closed Curves
The standard exam ple of a parameter representation in wh ich different t correspond to the same point on the curve is given by the formulas X = a COS I, y = a sin I, which describe the uniform motion of a point on a circle with 1 as the time. As t varies from - oo to + oo the point P = (x, y) describes the circle infinitely often in the counterclockwise sense. We can cause the points of the circle to be described exactly once by restricting 1 to any half-open interval of length 27T : IX
� t <
IX
+ 27T.
The end points IX and IX + 27T of the interval correspond to the same point on the circle. Here the end points of the parameter interval have no special geometrical significance for the curve. Generally, a pair of continuous functions x = cf(t), y = 1p(1) defined in a closed interval a � t � b will represent a closed curve if cf(a) = cf(b), 1p(a) = 1p(b). The closed curve will be si mple if different t-values with a � t < b correspond to different points (x, y). The point corresponding to t = a and t = b could be any point on the curve ; it is just the point at which we "break" the curve to make its points correspond to those of an interval on the axis. Closed Curr·es Represented by Periodic Functions
Just as in the example of the circle we can avoid d istinguishing any particular break by taking for cf(t) and 1p(1) periodic functions with period p = b - a. It is of value here to make some general remarks
Sec. 4.1
Theory of Plane Curves 337
about periodic functions to which we will turn more extensively in Chapter 8. A function f(t) is called periodic with period p if it i s defined for ali t and satisfies the equation f(t) = f( t + p). Thus, for example, the trigonometric functions sin t and cos t are periodic with period 2rr. (Any multiple 2nrr where n is an integer is then also a period . ) Geo metrically interpreted f(t) has the period p if a shift of its graph by p units to the right leads to the same graph again.
' p
Figure
4.10
Graph of a periodic function j(t ).
Since thenf(t) "repe �ts" itself, a functionf(t) of period p is determi ned for all t if it is known merely in a single interval a s t < h of length p = b - a ( Fig. 4. 1 0). Indeed for every t there exi sts a value 1' in the interval a s t ' < b such that t - t ' = np, where n is an integer [one only has to take for n the largest integer that does not exceed (t - a)/p]. Then f(t) = f(t ') i s known. As a matter of fact we can start with any continuous function f(l ) in a half-open i nterval a s t < b; the extended function will clearly be
f I
/
/
a-p Figure
4.1 1
/
/
,.. -'
/
a
I
//
/
b=a+p
/
"' _,
"
1 1 I
/
/
a + 2p
/.,., .....
I
/
/ /
a + 3p
Periodic continuation of a function ((! ) from the i nterval a
:S:
t
< b.
continuous for all 1 which are not of the form t = a + np with an integer n (Fig. 4. 1 I ) . For example, extending the function /( 1 ) defined by f(t) = t for 0 s t < I periodically, leads to a function of period p = I which we can call the "fractional part of t," and which i s discontinuous at points 1 which are i ntegers (Fig. 4. 1 2a). Generally, at t = a + np the periodi cally extended function f will have the value f(a) ; this will also be the limit ofjon approaching the point from the right, whereas the limit off from the left will be the same as at the point b. I n the case of greatest
338
Ck. 4
Applications in Physics and Geometry
interest to us at present we start out with a function defined and continuous in the closed in terval a � t � b which moreover has the same value in the end points f(a) = f(b). Continuing such a function periodically always leads to a function f( t) of period p = b - a which is continuous for all t. (Fig. 4. 1 2b.) Continuous periodic functions are ideal for representing closed curves C. Let C be given parametrically by x = cf>(t), y = tp(t), for , tp continuous i n the interval a � t � b and hav in g the same values in both end points . We can extend the defi nitio n of these functions to all values
-I
(a)f(t) = t, (b) .f(l) =
Figure 4.12 H e re
(b)
Periodic continuation of functions
21 - 21'.
f(t )
from the interval
0 � t < I.
of t i n such a way that cf> and tp have period b - a = p and are con tinuous for all t. F or any t the extended parameter representation only yields points of C, since we have t = t ' + np with n an i n teger and a � t ' � b . The point corresponding to t is then the same as the one corresponding to t', which lies on C. As t varies from - oo to + oo the point (x, y) traverses the curve C infi nitely often , just as in the circle x = a cos t, y = a sin t. Here the distinguished role of the parameter value t = a is removed. For any oc the whole curve is already repre sented by x = cf>(t), y = w(t) when t runs from oc to oc + p. A portion of the closed curve C corresponding to the parameter values t in an interval oc � t � p form s a simple arc if different /-values
Sec. 4 . /
Theory of Plane Curves 339
in that. interval lead to different points (x, y). The whole closed curve C is a simple c urve if different t in the same interval oc :5:: t < oc + p always lead to different points on C. Thus any closed parameter interval of length less than p gives a simple arc. Closed Cun•es Composed of Simple A rcs. Order of Points
The closed curves which we shall consider can all be decomposed into simple arcs. If the whole closed curve C is sim ple, it can be decomposed into two simple arcs t0 s t s t 1 and t 1 s t :5:: 10 + p which have only their end points P0, P1 in common . The sense of increasing t determines a positive sense or orientation on C by fixing a positive direction on each simple arc of C. Any two distinct points P0, P 1 on the simple closed curve C divide C into two simple arcs. In the sense of increasing t exactly one of the two arcs will have P0 as the initial point and P1 as the end point ; we will call it P0P1 : the reverse holds for the other arc. Orientation and Order
The positive orientation of C can also be characterized by an ordered triple of points P0P1 P2 of C if we specify that P2 does not lie on the
� - - - - -o- - � t t0 t1 t2 t0 + p
(a)
Figure 4.13
(b)
I I I I I
I
//
Pz ,.,. -- - ""'0.. .---.. ',
\
\ \ I I I
�\�1Po P1
--
Orientation of closed curves in the sense of increasing
I.
340 Applications in Physics and Geometry
Ch. 4
simple directed arc with initial point P0 and final point P1• The tri pl es P1P2P0 and P2P0P1 obtained by a cyclic permutation from P0P1P2 describe the same orientation (Fig. 4 . 1 3a). * Quite generally, any n distinct points on the oriented closed simple curve C always follow each other in a certain order P1P2 • • • P,. deter mined up to cyclic permutations', and divide C into directed simple arcs, P1P2, , P11_1P11, P.P1• We can always choose parameter values t1, t2, tn for the points P" P2 , P,. such that the t; form a monotone increasing sequence and are all contained in one and the same parameter interval of length equal to the period p (Fig. 4 . 1 3b). •
•
•
•
•
•
,
•
•
•
,
Orientation of Curves and A ngles
As already emphasized i n Chapter I we are forced to make use of the sign plus or minus to establish satisfactory relations between y
D Figure 4.14 Angle of i nclination
I of a direction
D.
geometric entities and analytic c on cepts expressed by numbers. Directed lines, such as the n umber axis, are the simplest instances. Which direction on a l ine we define as positive i s arbitrary at the beginning. A positive sense co rre spond ing to increasing t can be associated with any particular parameter representation x = at + b, y = ct + d o f the line. A line oriented in this way points in a certai n direction . Two parallel directed lines have either the same or the opposite d i rect io n . A d irection can also be determined by a ray issuing from a poi n t Po, that is, by a half-line which consists of the points on a line which "follow" a given point P0 in the po sitiv e sense. 1 That is, P,P, · tation .
· ·
P.P1, P,P,,
· · · P.P1 P2 , •
• •
,
P.P1
• • •
Pn- 1 give the same orien
Sec. 4.1
Theory of Plane Curves 341
Any direction in the plane can be represented by a ray from the origin or also by the point P on the circle of radius I about the origin that lies on that ray. If we represent this unit circle parametrically by x = cos t, y = sin t, we have associated with every direction certai n values t, differing from each other by multiples of 2rr . We call them the angles of inclination of the direction or the angles the direction makes with the positive x-axis. There is always exactly one angle of i nclination t for which 0 � t < 21r ( Fig. 4. 1 4). The angles between t wo directions are simply the differences of their angles of i nclination. M ore precisely, since the order in which we take the two di rections matters, we say that a direction with inclination t ' forms with a direction with inclination t" an angle a. = t ' - t" (Fig. 4. 1 5). rr
y
IY
Figure
4.15 Angle "' the di rection D' forms with the direction D".
Since t and t' can be changed by integral multiples of 21r, the same change is permissible for the angle one direction makes with another one. Sense of Rotation
We also say that the direction with angle of inclination t" passes into that with direction t' by a rotation through the angle a. . The i ntuitive i dea of rotation here i s that of a continuous motion, by which the direc tion with inclination t" goes into that with i nclination t' by passi ng th rough directions with all possible i nclinations t intermediate between t" and t'. We call the rotation positive or counterclockwise if a. = t' - t" i s positive, and negative or clockwise in the opposite case. Of course, there are many different rotations both clockwise and counter clockwise that will take a given direction into another given one unless we i nsist that the angle of rotation a. satisfies -1r < a. � 1r.
342
Applications in Physics and Geometry
Ch. 4
Ultimately then, the positive sense of rotation is associated with a particular parameter representation x = cos t, y = sin t of the circle which we have chosen. If as usual, the x-axis points to the ri gh t and the y-axis upwards, then the positive sense of rotation coin cides with the sense opp o s i te to that of the hands on a conventional clock. 1 Positit•e and Negative Sides of a Curt'e
A curve separates the points of the plane near one of it s p oi nt s P i nto two classes. Loca l l y at least we can d i s tingu i sh two "sides" of the curve. lf the curve C i s oriented, we can define a positive (or "left") and a negative (or r i gh t ) side2 as follows : Consider a ray issuing from P. We say that this ray points to the p o s itive side of the curve if there are points Q o n the curve arbitrarily close to P and fo ll o wi n g P in the sense given to the curve, such that the angle through which a line from P to Q m ust be rotated in the counterclock wise sense to reach the given ray, l ies between 0 an d TT ( Fig. 4. 1 6). The points on the ray c lo se to P are then said to lie on the po s i t ive side of th e curve. I n the opposite case the ray is said to point to the negative side of C, an d the points on it are said to lie on the Figure 4.16 Positive and negative side of the curve. If the curve C is negat ive side of oriented a simple closed curve, it divides all points arc. of the p l ane into two classes, those interior to C and those exterior to C.3 We say that C has the counterclock wise orientation if its i nteri or lies on the positive (that is, left) side (Fig. 4. 1 7). lf the closed curve C , h owever, consists of several loops, then it is not always possible to describe C so t h at all enclosed regions are on the positive side of C (see Fig. 4. 1 8). "
"
1 This sense, in turn, is suggested by the mot ion of the shadow on t he ground
in a sun dial in the northern hemisphere. ' The terms "left" and " right" �ide correspond to the ordinary usage of the words "left bank" and "right bank'" for a river oriented by its direction of flow. 3 These concepts as well as the d ivision of the plane by a simple closed continuous curve into two parts are anal y zed precisely in topology and must be accepted here on an intuitive basis.
Sec. 4. /
Figure 4.17
Theory of Plane Curves 343
S i mple closed curve with counterclockwise orientation.
Figure 4.18
+
e. Derivatives, Tangent and Normal, in Parametric Representation
Direction and Speed
For a curve C given m parameter representation with the time parameter t x = x(t) = ,P(t}, y = y(t) = 1j!( t ) we denote the derivatives, as Newton did, by a dot : i:
y = d1j! = �= d = ¢, dt dt The deri vatives i:, li are often conveniently visualized as the "velocity components" or the " speeds" of the coordinates of a point P moving along C. Whenever i: � 0, it is possible to represent the corresponding portion of C by an eq uation y = f(x) by first calculating t as a function of x fro m t he first equation and then substituting the resulting expression for t into the second equation. By the chain rule of differentiation and the rule for the derivative of the inverse of a function (see p. 207) we find then for the slope of the tangent to the curve dy
dy = dy .!!!_ dt dx dx
The equivalent formula dxfdy
=
= .!!!.. = � i dx
dt :ify holds if y � 0.
344
Applications in Physics and Geometry
Ch. 4
Unless the contrary is stated we always assume that i a n d y do not vanish simultaneously or, concisely written, we assume i;2
+
y2
'F
0.
Then the tangent always exists ; 1 it is h o ri zo n tal if y = 0 and vert ic al if i = 0. For the cycloid, for example, [see Eq . ( 1 ) , p. 329] we have
i = a( l
- cos t) = 2a
y
=
a s in t
dy dx
=
cot
-
t
-
2
=
sin2 � ,
2a sin .!. co s .!. 2 2
,
.
These formulas show that i2 + y 2 -F 0 except fo r t = 0, ± 2rr, ± 4rr, . . . . Moreover, the cycloid has a cusp (that is, a point where it reverses direction), with a vertical tangent at those exceptional points at which it also meets the x-axis, that is, when y = 0; for on approach ing these points, the d erivative y ' = y/i = cot (t/2) becomes infin ite. Tangent, Normal, and Direction Cosines
The equation of the tangent to the curve at the p o i n t x, y is
;I are the "running" coordinates corres p onding to an a rbitrary point on the tangent, whereas x, y, and dyfdx h ave the fixed values belonging to the point of contact. Substituting yfi for dyfdx we can write the eq uation of the tangent i n the form
where � and
(5)
(� - x)y
-
( 1) - y):i
=
0.
Exactly the same eq u ation is obtained under the a ss u mpt i on y -F 0 ; we on l y have to express x as a fu nction of y . I n the exceptional points where both i and y vani s h for the same t t he equation becomes meaningless, since it is satisfied for all ; 1) . ,
,
1 We observe that the condition x' + y' � 0, although sufficient, is not necessary to guarantee a nonparametric representation. Thus we may define the curve y = x' by means of the parametric equations x = r • , y = r • . At the origin of the t-axis, the condition of positivity for x' + y' fai ls, but still the curve has a definite and well defined nonparametric representation.
Sec. 4. 1
Theory of Plane Curves 345
The norma/ to the cun•e, that is, the straight line through a point of the curve perpendicular to the tangent at that point, has the slope - dx /dy Th is leads to the equation .
(� - .T) i + (17 - Y)!i = 0
(6)
for the normal. If a point of C corresponds to several values of t, then in general a different tangent exists for each of the branches of the curve passing th rough the point, or for each value of 1. For example, the curve x = 1 2 - I , y = 13 - I ( Fig. 4.9, p. 335) passes through the origin for I = - I and t = + I . For t = - I we find for the eq uation of the tangent � + 'YJ = 0, whereas the tangent for t = + I is given by � - fJ = 0. From the definition of derivative we have dy - = -!i = tan :i dx
:x ,
where :x is the angle the tangent makes with the x-axis. This means a rotation by the angle :x applied to the x-axis (countercl ockwise if :x > 0, clockwise if :x < 0) will cause it to be paral lel to the tangent. Rotations by the angles :x ± rr, :x ± 2rr, . . . will then also make the :r-axis parallel to the tangent. Hence the angle :x is determined only to within a multiple of rr, whereas tan :x is determined uniquely. From the relations yf:i = (sin :x)/(cos :x) and :i2 + :t? ;'= 0 we fi nd cos :x = ±
I -v x"
+ !?
,
.
S ill :X
= ±
if
J i:" + :¥"
-=
(where the same sign must be taken i n both form ulas). We call cos and sin :x the direction cosines of the tangent. '
:x
Assigning Directions t o Tangent and Normal
The two possible choices for the d irection cosines correspond to the two directions i n wh ich we can traverse the tangent ; the corresponding angles. 11. differ by an odd multiple of rr . One of the two directions on the tangent corresponds to increasi ng t, the other one to decreasing t. Assume that the sense on the curve is that of increasing t : Then, by defin ition, the positive direction on the tangent, or the one that corre sponds to increasing val ues of t, is the one that forms with the positive ' One t hinks here of sin with the tangent.
oc
as cos {1, where fl
= rr/2
-
oc
is the angle the y-axis forms
346
Applications in Physics and Geometry
Ch. 4
x-axis an angle IX for which cos IX has the same sign as x and sin IX the same sign as y . The direction cosines of that direction on the tangent are then, without ambiguity, y . COS IX = J S i n iX = J (7) , 2 2 j;2 + y2 y j; + .
I f, say, x = dxfdt > 0, then the direction of increasing t on the tangent is that of increasing x; the angle that direction forms with the positive y
Tangent
a
�0�------��--L---� x {3
Figure 4.19
Positive tangent and normal of an oriented curve.
x-axis has then a positive cosine. Similarly, that normal direction obtained by rotating the direction of the positive tangent corresponding to increasing t in the positive (counterclockwise) sense by 7T/2 has the unambiguous direction cosines i -y _ J _ x _ z_ sin IX + = COS IX + -. = J , y_·z j;2 + y2 2 +
( 7T)
( �)
__
It is called the positive normal direction and points to the "positive side" of the curve ( Fig. 4 . 1 9). I f we introduce a new parameter T = x(t) on the curve, then the values of cos IX and sin IX stay unchanged if dT/dt > 0 and they change sign if dT/dt < 0 ; that is, if we change the sense of the curve, then the positive sense of tangent and normal likewise is changed .
Sec. 4 . 1
Theory of Plane Curves 347
Criticill Points
If x and y are continuous and x2 + y 2 > 0, the quantities cos IX and sin IX wh ich determine the direction of the tangent will vary continuously with t. The tangent, whose equation is (; - x) sin IX - (1] - y) cos IX = 0,
then changes continuously along the curve, as does the normal. lf both x and y vanish for a certain value of t, the direction cosines of the tangent are not defined by our formulas ; a tangent may fail to exist altogether or it m a y not be determined uniquely. Such a point is called a "critical" point or a "stationary" point. We illustrate by examples various possibilities that arise at critical points. One example is furnished by the curve y = lxl with the parameter representation x = 1 3 , y = l t l 3 ; this curve has a corner for t = 0 although both x and y stay continuous. In the example of the cycloid, discussed on p . 344, the " stationary" points at which x = y = 0 corre spond to cusps. On t h e other h and, the vanishing of x and y i n some cases is merely inherent in a specific parameter representation and not con nected with the behavior of the curve, as for the straight line repre sented by x = t3, y = 1 3 for the parameter value t = 0. Corners
Curves conststmg of several smooth arcs meeting at corners are represented conveniently in parameter representation by functions x(t), y(t) wh ich are continuous but have derivatives x, if with jump dis conti nuities. This is illustrated by the trivial example of the broken y
Figure
4.20 Graph
of
x = t, Y =
Ht + I t ! ) .
348 Applications in Physics and Geometry
Ch. 4
line represented by X =
and
f,
X = !,
y=O
for t � 0
y = t
for t � 0.
Here x = I , y = 0 for t < 0 and x = I , y the tangent is indeterminate (see Fig. 4 . 20).
=
1 for t > 0. At t = 0
f. The Length of a Curve
The Length as an Integral
Two different types of geometrical properties or q uant1t1es are associated with curves. The first type depends only on the behavior of the curve in the small, that is, in the immediate neighborhood of a point ; such properties are those which can be expressed by means of derivatives at the point. Properties of the second type or properties in the large depend on the whole configuration of the curve or of a portion of the curve, and are usually expressed analytically by means of the concept of integral. We shall begin by considering a q uantity of the second type, the length of a curve. Of course, we have an intuitive notion of what we mean by the length of a curve. However, j ust as in the classical case of circular arcs, a precise mathematical meaning m ust be given to the intuitive concept. Guided by intuition we define the length of an arbitrary curve as the limit of the lengths of approximating polygons, in particular, inscribed polygons. The lengths of polygons, in turn, are immediately defined as soon as a unit of length is chosen . The final result will be the expression of length by an integral . We assume our curve given in the form x = x(t), y = y(t), rx � t � {3. In the interval between rx and {3 we choose intermediate points t1, t 2 , , t,_ 1 such that .
•
•
!X = t0 < ! 1 <
fz
< · · · < t ,._ 1 < t ,.
=
{3.
We join the points P0, PI > . . . , P,. on the curve corresponding to these values t, in order, by line segments, thus obtaining an i nscribed polygon. The length of the perimeter of this inscribed polygon depends on the way in which the points t,, or the vertices P, of the polygon, are chosen. We now let the number of the points t, increase beyond all bounds in such a way that at the same time the length of the longest subinterval (t,, ti+ 1 ) tends to zero. The length of the curve is then defined to be the l imit of the perimeters of these inscribed polygons, provided that such
Sec. 4. 1
Theory of Plane Curves 349
a limit exists and is independent of the particular way in which the polygo ns are chosen. When this assumption (assumption of recti fiability) is fulfilled , we can speak of the length of the curve. We assume that the functions x(t) and y(t) have continuous derivatives :i:(t) and y(t) for IX � t � {3. The inscribed polygon corresponding to the subdivision of the t-interval by points I; with !:J. t; = t; H - t; has vertices P; = (x(t; ) , y(t ; )) ; its total length is given by the expression n -1
S
n-l
= L P;Pi+1 = L J [x( ti+1) - x(1;)) 2 + [ y(t;H ) - y( t;W i=O i=O
according to the theorem of Pythagoras (cf. Fig. 4 .2 1 , p. 356). By the mean value theorem of differential calculus where �; and Y/; are intermediate values in the interval t; < t < t;H· This leads to the expression n-1
sn
= L J [x2( 0J2 + [?/Cn.) J2 tJ.r. i=O
for the length of the polygon, where we have made use of the fact that the differences !:J.t; are positive. If the number n of points of subdivision t ; i ncreases beyond all bounds while at the same time the largest value !:J. ti tends to zero, the sum sn tends to the integral
L=
J:J x2 +
y 2 dt.
This fact is a direct consequence of the existence theorems for integrals in Chapter 2.1 This proves that for continuous i, y the curve actually has a length and that this length is given analytically by the expression (8 ) The same is true if i and y are allowed to be discontinuous at isolated points, where then the curve may not have a unique tangent ; the 1 Since the intermediate points ;; and I) ; need not coincide, we make use of the more general approximating sums that were shown to converge to the integral on p. 1 95 .
3 50
Applications in Physics and Geometry
Ch. 4
integral of course must then be considered as an "improper" one (see Chapter 3, p. 30 1). More general "rectifiable" curves, for which our i ntegral is meaningful, will not be discussed in this volume. A lternative Definition of Length
We add an interesting observation : The perimeter S of any inscribed polygon " can never exceed the length L of the curve. (In particular, the distance of the end points of the curve cannot exceed L; for the straight line joining the end points is the shortest curve joining those points.) Indeed we may obtain L as limit of the perimeters of a special sequence of inscribed polygons, in which we start with the polygon " of perimeter S and obtain the following ones by adding successively more and more vertices. Inserting an additional vertex between two successive vertices of an inscribed polygon can never lead to a decrease in perimeters, because one side of a triangle can never exceed the sum of the other two. Thus L is the limit of a non decreasing sequence of perimeters that starts with S. Hence S � L. Instead of defining therefore L as limit of the perimeters of a sequence of i nscribed polygons corresponding to finer and finer subdivisions of the /-interval, we could also have defined L as the least upper bound of the perimeters of ail i nscribed polygons. It is i nteresting that the length can be defined without formally invoking any passage to the limit. lm·ariance of Length under Parameter Changes
From its definition it i s clear that the length L of a curve c cannot depend on the particular parametric representation we use for C. Hence, if we introduce a new parameter = x(t), where d-rfdt > 0, our integral formula for L must give the same value whether t or is used as parameter. This ca n be verified im mediately from the chain rule of differentiation and the substitution law for i ntegrals. We have indeed
-r
-r
i:2 + !/ )(ddtx)2 + ( ddty)2 = J ( dd-rx ddt-r)2 + (dd-ry �)2 dt J ( dx)2 (dy)2 d-r ; = d-r d-r dt hence, if X( ) x(fJ) b, P - P) dx 2 dy 2 d-r J J x2 + 1l dt J (-d-r) (-d-r) -dt dt = rJ(�:r + (��r d-r. J
=
<X
=
L=
a,
+
=
+
=
•
•
Theory of Plane Curves 3 5 1
Sec. 4. 1
so th � t the expression for length based on the parameter -r leads to the same value L. If, instead, d-rfdt < 0 we find similarly
s: �x2
f J (�r (�r d-r = J r e:r (��r d-r ;
+ y2 dt = -
+
+
the right-hand side is again the correct integral for the length of C referred to the parameter -r since now b < a because x(1) is a decreasing function. For a curve given nonparametrically by a function y = f(x), a � x � b, we can introduce x as parameter 1. Then x = I , y = dyfdx. The length of the curve is then given by
L=
(9)
f j, (��r ax. +
Examples. As an example we find for the length of a segment of the parabola y = �x2· corresponding to the interval a � x � b :
far
L=
far ar
f� l + x2 dx.
Here the substitution x = sinh 1 (see Chapter 3, p. 273) leads to ar
sinh b
sinh
a.
cosh 2 t dt = !
= Ht
Hinh b
(1
sinh
lar ar
+ cosh 21) dt
a.
+ sinh t cosh t)
s i n h '' Sinh a
= ! ( a r sinh b + b� l + b 2 - a r sinh a - a � ! + a 2). For a curve given by an equation r = r(O), rx � 0 � {3 in polar coordinates, we have the representation x = r(O) cos 0, y = r(O) sin 0. Choosing 0 as parameter, we have x
= ; cos 0 - r sin 0,
y = ; sin 0 + r cos 0,
This leads to the expression ( 1 0)
L=
f Jr2 (:;r +
dO
for the length of a curve in polar coordinates. We have, for example, for the circle of radius a about the origin, the equation r = constant = a 0 � 0 � 27T. This gives for the total length of the circle
{2> L = o a d0 = 21Ta . J
352 Applications in Physics and Geometry
Ch. 4
Additivity of Length Let C be a curve given by x = x(t), y = y(t), ex � t � {3, where and iJ are continuous. Let y be any intermediate value between ex and fl. From the general rules for integrals we have x
f1 1x• x• x• J p �J P 1•
v
+ 1 l dt =
•
+ 1l dt +
v
1
+ 1l dt.
The integrals on the right, respectively represent the lengths of the portions i nto which C is divided by the point corresponding to t = y. Hence the length of the whole curve equals the sum of the lengths of its parts. It is not necessary that and y are continuous. The integrals exist just as well when and y have a finite number of jump discontinuities, as would occur in a curve with corners. The total length of the curve is then the sum of the lengths of the smooth portions between the corners. Even more singular behavior of and y is permitted as long as the expression for the length is meaningful as an i mproper integral.
x
x
x
g . The Arc Length as a Parameter
We have seen that one and the same curve permits many different parameter representations x = x(t), y = y(t). Any monotone function of t can be used as parameter instead of t. For many purposes, however, it is of advantage to refer curves C to some "standard parameter" which in some way is distinguished geometrically. The abscissa x or the polar angle () are not suitable for that purpose if curves are to be described in the large ; moreover, they depend on the choice of coordinate system. The possibility of measuring lengths along a curve provides us with a natural geometrically defined parameter to which points P of a rectifiable curve can be referred, namely, the length of the portion of the curve between P and some fixed point P0. We start out with an arbitrary parameter representation x = x(t), y = y(t), ex � t � f3 of C. Differentiation with respect to t i s indicated by a dot. We introduce the "arc length" s by the indefinite integral (I I)
s =
J�x•
+ y 2 dt
or more precisely s as a function of t by
( l l a)
s = s(t) = c +
it '
y x2(-r) + y2(-r) d-r,
to
Sec. 4 . 1
Theory of Plane Curves 3 5 3
where c is a constant, !0 a value between oc and {1, and where we have writteh -r for the variable of integration to distinguish it from the upper limit t. Clearly, for any values 11 and ! 2 in the parameter interval the difference ( 1 2)
is equal to the length of the portion of the curve bounded by the points corresponding to t = 11 and I = 1 2 , provided !1 < ! 2 • For ! 1 > t 2 the difference s ( t2 ) - s(/ 1 ) is the negative of the length of that portion. Thus the knowledge of any i ndefinite i ntegral s permits us to calculate the length of any part of the curve. The Sign of A rc Length
lf the constant c has the value 0 we can interpret s(t) itself as the length of the arc of the curve (or the "distance along the curve") between the point P0 with parameter !0 and the point P with parameter I ; here the length is counted positive in the case where the arc with initial point P0 and end point P has the orientation corresponding to i ncreasing 1.1 The i ntegral form of the definition of s is equivalent to the relation
ds = J (dx)" (dy)". dt dt
( 1 2a )
+
dt
.
Using the symbolic notation for differentials (p. 1 80) ds = (dsfdt) dt, etc., we can write th is relation in the suggestive form ds
= .Jdx 2
+ dy 2
for the "element of length" ds.
Speed of Motion along a Curt•e If t is interpreted as the time and x(t), y(t) as coordinates of the position of a moving point at the time t, we have i n s
=
ds = lim s(t + h) - s(t) dt h�O
h
the rate of change of the distance moved by the point along its path with respect to the time, that is, the speed of the particle. For a particle 1 Notice that the variable s is not completely unique ; it depends on the choice of Po and c and also on the orientation of the curve induced by the parameter t. How ever, any other arc length is expressible in terms of s in the form (s + constant) or ( -s + constant).
Ch.
354 Applications in Physics and Geometry
4
moving with uniform speed along the curve s is a constant and s is a linear function of the time t . If our usual assumption is satisfied, we have ds/dt � 0 and can introduce s itself as parameter. Many formulas and calculations then simplify. The quantities dx dx dt i = = --:::=-...,., ds dt ds 2 .Jx + il ' -
- -
dy dy dt = = dt ds ds -
- -
if
,j i2 + il2
--=====
are then just the direction cosines of the tangent pointing in the direction of i ncreasing s (see (7), p. 346). The relation
( 13 )
e:r + e�r
=
1
characterizes the parameter s as the arc length along the curve. h. Curvature
Definition by Rate of Change of Direction We discuss next a basic concept which refers only to the local behavior of a curve in the neighborhood of a point, the concept of curvature. As we describe the curve, the angle oc of inclination of the curve will vary at a definite rate per unit arc length traversed ; this rate of change of oc we call the curvature of the curve. Accordingly the curvature is defined as
( 1 4)
doc
I< = - .
ds
Parametric Expressions. Let the curve be given parametrically by functions x = x(t), y = y(t) having continuous first and second derivatives with respect to t, for which i 2 + if 2 -,6 0. In calculating the rate of change of the direction angle oc at the point P we have to take into account that oc is not defined uniquely. However, the trigonometric function of oc , tan oc = iJ/i (or cot oc = i/Y for i = 0) has a definite value. In forming docfds we can always assume that the parameter values belonging to points in a neighborhood of P all lie in an interval throughout which one of the quantities i, iJ stays different from zero. If, say, i � 0 we can assign to oc a value that varies continuously with t
Sec. 4.1
Theory of Plane Curves 355
throu�hout the interval by taking 01.
= 01.(1) = arc tan fl.. + 111r , i
where n is a fixed, possibly negative integer, and "arc tan" stands for the principal value of the function (cf. p. 2 1 4), lying between - 1rj2 and 1rj2. Similarly, if iJ � 0 in the interval we can take for 01. the expression :i
7T
;i;
01.(1) = arc cot - + 111r = - - arc t an - + 111r . 1 2 iJ iJ In either case we find by direct differentiation for any parameter representation - x y ri _ - ddt01. _- xi} i 2 + ?? . Since (see ( 1 2a), p. 353) also . ds . ., s = - = v; x -' ' • + ·v
dt
.
we obtain for the curvature d01.jds = rifs of the curve the expression
( 1 5)
K =
d Ot. = � = :i;ij - y:i ds s ( x2 + ?/)' '
Choosing in particular, the arc length s as the parameter t we have i; 2 + y2 = 1 [see Eq . ( 1 3) , p. 354) and hence we obtain the si mplified result
( ! Sa)
K = Xjj - yi.
Sign an d Absolute Value of Curvature
Intoducing a new parameter T = T(t) instead of t does not affect the direction of the tangent, and hence, does not affect changes in 01.. Similarly, the absolute value of the difference of the s-values in two points has a geometric meaning independent of the choice of parameter, namely that of distance measu red along the curve. However, the sign of the difference must always be taken as the same as the sign of the difference of the corresponding parameter values, since we defined s as 1 We could define rx(t) as a continuous function for all parameter values t by d is secting the whole parameter interval into su bintervals in each of which either i: >" 0 or y ,e 0. In each of the subintervals we can define then rx(l) by one of the above expressions, choosing for each interval the constant integer n in such a way that the values of rx in the common end point of two adjacent intervals, as determined from the expressions for those intervals, coincide.
356 Applications in Physics and Geometry
Ch. 4
y
--�0�------� x a
Figure
t;
4.21 Rectification of curves.
y
Figure
4.2l(a) Curvature
K = lim
< 0.)
!la.j!ls of a curve. have
K
( I n the case illustrated
we
Sec. 4.1
Theory of Plane Curves 357
an increasing function of t. Thus the absolute value of the curvature I K i = i dcxfdsi does not depend on choice of parameter, whereas the sign o f K depends on the sense on the curve corresponding to increasing t. Obviously, K > 0 means that ex increases with s, that is, that the tangent turns counterclockwise as we proceed along the curve with increasing s or t (see Fig. 4. 2 l a). In this case the orientation of the curve C is such that the positive side of C also i s the "inner" side of C, that is, the side toward which C curves. y
�0�-------+ x Figure
4.22
Graph of a c o nvex fu nction f(x) ( left) and concave fu nction (right).
If the curve is given by an equation y = f(x), we have, using x as parameter, " y K = ( 1 + 2)% ' ( 1 6) y' " where y ' and y are the derivatives of y with respect to the variable x . Here the sign of the curvature is that corresponding to increasing x. Obviously, K is positive for y" > 0 ; i n this case the tangent turns counterclockwise as x increases ; we call the function f(x) convex. The portion of the curve joining any two points lies below the straight line joining them. For y " < 0 the tangent turns clockwise for increasing x, and the function / i s called concave. (Fig. 4.22) Here the curve lies above the chord joining two of its points. The intermediate case where the curvature has the value zero corresponds (generally speaking) to a point of inflection at which y " = 0 (see p. 237). x =
Examp les. For the curvature of the circle of radius a given by a cos t, y = a sin t we find the constant value If a from the
358
Applications in Physics and Geometry
Ch. 4
general formula ( 1 5). Thus the curvature of a circle described in the counterclockwise sense is the reciprocal of the radius. This result assures us that our definition of curvature i s rea lly a suitable one ; for in a circle we naturally think of the reciprocal of the radius as a measure of its curvature. A second example is the curve defined by the functio n y = x3 • The curvature is 6x
For x < 0, t he fu n ctio n y = x3 is concave, since K < 0, and the tangent is turning in a clockwise sense, whereas at x = 0, we h ave a point of inflection, and for x > 0 the function becomes convex. A function whose curvature is i dentically equal to zero i s a straight line as is easily seen by our definition, and the straight line is the o nly such curve.
Circle of Curvature and Center of Curvature
We introduce p = 1 /K. The quantity I pi = 1 / I K I is called the radius of curvature at the point in q uestion. (It i s infinite at a point of inflection where K = 0.) For a circle the rad i us of curvature at any point is j ust the radius of the circle. To any point P = (x, y) of the curve C we assign a circle tangent to C and P and having the same curvature as C when we traverse the curve and the circle in the same sense at P. This circle is called the circle of curvature of the curve C at th e point P. lts center is the center of curvature of the curve C corresponding to the poi nt P (Fig. 4.23). Since C and the circle have the same radius of curvature the radius of the circle must be the radius of curvature I p i of C, and the center (�. rJ) of the circle must lie on the normal of C at P, a n d a d istance I pi away from P. Since C a nd the circle curve toward the same side, the center lies along the normal direction to the curve at P, on the positive or negative side according as the curvature K i s positive or negative. The d irection from P to the center of curvature forms an a n gle oc + rr/2 with the p os iti ve x a x is i f K > 0. Thus, if �. rJ are the coor dinates of t he center of curvature and x, y those of P, we have [see Equation (7), p. 346] -
,
Sec. 4. 1
Theory of Plane Curves 359 y
(�. '1)
r
Figure 4.23 Circle of curvature r and center of curvature ($, '1) corresponding to point P of curve C.
Hence for K > 0, (1 7)
l f arc length s is used a s parameter 1, w e obtain the simple expressions ( 1 7a)
g = X - p y,
1)
= y + pi.
The same formulas for g, 1) are obtained for K < 0, in which case the radius of curvature is - p and moreover the direction from P to the center forms an angle cc - 77/2 with the positive x-axis. Circle of Curmture as Osculating Circle
Formulas ( 1 7) give an expression for the center of curvature in terms of the parameter t of the point P on the curve. As t ranges over all values in the parameter interval the center of curvature describes a curve, the so-called el'Olute of the given curve ; since, with x and y, we have to regard :i, y, and p as known functions of t, the foregoing formulas give parametric equations for this evolute. Examples and a discussion of geometrical properties of the evolute will be found in Appendix I, p. 424. Any two curves are said to "osculate" at a point P or to have "con tact of order two" at P, if they pass through P, have the same tangent
360
Applications in Physics and Geometry
Ch. 4
at P, and also the same curvature, when oriented the same way. Obviously, two osculating curves have the same circle of curvature and center of curvature at P. If the curves are given by equations y = f(x) and y = g(x) in nonparametric form, it is easy to express the condition that they have a point of contact P and the same tangent and curvature at P. If x is the abscissa of the point of contact P, we have f(x) = g(x), f'(x) = g '(x) ; the equality of curvature is expressed by f"(x)
[1 + rex)]}>
=
=
g"(x) [1 + g'2(x)] % '
and hence also f"(x) g"(x). Thus the condition for a point of contact with equal curvatures is that the values off and g together with those of their first and second derivatives agree at the point . Consider a curve C : y = f(x) and its circle of curvature r at P represented by y g(x) in a neighborhood of P. Since the circle r coincides with its circle of curvature, we see that C and r have the same circle of curvature, hence osculate at P. Consequently, at the point of contact f(x) g(x), f'(x) g'(x), f"(x) = g"(x). We say this circle is the "best fitt i ng" circle to the curve at the point P of contact, since no other circle meeti n g the curve at the point of contact has "contact of order two" with C at the point . The circle of curvature is the osculating circle. (See also Chapter 6, p. 459.) Inciden tally , just as the tangent to a curve is the limit for P1 P of a line through two consecutive points P and P1 on C, one can show that the circle of curvat ure at P is the limit of the ci rcles through three points P, P1 , P2 for P1 --+ P and P2 --+ P. The proof is left to the reader. (See Problem 4, p. 437.)
=
=
=
--+
i. Change of Coordinate Axes. lnvariance
Properties inherent in a geometrical or physical situation do not depend on the specific coordinate system or "frame of reference" with respect to which they are fo rm u l a ted ; the intrinsic character of prop erties such as distance or length or angle must be reflected in state ments showing that the respective formulas remain unchanged or are invariant if one passes from one coordinate system to another. A few brief remarks concerning this subject are appropriate in this section. We use the general equations connecting the coordinates x, y of a point P in one coordinate system with the coordinates �. r; of the same point P in any other system. The relative position of the second set of coordinate axes to the fiist set is characterized by the coordinates a, b that the origin of the second system has in the first system, and by the
Sec. 4. 1
Theory of Plane Curves 361
angle y which the pos1ttve �-axis makes with the positive x-axis.l The coordinates (x, y) and (�, r;) of the same point in the two systems are (cf. Fig. 4.24) connected by the transformation x = � cos y - r; sin y y = � sin y + r1 cos y
( 1 8)
+
+
a, b.
For y = 0 no rotation of the axes but only a parallel displacement or translation i s involved, and the formulas take the simple form x = � + a, y = r; + b . y
y
: � : ·= , .,. : '"' : : 'Y ------L- - - -----� - - - - \ . ' '' : I '
b
' ----------_. % ' --� ��----------& X a
-+
0
----
----------
Figure
'
4.24 Change of coordinate axes.
Solving for �. r; in terms of x, y we find
( 1 8a)
� = (x - a) cos y + (y - b) sin y, = - (x - a) sin y + (y - b) cos y.
r;
If x and y are functions of a parameter t defining a curve, we obtain i mmediately from these formulas expressions for � and r; as functions of t, giving the parameter representation of the same curve in the �.r; system. Differentiating with respect to t (the quantities a, b, y which fix the relative position of the two coordinate systems do not depend on t) yields the transformation of the "velocity components," that is,
1 W e restrict ourselves to "right-handed" coordinate systems in which t h e positive direction of the second axis of a system is obtained by a counterclockwise 90° rotation from that of the first axis.
362
Applications in Physics and Geometry
Ch. 4
for the derivatives of the coordinates with respect to t, x = � cos y - iJ sin y,
We confirm
xz
+
yz
y = � sin y + ·� cos y. 1
=
�2 + il.
Thus the expression J x2 + y2 has the same value in all coordinate systems ; this in variance property is, of course, obvious from the interpretation of this quantity as rate of change ds/dt of the length along the curve with respect to t. The reader may verify by an easy y
/"
/
/
/
/
\ / /_ _ _ .�.-- -
Figure 4.25
/
/ ---
(�. '1)
-0��--------------------------• x
D i sp
lacem e n t of point P from pos it io n (x, y) to position (�. 'J).
calculation that also the expression K = (xfi - X1j)(x2 + !?)-h for the curvature is i nvariant. (This, of course, follows also directly from the fact that the angles the tangent makes respectively with the � and x-axes differ only by the constant value y, so that K = d(/./ds cannot change.) Equations ( 1 8) relating the coordinates x, y to the coordinates �. rt are often interpreted in a different way as describing a displacement. In this i nterpretation the points P are shifted instead of the coordinate axes (Fig. 4.25). Only one coordinate system is used. The poin t with
In some physical applications, where t stands for time, the relative position of the two coordinate systems also depends on time ; let the quantities x, y stand for the coordinates of a particle in a coordinate system that is at rest, whereas (: , 'I are the coordinates of the same particle referred to a moving coordinate system, for example, axes that are attached to the moving earth. The functions x(t), y(l) describe the path of the particle as it looks to an observer at rest, whereas ;(t), 7](1) describe the path as it looks to a moving obserYer. The formulas connecting :i:, '!i with �. •i have to include then also the obvious lerms arising from differentiation of a, b, and )' · 1
Sec. 4. 1
Theory of Plane Curves 363
coor�inates (x, y) in that system is mapped onto the point with co ordinates (�, 'f/) in the same system. lnvariance of length or curvature of a curve now means that t hese q uantities do not change when the whole curve u ndergoes a rigid motion. * j. Uniform Motion in the Special Theory of Relativity As pointed out on p. 234 there are far reaching analogies between the trigonometric and the hyperbolic functions which have t heir geometric counterpart in the correspondence between properties of ellipses and hyper bolas. The relationship will become clear when we shall be able to define the trigonometric functions for an imaginary argument and to verify that cos (it) = cosh t, sin (it) = i sinh t in Section 7 .7a. As an application of this analogy we consider the "hyperbolic rotations" of the plane which can be i dentified with the Lorentz-transformations of a line in Einstein's special theory of relativity. We saw i n ( 1 8a), p . 3 6 1 , that a rotation of coordinate axes by an angle y which leaves the origin fixed can be described by the equations ( 1 8b)
� =
x
cos y + y sin y,
YJ =
-x
sin y + y cos ;•
connecting the coordinates x, y of a point P i n t he first system with its co ordinates �. T} in the second system. The distance of P fr o m the origin is given by the same expression in both systems :
OP = v x2 + y2 = v ;2 + '12 ·
This fol lows a lso immediately from the transformation equations if we make use of the i dentity cos2 y + sin 2 y = I . We now consider the analogous transformation with coefficients that are hyperbolic instead of trigonometric functions : ( 1 9)
; =
x
cosh oc
-
t sinh oc,
T =
-x
sinh " + t cosh oc ;
these formulas can be obtained from the formulas ( 1 8b) for rotations by taking for the rotation angle y and the y-and 17-coordinates, pu re i maginary quantities : y
=
ioc,
y = it,
1]
= iT.
We notice that for a real value of oc (which would mean an imaginary angle of rotation y in the original interpretation) formulas ( 1 9) define � and T as real l inear functions of x and t. These functions have the special property that ; 2 - T2 = (x cosh " - t sinh oc)2 - ( -x sinh oc + t cosh =<)2 =
x2 - t2
as a consequence of the identity cosh2 " - sinh2 " = I . (This fol lows, of course, also from the observation that x2 - t2 = x2 + y 2 is the square of the distance from the origin in the x,y-plane.) We now i nterpret t as the time and
364
Applications in Physics and Geometry
Ch. 4
x as a space-coordinate describing the location of a point in a one-dimen sional space, that is, on a straight line. Any event takes place at a certain point at a certain time. These two pieces of information are provided by the two numbers x, 1 giving respectively the (signed) distance x of the point from the origin 0 and the time t that has elapsed from the time 0. In the theory of relativity we take the point of view that the measured values of this distance and of elapsed time depend on the frame of reference used by the observer, that is, on the special coordinate system in the space-time continuum. The quantities �. obtained from the formulas ( 1 9) will describe the same event in a different frame of reference in which distances and lengths of time inter vals can have different values. The quantity that is unchanged in the transition from one reference frame to another one (known as "Lorentz transformation") is T2, v =v
T
x2 12 �2 � _
_
the "space-time distance" of the event from the ongm . For an observer using the second system the quantity is the space distance measured from the origin = 0. That origin is a point for which
x/t
�
x
cosh "'
- I
sinh "' = 0
or = tanh rt.. Thus the origin of the second system is a point which in the first system appears to move with uniform velocity v = = tanh "' relative to the origin of the first system. Hence the Lorentz transformation relates the val ues of distances and times as they appear for observers in two systems moving with constant velocity v relative to each other. Here sinh
dxfdt
e-a.
v = -= --cosh "' e• + e- • ex.
ea -
lies necessarily between - I and + I so that we are restricted to relative velocities of the two systems that lie n umerically below the value I . The value I here represents for suitable choice of units the velocity c of light which cannot be exceeded by v. For a constant u the equation x = ut corresponds to a point which in the first system moves with the velocity u, starting at x = 0 at the time = 0. I n the second system the same point will have the velocity
d� (d�) (dT). I
w = d,- = dt
.di
=
I
u - tanh "' u-v = - u tanh "' I - uv ·
t
This result, valid in Ei nstein's special theory of relativity, differs from what we would obtain in classical kinematics where the velocity w of a point with respect to a system moving with velocity v would simply be given by w = u - v. The relativistic formula shows that w = u when u = + I or - I ; this corresponds to the fact suggested by the famous M ichelson-Morley experiment that the velocity of light is the same for observers moving with different velocities.
Sec . 4.1
Theory of Plane Curves 365
k Integrals Expressing Area within Closed Curves . .
In Chapter 2 the concept of integral was motivated by reference to "area under a curve," that is, the area of a strip of special shape. This specialization to areas under a curve is not quite satisfactory since the areas actually encountered most frequently are those of domains inside closed curves C, and are of more general shape than the strips whose area can be represented by integrals of the form Je... f(x) dx. The Basic Formula
We shall now derive an elegant general integral representation for the area bounded by a closed curve C whi c h is given in parametric representation, by breaking up the area into special strip areas. This representation will be independent of the parameter representation and likewise independent of the coordinate system. Furthermore, it will express the oriented area within the curve in accordance with the sense of direction assigned to the boundary C; that is it will assign to an area within a simple closed curve C the negative or positive sign accord ing as the sense of the boundary curve is clockwise or counter clockwise. Assume that the simple closed oriented curve C is given by = x(t), y = where t varies over the interval � � fJ and the sense of increasing determines the sense on C. We assume that x and y are continuous functions of t (with the same value at = and 1 = fJ) and that their first derivatives and y are continuous, with the possible exception of a finite number of jump-discontinuities if C has corners. Under these assumptions we shall prove the basic formula (20) A = -fp y:i dt =fp y dt = .! JP(xy - y:i) dt 2 for the oriented area A within C. That the three integral representations in the formula are equivalent follows directly if we integrate the first one by parts and use the perio dicity conditions x(oc) = x(fJ), y(r:x) y(fJ) ; the third, more symmetric representation is just the arithmetic mean of the first two. The expressions (20) do not depend on the location of the coordinate system in the plane. In fact, the symmetric expression A = � JP(xy - y:i) d t 2 x
y(t) ,
r:x
t
:i
1
x
a.
1%
a.
=
•
t
r:x
366 Applications in Physics and Geometry
Ch. 4
shows clearly that the value of A is independent of the choice of the coordinate system. As we saw on p. 361 a change of coordinates from an xy-system to a �7)-system is achieved by a substitution of the form = � cos y r; sin y + a, y = � sin y + 7J cos y + b,
x
-
with constant a, b, y. Differentiation of these formulas w i th respect to t yields x = � cos y - ij sin y, = � sin y + ij cos y
y
and consequently,
xy - yx = �ij - 7J � + ay - bx.
Thus the express i on xy - yx is invariant under rotations about the origin (that is, when a = b = 0). Even when a or b do not vanish, the value of the integral for A is not affected, since
fP(ay - b x) dt (ay - bx) IP. =
..
=
0
for t h e closed curve C. Proof of the Basic Formula (20). Line Integrals over Simple A rcs. The basic formula (20) is proved in some easy steps. First, let C be a simple oriented arc with initial point P0 and final point P1 . Let x(t), y = y(t) be any parameter representation of C t = t0, t 1 . (Here t0 may be with P0, P1 corresponding respect ively larger or smaller than t 1 .) Then the integral
x
=
to
A = -
l t, to
dx y - dt dt
depends only on C and not on the particular parameter representat i on. This is an obvious consequence of the substitution rule ; if we i ntrod uce a new parameter -r by the monotone function -r = x(t) where -r0 = xCto ), Tl = x(t l ) the COrresponding integra) is1 -
it' dx d-r lt1 dx '1 dx Jro y -d -r d-r - y -d -r -d dt - y -d dt = A. =
=
to
I
to
I
1 We assume not only that x(t), y(t) but also T(t) are continuous functions and that their derivatives are continuous with the possible exception of a finite number of jump-discontinuities.
Sec. 4.1
Theory of Plane Curves
367
It is therefore justified to drop from the expression for the integral A the reference to any special parameter t and simply to write
A =
Ac
=
-
Ly dx.
Here A c for a simple oriented arc C is to be computed by referring the arc to a parameter t, using dx = (dxfdt) dr, and taking as limits for the t-integration the parameter values for the end points of C in the order determined by the orientation of C.1 ]f C' is the arc obtained from C by changing its orientation, that is, t he arc with initial point P1 and final point P0 we h ave, using the same parameter representation for C',
A c· = -
ltoy -dx dt = + lt'y t,
dt
to
dx - dt = dt
-
A c.
Hence changing the orientation of an arc C changes the si g n of the integral A c. If the oriented si m ple arc C is broken up into or i e nted subarcs C1 , C2 , , Cn, each with the same orientation as C, we obviously h ave •
•
.
A c = A c,
+ A c, + . . + A c,,.
For in a p arameter representat ion of C where, say, the sense of C is that of increasing I, this decomposition corresponds to a s ub division of the parameter interval /0 � t � tn for C into subintervals t0 � t � 11, 11 � t � 1 2 , , tn-I � t � t, corresponding to C, , . . . , C,. The resul t then follows from the additivity of integrals. The additivity of the integrals Ac makes it much easier to compute the value of A e in cases where C consists of several smooth arcs C1 , C2 , , each with its own parameter representation. We do not need to construct artificially a common parameter representation for the whole curve C, but i n stead compute each A c , separately from its param eter representation and then take the sum. M oreover, the A c , can be added in any order ; we only have to make sure that all Ci have the same orientation as C. •
•
•
•
•
•
The Basic Line integral for Closed Curves
We can n ow define Ac for any oriented , simple closed curve C by breaking up C into simple arcs C1 , . . , C, with orientations agreeing .
1 The integral
LY
dx is an example of the general line integrals
which will be discussed in Volume I f .
i/
dx
+ q dy
368
Applications in Physics and Geometry
Ch. 4
with that o n C and fo rmi n g the sum of the A c 1 If the whole closed curve C has the parameter representation ; = x(t), y = y(t) for oc � t � {J, where the sense of increasing t gives the orientation of C and where t = oc and t = {J correspond to the same point then A c is again gi ve n by .
Ac = -
J
P
•
dx y - dt. dt
In the same way we can de fi n e Ac for n o n s i m p le oriented curves C by d ec om p os i ti on into simple oriented arcs, even when C consists of several disjoint pieces, as long as each portion of C has a definite sense.
The Basic Integral as Area A c for a cl osed area within C.
We now turn to the
main p o i n t ;
that i s , we identify the expres s i o n s
curve w i t h t h e i ntuitive geometric quantity
of oriented
y
�--��---------��----� % 0
a
b
Figure 4.25(a)
We
Area
of a "cell."
con sider fi rst a d o m a i n G bounded fro m above by a n arc C, :
ll = g(x) for a � x .::;; b ;
below by an arc C3 :
y
= j(x) for
a .::;; x .::;; b :
and l ate ra lly by l i ne segments C2 , C4 given by x = a a n d x = b ( Fig . 4.2Sa). Here C2 and C4 are permitted to shrink into poin ts . 1f we give to
1 That the value of Ac obtai ne d in t h is fashion does not depend on the particular way in which we d ivide up C into simple arcs follows easily : first the addi t ivity property of A for simple arcs shows that refin i ng a g i ven subd ivision by i ntroducing additional d iv i din g points does not chan ge t he resulting value of Ac; moreover, any two subdivisions can be replact"d by one t ha t is a refinement of both without c han g in g the value of A c.
Sec. 4.1
Theory of Plane Curves
369
C the counterclockwise orientation, the arc C1 will be described in the " sense of decreasing x, and the arc C3 in that of i ncreasing x . I n forming Ac as the sum of the four A c, the p ortions C2 and C1 along which x is constant make no contribution since there d.rjdt 0. Using x a s parameter on the arcs C1 and C3, we find
=
A c = Au,
+
A c, =
- f"g(x) d.r - J "f(:r) d.r u
"
= J,,g(x) dx - J 'f( .r ) dx. x.
n
This clearly is the positive area of the domain G, if G l ies completely above the .r-axis, b eing the difference of the areas lying respectively below the curves C1 and C3• We can always guarantee that G lies above the axis by by + c with a suitable constant c , that is, by a translation i n the y direc t ion. This does not change areas
replacing y y
-
and also does not affect the value of A e
= - J.{ y dx or a closed curve f
C as we saw before Hence for domains G of the type described which have a boundary C intersected i n no more than two points by parallels to the y-ax is, the integra l Ac represents the area, tak e n positive i f C is oriented counterclockwise, negative i f clockwise. We obtain the same result for areas bounded by a curve C intersected by parallels to the x-axis i n at most two points ; we have only to write A I ' in the form .
r X dy and to i nterchange X and Jr·
y in the preceding argument. We call domains G of one of these two types "cells." We shall talk of "oriented cel ls" when their boundary curves are given one or the other o r ientatio n .
We now consider a domain G with the oriented boundary C, which i s composed of a nu mber of simple cells G1, G 2 , , G with the boundaries cl , . . ' en respectively ; all these cells are assumed to have the same orientation, say counterclockwise. Then, as ind icated in Fig. 4.26, the parts of the boundaries of the cells which are common to two adj acent cells are descri bed in a different sense accord ing as they are considered boundary arcs of one or the other of the •
•
.
n
.
adjacent cell s . Therefore, if we add the inte rgrals
A (',
= - )c(
..
11 dx
for
the different cells, the c o ntribution s of all the interior cell b � undaries cancel out and we obtain
where
A
A = �1Ac, = ;�( - L,y dx) = - Ly dx = A u
i's the oriented area of the total domain G.
370 Applications in Physics and Geometry
Ch. 4
Thus the formulas (20) for the area A of an oriented domain G within closed curve is proved for all domains which can be decomposed into simple cells, for example, by drawing parallels to the coordinate axes. For all domains that we shall encounter, this assumption will be obviously satisfied, as for example, polygonal domains. a
y
Figure 4.26
Decomposition of oriented domain into oriented cells.
Supplementary Remarks Finally, it might be added that the validity of the formula for area follows in the same way, even for multiply connected domains, such as ring-shaped domains, which can be decomposed into a finite number of simple cells. Then all the boundary curves have to be described con sistently in such a sense that the interior of G is always either on the "left" side or always on the "right" side. The formulas for A remain meaningful even when C is not a simple curve but is allowed to intersect itself, dividing the plane into more than two regions. In this case we may consider the formula as a guide to interpreting area suitably as an additive combination of the oriented areas of the various connected pieces of the plane bounded by C. We shall discuss this matter in Appendix II to this chapter. Examples . As an example we can find the area enclosed by the ellipse x2/a2 + y2fb2 = I . Using the counterclockwise orientation for
Sec. 4. 1
Theory of Plane Curves
311
the �llipse, we find from the parameter representation x
that
= a cos t,
A
1 =-
y = b sin t
12"(xy -
for 0 � t � 21T
1 yx) dt = -
2 0
2
12' 0
ab dt = 1ra b.
Area in Polar Coordinates. To express area in polar coordinates r and 0, we consider first the area A of the region bounded by a curve segment r = f(O) and the radii 0 = oc and 0 = {J. We assume that oc < {J and that 0 can be used as parameter along the curve (that is, that different points have different polar angles). We use for A the expression A =
� Jcx
dy -
y
dx) = � J(xy - yx) dt ,
which then has to be extended over the curved part of the boundary and over the two radii. On the radii 0 = oc and 0 = {J we can use r as parameter and find from = r cos 0, y = r sin 0, and 0 = constant that x = cos 0, = si n 0, and thus xy - yx = 0. On the curved part we use 0 as parameter. Then
x
y
.
x
and thus x,lj (2 1 )
dr . . y = - sm () + dO
dr . = - cos 0 - r sm 0 , dO
- yx = r2•
A =
r
cos 0 ,
Consequently,
l
2
f pr 2
dO =
•
!
2
fPF(O) dO. •
For a simple closed curve C which contains the origin in its interior and is intersected by every ray from the origin i n exactly one point we can use 0 as parameter for 0 � () � 21r and find for the enclosed area (22)
1 A =-
2
12" d . r2 O
0
Formula (2 1 ) for area in polar coordinates can also be derived directly from the definition of integrals. For that purpose we divide our domain into sectors by drawing radii from the origin (Fig. 4.27). Each sector is described by inequalities 0;- l < 0 < o,,
0 < r < f(O).
Obviously, the area of the sector lies between the areas of the inscribed and circumscribed circular sectors ; the area of a sector of the domain
372 Applications in Physics and Geometry
Figure 4.27
Ch. 4
Area in polar coordinates.
is then equal to �r2(6; - 6;_1), where r lies between the largest and smallest values of /(6) for the interval 6;_ 1 < 0 < 6;. As we refine the subdivision, the sum of the areas of the sectors of our domain clearly
1 1�
converges to the integral -
2 ,
Area in a Lemniscate
r 2 d6.
As an example of Equation (2 1 ) we consider the area bounded by a loop of a lemniscate. The equation of the lemniscate (cf. p. 1 03 ) is r 2 = 2a 2 cos 20 ; one loop is obtained by having 0 vary from - rr/4 to + rr/4. This gives us the expression a2
f•l•
cos 26 dO
- rr / 4.
=
a2
for the area. Of course, the othe r loop has the same absolute but negative value of area. A rea Bounded by a Hyperbola We now consider the area of a sector bounded by the hyperbola x 2 - y2 = I , which we computed already on p. 234 in a rather cumber some fashion (see Fig. 3 . 12). For the hyperbola (or rather for its right-hand branch) we have the parameter representation x = cosh t, y = sinh t. We find indeed for twice the area bounded by the hyperbola and the radii leading to the points with parameters 0 and t the value
2A
=
=
f(xy - yi) d-r
fd-r
=
t.
=
f
ccosh2 -r - sinh2 -r) d-r
(There is again no contribution to the integral from the radii.)
Sec. 4.1
Th�ory of Plane Curves
373
I. Center of Mass and Moment of a Curve
We now turn to some ideas arising in mechanics. We consider a m ,. and system of n-particles in a plane having the masses m 1 m 2 , the respective ordinates y1, y2, • • • Y n · We then call ,
•
•
•
,
,
n
T = L m ,y, = m 1y 1 + m 2y 2 + \1= 1
· · ·
+ m ,. y .
the m o m en t of the system of particles with respect to the x-axis. The expression 'Y/ = T/ M, where M denotes the total mass m 1 + m 2 + · · · + m, of the system, defines the height of the center of mass of the system of particles above the x-axis, or its ordinate. It i s just the weighted average of y 1, y2, • • • Y n using the "weight factors" m 1 , m2, m " (see p. 1 42). Hence 'Y/ is the average height of the masses. We define similarly the moment with respect to the y-axis and the abscissa of the center of mass. We can now easily extend these definitions of the moment to a curve along which a mass is uniformly distributed. and thus define the coordinates ,; and 'Y/ of the center of mass of such a curve. (The assump tion of a constant density, say fl , along the curve is not essential : Any continuous distribution could be discussed equally well.) In a procedure typical for mechanics we start with a system of a finite n umber n of particles, and then pass to a limit for n -+ ro. For this purpose we i ntroduce the length of arc s as a parameter on the curve, and subdivide the curve by (n I) points of division into arcs of , �S11 • We represent the mass 1� � s; of each arc lengths �s 1 , �s2, �s; as if it is concentrated at an arbitrary point of the arc, say that with the ordinate Y; · By definition the moment of this system of particles with respect to the x-axis is ,
•
•
•
,
-
•
•
•
If now the largest of the quantities �s; tends to zero, this sum tends to a limit given by the integral (23)
T = f1
fsl ds f x 1 .jxo y
= f1
2 y I + y' dx,
so
which is therefore naturally accepted as the definition of the moment of the curve with respect to the x-axis. Since the total mass of the curve is equal to its length multiplied by fl· f1
f..ds ••
= f1 ( S 1 - S0 ) ,
374 Applications in Physics and Geometry
Ch. 4
we are immedi ately led to the following expressions for the coordinates of the center of mass of the curve : t · -
(24)
f..
x ds
n =
f..y ••
ds
. s l - So sl - S o These statements are actually definitions of the moment and center of-mass of a curve ; but they are such straightforward extensions of the simpler case of a finite number of particles that we naturally expect that-as is actually the case-any statement in mechanics involving the center-of-mass or the moment of a system of particles will be valid also for continuous mass distribution along curves. · ·--
'
--
m. Area and Volume of a Surface of Revolution
Guldins Rule If we rotate a curve y = f(x) for which f(x) � 0, about the x-axis, it describes a so-called surface of revolution. The area of this surface, whose abscissas we suppose to lie between the bounds x0 and x1 > x0, is obtained by a discussion analogous to that above. For i f we replace the curve by an inscribed polygon, instead of the curved surface, we have a figure composed of a number of thin truncated cones. intuition suggests that we should define the area of the surface of revolution as the limit of the areas of these conical surfaces when the length of the longest side of the inscribed polygon tends to zero. From elementary geometry we know that the area of each truncated cone is equal to the length of the slanted straight gene rating side multi plied by the circumference of the circular section of mean radius. (Fig. 4.28). I f we add these expressions and then carry out the passage to the limit, we obtain the expression
s
= 27T r·· y ·./1 + y '2 dx = 27T'f} ( .1 1 - S o) J xo so for t h e area. Expressed in words, this result states that the area of a surface of revolution is equal to the l en g th of the generating curve m u lt i p l ied by the distance traversed by the center of mass (Guldin 's rule). In the same way we find that the volume interior to the surface of revolution and bounded at t h e ends by the planes x = x0 and x = x1 > x0 is given by the expression
(25)
(26)
A =
27T ··· y ds
V=
7T
J"•y2 "•
dx.
Sec. 4. 1
Theory of Plane Curves
y
375
0
Figure 4.28
Area
of surface of revolution.
This formula is obtained by following the intuitive suggestion that the volume in q uestion is the limit of the volumes of the earl ier mentioned figures consisting of truncated cones. The rest of the proof is left to the reader. n. Moment of Inertia I n the study of the rotation of an object an i mportant role is played by certai n quantities cal led moments of inertia. These expressions will be briefly mentioned here. We suppose that a particle m at a distance y fro m the x-axis rotates uniformly about that axis with angular velocity 01 (that is, in unit time it rotates through an angle w). The k inetic enerKY of the particle, expressed by half the product of the mass and the square of the velocity, is obviously
We call the coefficient of � w2 , that is, the q uantity
of inertia of the particle about the x-axis.
Simi larly, if we have n-particles with masses ordinates y1, y2 , , y,. , we call the expression •
•
2 my ,
m1 , m 2 ,
the moment •
•
•
,
m ,. and
•
T = I m;y/ i
the moment of inertia of the system of masses about the x-axis. The moment of i nertia is a quantity that belongs to the syste m of masses i tself, without reference to its state of motion. Its i mportance l ies in the fact that under rigid rotation of the system about an axis, without change of the distance between pairs of particles, the kinetic energy is obtained by m ultiplying the moment of inertia about that axis by half
376 Applications in Physics and Geometry
Ch. 4
the square of the angular velocity. Thus the moment of inertia about an axis plays the same part in rotation about an axis as is played by the mass in rectilinear motion. Suppose now that we have an arbitrary curve y = f(x) lying between the abscissas x0 and x1 > x0, along which a mass is uniformly dis tributed with unit density. In order to define the moment of i nertia of this curve we proceed just as in the preceding section, arriving at an expression for the moment of inertia about the x-axis, (27)
For the moment of inertia about the y-axis we have correspondingly (28) 4.2
Examples
From the great variety of plane curves we choose a few typ ical examples to illustrate the concepts discussed. a. The Common Cycloid
-
From the equations (cf. ( I ), p. 329) x = a( t sin t), y = a( l - cos t), we obtain :i = a( l cos t), y = a sin t, and find for the length of arc
-
s =
r· Jo Jx2 + !i' dt
=
r· J o )2a 2 ( 1 - cos t) dt.
Since 1 - cos t = 2 sin2 t/2 the integrand is equal to 2a sin t/2, and hence for 0 � oc � 217' s =
l•
.
t
/" (
2a o sin (t/2) dt = - 4a cos 2 o
=
oc
4a 1 - cos -
.
2
)
=
oc
Sa sin 2 - . 4
If, i n particular, we consider the length of arc between two successive cusps, we must put a: = 217', since the interval 0 � t � 217' of the values of the parameter corresponds to one revolution of the rolling circle. We thus obtain the value 8a ; that is, the length of arc of the cycloid between successive cusps is equal to four times the diameter of the rolling circle.
Sec. 4.2
Examples 377
Similarly, we calculate the area bounded by one arch of the cycloid and the x-axis: I = Jor2 · yx dt = a2Jr0b( I - cos t )2 = a2J(02•( I 2 cos t + cos2 I ) dt = a 2 ( I 2 sm t + 2t + sin4 21 ) 12()' = 3a21T. This area is therefore three times the area of the roll ing circle. For the radius of curvature = 1 / IKI we have by Eq. ( 15), p. 355, �' = (x2yx. +. !y:rl.) . = -2a .)2( 1 - cos I) = -4a I si n -2t I at the points t 0, 1 = ±27T, this expression has the value zero. These are actually= the cusps, where the cycloid meets the x-axis at right angles. The area of the surface of revolution formed by rotating an arch of the cycloid about the x-axis is given according to our formula (25), p. 374, by = f.� ds = 27Ti b a( I cos 1) 2a sin -1 !. dt = 16a27TJrn·sin3 = 8a27TJe·sin3 o = 16a27TL'( l cos2 sin The last integral can be evaluated by means of the substitution cos = we find ) 1· = -( 1 64a27T . A = l6a21r -cos + - cos3 3 3 As an exercise the reader may calculate for himself the height of the center-of-mass of the cycloid above the :r-axis and also the moment of inertia T,.. The results are 'Y) = -43 a = hs -A and dl
-
•
-
-
--
l pl
;
p
.
A
h
0
•
y
.
·
-
0
dt
u du
2
u)
-
u
2
u du.
r;
u
u
0
'Y)
378 b.
Applications in Physics and Geometry
Ch. 4
The catenary
The catenary! is the curve defined by the equation y = cosh x. The length of the catenary between the abscissas x = a and x = b is b + sinh2 x dx = cosh x dx = sinh b - sinh a . s=
J.
f.\11
For the area o f the surface of revolution obtained by rotating the catenary about the x-axis, the so-called catenoid, we find b 1 + cosh 2x A = 27T cosh2 x dx = 27T dx 2 = 1r(b - a + ! sinh 2b - ! sinh 2a).
Jb
J
a
a
From this we further obtain the height of the center-of-mass of the arc from a to b : A b - a + ! sinh 2 b - ! sinh 2a 1) = - = . 2(sinh b - sinh a) 27Ts Finally, for the curvature we have = K
y"
( I + y' 2/ 1
=
cosh x 3 cosh x
=
cosh2 x ·
c. The Ellipse and the Lemniscate
The lengths of arc of these two curves cannot be reduced to elemen tary functions, but belong to the class of "elliptic integrals" mentioned on p. 299. For the ellipse y = (b/a)J a2 - x2 we obtain
1 Jj�· -
(a2 - b2)x2
a2 - x2
s = -
a
dx = a
J J 11 --
1'/ 2e
e
---
d�,
where we have put xfa = �. 1 - b 2/a2 = 17 2 . By the substitution � = sin this integral can be expressed in the form s
J
= a J t - 17 2 sin 2 q, dq,.
1 The name derives from the fact that a chain suspended from its ends will assume the shape of this curve. Curioll51y enough the same curve arises in a quite different physical application. A soap film, bounded by two circles in space that lie in parallel planes and have centers on the same perpendicular to those planes, has the same shape as the surface of revolutwn obtained by rotating the catenary about the x-axis.
Sec. 4.3
Vectors in Two Dimensions 379
Here, to obtain the semi perimeter of the ellipse, we must let x traverse the i nterval from -a to +a, which corresponds to the interval - 1 :5: � :5: + 1
::_
::_ .
- 2 -< :5: + 2
or
For the lemniscate, whose equation in polar coordinates r, t r 2 = 2a2 cos 2t, we similarly obtain s
If we
=
fJ 1-J
r2 + f 2 dt =
J) ;-J
IS
s i n2 2 1 d 2 a 2 cos 2 t + 2 a2 t
dt = a , 2 ---= = a....; 2 J cos 2 t Jt
dt
cos 2 1
- 2 s i n2 t .
i ntroduce u = tan t as an i ndependent variable in the last integral, we h ave du u2 S ill t = --- ' dt = , 1 + u2 1 + u2 and consequently,
.2
s
J-JJ I
= a 2
--
du - u4
In a complete loop of the lemniscate u runs from - 1 to + I , and the length of arc is therefore equal to a Jl
f" 1 du I -1 vI 1 u 4
•
-
a special elliptic integral which played a great part in the researches of Gauss. 4.3
Vectors in Two Dimensions
For the discussions of plane curves and of many other topics in geometry, mechanics, and physics, vector notation constitutes a convenient and almost indispensable tool. We shall develop and apply in this chapter the concept of a vector i n two dimensions, leaving extensions to higher dimensions to Volume I I . Intuitire Explanation
M any mathematical and physical objects are characterized com pletely by a single n umber, called a "scalar" since it measures the object on a given scale. Examples are angles, lengths, areas, times, masses, and temperatures. There are other objects, however, for which such a characterization i s not possible, for example, the shape of a
380 Applications in Physics and Geometry
Ch.
4
triangle, the location of a point in space, the acceleration or direction of motion of a particle, and the state of tension in a body. Several numbers are required to identify each of these objects. Gradually, mathematical concepts beyond the continuum of real numbers have been developed which permit us to represent such objects by a single symbo!,l Vectors i n a plane are objects that can be described by two items of information : A length and a direction. Of this type are, for example, the relative position of two points, the velocity and acceleration of a particle, and the force acting on a particle. Geometrically, or intuitively, a vector is essentially a directed straight line segment i n the plane (or in space ) , characterized by its length or magnitude a nd by it s direction. Ordinarily, vectors are indicated by arro ws of the given length and pointing i n the given direction. Unless a restriction is explicitly i mposed , the vector is "free", that is, the location of the beginning of the directed line, or arrow, is not an inherent part of the s pec i ficati on s for the vector. W h il e p h y s ical concepts, such as velocity, acceleration, and force, are p r ima ry instances of vectors in applications, we shall define vectors geometrically, by means of "translations" or "parallel displacements." Vector analysis starts simply by giving a name, "vector," to such d irected line segments or parallel displacements. However, its decisive s ign ifica nc e is not that a u n i fyi ng name was introduced , but that these entities, the vectors, (similarly as th e complex numbers) can be combined with each other or with scalars (that is, o rd inary numbers) by a set of rules, called vector algebra or vector analysis, in ways that have na tu ral interpre tati on s in the various appl ica ti on s , as, for example, the super position of two velocities or the work done by a displacement against a fo rce. In the i n tu iti vely appealing language of vectors many mathe matical and physical relations can be expressed concisely and clearly.
2
a. Definition of Vectors by Translation. Notations
The si mplest type of transformation of the plane is a translation or parallel displacement. A translation shifts or maps any point P = (x, y) into the point P' = (x ' , y ' ) w i th coordinates x'
= x + a,
y'
=
y + b,
1 Of course, complex numbers " + bi = z are such symbols representing the pair of real numbers, a, b; it is indeed someti mes convenient to use complex numbers rather than vectors. ' Vectors are insufficient for some purposes ; to describe, for example, tensions or curvature of spaces, more general entities called "tensors" are used.
Sec. 4.3
Vectors in Two Dimensions 3 8 1
where a and b are constants. The translation is completely determined by the ·constants a and b which we call the components of the trans lation. We shall use the term "vector" as another name for "trans lation." Employing boldface type to denote vectors or translations we write R = (a, b) for the vector with components a, b ( Fig. 4.29). y
6
4
2 --o=+----�2�----o4----�6�----o8----�1�0-----1o2---- x a
Figure 4.29
= fiP· =
The translation x ' = .r + 2,
otr = (2, t ).
y
'
=
!I
+
1
corresponding to the vector
The components of the vector R are determined by one pair of corresponding points P = (x, y) and P' = (.c', y'), since then a =
x'
-
x,
b
=
y' - y.
Clearly, for any points P and P' i t is always possible to find a t � lation R which takes P into P'. We denote it as the vector R = PP' . Any ordered pair of points P = (x , y), P' = (x', y'), that is, any oriented
line segment, thus determines the vector R = W = (x' - x, y' - y). We observe that a second pair of points Q = (� , rJ), Q' = W, r / ) defines the same vector if �' - � = x' - x and r / 'fJ = y' - y ; the same translation R takes then P into P' and Q i nto Q ' . Vectors R are determined by two numbers, the components, just as points are by two coordinates in the plane ; the basic distinction is that a vector i s represented geometrically b y a pair of points. I n the representation -
= W we call P the initial point and P' the end point. For given R one of the points, say the initial point P = (x, y), can be chosen arbitrarily ; the end point P' = (x', y') is then determined uniquely
R
382
Applications in Physic.< and Geometry
Ch. 4 P"
p
Figure 4.30
-- - - - - - - ------
a
Components a, b and length
r
I I I lb I I I I
�
of a vector
R
= Pr.
by the relations x' = x + a, y' = y + b. Interchanging i nitial and end point leads to the opposite vector n = ( - a, -b). If we choose for the i nitial point the origin 0 = (0, 0) , we can associate uniquely a vector R with every point Q = (x, y) by taking R = OQ. The vector R with the fixed i nitial point 0 is then called the position vector of Q. The components of the position of Q are simply the coordinates x, y of Q. The vector R with components a = 0, b = 0 is called the null vector and is denoted by 0. It corresponds to a translation that leaves every point fi xed : 0 = (0, 0) = PP. �
The distance r of two points P = (x, y), P' = (x' , y') depends only on the vector R = (a, b) = PP', since r = -/(x' - x) 2 + (y' - y) 2 = J a2 + b 2 •
We call it the length of the vector R and write r = [ R [ . The length of R is always a positive number unless R = 0 (see Fig. 4 . 30) . We define the product of a vector R = (a, b) by a number or a "scalar" ). as the vector R* = AR = (la, l b) For A = - I we have i n R * ( Fig. 4 . 3 1 )
=
.
( - a, - b) the vector opposite to R
.
Figure 4.31
Scalar multiples of a vector R.
Sec. 4.3
If
Vectors in Two Dimensions 383
R =;= fiP = (a, b) with P = (x, y), P' = (x', y'), we can represent R
= A as W, where r = (x", y") = (x + Aa, y + Ab) (see Fig. 4 .32). For a = b = 0 we have, of course, P" = P' = P. For a and b not both zero the point P" = (x", y") = (x + Aa, y + Ab) traverses for varying A the whole line
R*
x"b - y"a = xb - ya.
The value A = 0 gives P" = P, whereas A = I gives P" = P' . Thus P" lies on the line through P and P' ; for A > 0 the points P" and P' lie on the same side of P, for A < 0 they lie on opposite sides. fY' =
P=
(x.y)C�----L-:a
Figure 4.32
I
_ _ _ _ _ _ _
I I I I I I I I I I I I
=J
(x'', y")
,\b
The vector relation R • = PP�' = A ff' for ?. = l
The two vectors R = (a, b) and R * = (a *, b *) are said to have the same direction if R * = AR with a positive A and opposite directions i f A < 0 . I f R = 0 , t h i s means that also R * = 0 . I f R ¥- 0, the necessary and sufficient condition for R * to have the same d irection as R is that b* b a a* = ../a2 + b2 ../a•2 + b*2 , .Ja2 + b2 ../a*2 + b*2 We call the quantities 7J =
b = -b b =.Ja2 + b2 [ R [
r
which determine the direction of the vector R the direction cosines of R ; they are, o f course, n o t defined for R = 0 . Since � 2 + 7J 2 = I , we IX can always find an angle IX and a corresponding angle {3 = 71'/2 such that � = COS IX, 7J = sin IX = cos {3. -
384 Applications in Physics and Geometry
Ch. 4
The angle ex is called a direction angle of R (Fig. 4.33). It is determined uniquely only to within an even multiple of rr . For R = cos ex =
x' - x
---
r
sin ex =
'
y' - y .
PfY we have
r
Obviously, ex is the angle between the positive x-axis and the line from P to P'. More precisely a rotation of the positive x-axis about the origin by the angle ex (counted positive if we turn counterclockwise, negative if clockwise) will give the axis the direction from P to P'. P'
b
a-
a
Figure 4.33
D i rection cosines � .
'I•
X
2?r
and d i rect ion angles for a vector
ff'.
The opposite vector R = ( -a, - b) has direction cosines - � and -1} and di rection angles differing from ex by an odd multiple of rr . -
If the initial point P of the vector R = PP' is the origin, the direction angle ex of R is simply the polar angle () of P'. b. Addition and Multiplication of Vectors
Sums of Vectors
Vectors have been defined by translations, that is as certain mappings of points in the plane. There is a perfectly general way of com bining any two mappings by applying them successively. If the first mapping carries a point P into the point P' and the second one carries P' into P", the combined mapping is the one that carries P into P". In the case of two vectors R = (a, b) and R * = (a* , b*) the vector R will map the point P = �x. y) onto the point P' = (x + a, y + b) and R * will map P' onto P" = (x + a + a *, y + b + b*). The
Sec. 4.3
Vectors in Two Dimensions 385
resulting mapping from P onto P" is again a translation ; we call it the ' or the resultan t of the vectors R = ""fJP and R * = ff, and denote i t by R + R * ( Fig. 4. 34). 1 The components of the sum are a + a * and b + b * . Thus our definition of the sum of two vectors is
sum
or, if we describe the vectors by their components,
(a, b) + (a * , b *) = (a + a * , b + b *) . If R * is taken from the same i nitial point a s R, say R * = PP"', the points P, P"', P", and P' form the vertices of a parallelogram. The
p Figure
4.34
Addition of the vectors pP'
=
(a, h) and p;t:;
=
(a• , b*).
two sides from P represent the vectors R and R * ; the s u m R + R * is represented by the diag o na l from P ("parallelogram construction" for the sum of vectors). Sums of vectors obey the commutative and associative laws of arithmetic, since addition of vectors j ust amounts to add ition of corresponding components ( Fig. 4.35). They obey moreover the distributive laws for multiplication of a sum of two uectors by a number A. and of a L'ector by the sum of two numbers l, p, :
A.(R + R *) = A.R + A.R *,
(A + p,)R
=
A.R + p R . 2
This "sum" is real ly the "symbolic product" of the two mapp i n gs as defined on p. 5 2 . The sum notation is here more natural because it corresponds to addition of the components. 2 To distinguish vectors from numbers in a n equation we al ways let the n u m ber precede the vector in writing products ; the combination R). will not be used , although it could be defined by ?.R = R?.. 1
386
Applications in Physics and Geometry
Figure 4.35
(R
+
R*)
+ R**
=
R + (R•
+
R•*)
Commutative and associate laws of vector addition .
w
p
Ch. 4
- -- --
-- - -
Figure 4.36
--
-- -- -
p·
0
ff' = OP' - OP.
p
Figure 4.37
PQ = iJ + AD+ BC+ · · · + FQ.
D
Sec. 4.3
Vectors in Two Dimensions 387
These�es pe�t us to express a vector "fiP' i n terms of the position
vectors OP and OP' of the points P and P' (Fig. 4. 36) :
i>P = ro+ or = DP + ro = w - oP: It is i mportant to realize that generally if we go from a point P to a
. , £, F, then the vector PQ is the "AB, BC, . . . , Ei, FQ (Fig. 4 . 37).
point Q by way of points A , B, C, .
sum of the vectors PA,
.
Angle betll'een Vectors The angle () formed by a vector R • = (a*, b*) with the vector (a, b) is defined as the difference of their directi on angles : 0 = oc * - oc. (It is assumed here that neither R nor R * is a zero vector.) The angle 0 again is determined only to within integer multi ples R =
Figure
4.38 A ngle e the vector R* forms with R.
of h (Fig 4.38) A rotation by the angle 0 (with the sign of () ind icating the sense of rotation) will take the di rection of R into that of R *. The quantities cos () and sin 0, wh ich are determ ined uniq uely , can be expressed i mmediately in terms of the direction cosines of R and R * :
- oc) = cos oc cos oc* + sin oc sin oc* aa* + bb* Ja2 + b2Ja•2 + b*2 ' sin 0 = sin (oc* - oc) = cos oc sin oc* - sin cos oc* ab* - a*b
cos 0 = cos ( oc*
oc
The denominator i n each expression is just the product rr * of the length of the vectors. We introduce the expressions occurring in the numerators as "products" of the two vectors.
388
Applications in Physics and Geometry
Ch. 4
Inner Product and Exterior Product of Two Vectors We define the "scalar" or "inner" or "dot" product of the vectors R = (a, b) and R * = (a * , b * ) by
R · R* = aa* + bb * = rr* cos 0, and the "outer" or "exterior" or "cross" product by R
x
R* = ab * - a *b = rr * sin 0.1
As immediately confirmed inner and outer products obey the dis tributive and associative laws:
R · (R * + R * *) R X (R * + R * *) ?.(R · R*) A(R X R*)
Q•
= R · R * + R · R * *, = R X R * + R X R* *' = (?.R) · R * = R · (AR *), = (?.R) X R * = R X (AR *). Q
Figure 4.39 The vector product triangle PQ Q * .
R
x
R*
=
I R [ I R * I sin e as t w ice the area of t he
The commutative law of multiplication also holds for inner p roducts
R · R* = R * · R ; for exterior products however, the sign is changed if the factors are interchanged : R X R * = - R * X R. Giving R and R * the same initial point, R = PQ, R * = PQ* we can interpret R · R • as the product of the projection r* cos 0 of the segment P Q * onto the segment PQ, with the length r of that segment. The outer prod uct R x R * is simply twice the area of the oriented triangle PQ Q * , taken with the positive sign if the vertices PQ Q * are in counterclockwise order , with the negative sign if in clockwise order ( Fig. 4. 39).
1
With our definition both inn er and exterior products are actually "scalars." The term "sca lar product" is re se r ve d for t he inner product because in three d imensions t he analogue of the exterior pr;,duct is a vector.
Vectors in Two Dimensions 389
Sec. 4.3
For any vector R
=
(a, b) R
·
R =
a2 + b 2
=
IRI2
R = 0.
is the square of the length of the vector. Thus R R is positive unless On the other hand, R x R is always zero. The condition for two nonzero vectors to be orthogonal to each other is that R R * = 0 while they are parallel (that is, have the same or opposite directions) if R X R* = 0. ·
·
Equation of Straight Line We can easily write the equation of a line thro ugh two poi nts and that of a line through a given point with a given direction. in vector
Figure 4.40
0
Line in vector notation.
notation. I f P = (x, y), P0 = (x0, y0) , a nd P 1 = (.r 1 , .'ld are th ree points with P0 ""' P1 then P l ies on the line through P11 and P1 if the vectors
�
�
P0P and PoP1 are parallel, that is, P0P
X
P0P1 =
0.
I f R = oP, R0 = �. and R 1 = oP. are the position vectors of the th ree points, the condition takes the form or
(R1 - R0) X R
= R1
X R0.
Substituting the coordinates of the points for the position vectors, we o btain the equation of the line i n the usual form ( Fig. 4 .40) : (x1 - X0)!1 - ( y1 - Y0)X
=
X1 Yo - Y1Xo.
390 Applications in Physics and Geometry
Ch. 4
Instead of prescribing two points of the line we can prescribe one point P0 and require that the line is to be parallel to a vector S = (a, b). Obviously, the equation of the line is then or
(R - R0) (x - x0)b
X
S=0
- (y - y0) a =
0.
For S = P0P1 we obtain the previous equation. The distance d of the line from the origin can also be expressed in �r notation. Obviously, d multiplied with the length of the vector P P0P1 is twice the area of the triangle OP0 1 • Hence R 1 (Mo Q?;) R0 X 1 d = IP P I X R I 0 , , - R ol =
Xo Y ! - x ,yo . ,J( x , - Xo)2 + (y, - Yo)2
-
Here d is taken with the positive sign if the points 0, P0, P1 follow each other in counterclockwise order.
Coordinate Vectors. A vector R in the form (29)
R
=
(a, b) trivially can be represented
=
a i + bj,
where we denote by i and j the "coordinate t'ectors" (30)
i
=
( 1 , 0),
j
=
(0, 1 ) .
I n this way R is split i nto two vectors a i and bj pointing respectively in the direction of the x-axis and y-axis. The components a and b of R are just the (signed) lengths of these two vectors. In appl ications one is often called upon to represent a vector R as resultant of vectors with two given orthogonal (that is, mutually perpendicular) directions. For that purpose it is best to introduce two unit vectors (that is, vectors of length I) I and J with the given directions. The required decomposition of R is achieved if we can represent R i n the form R = A l + BJ (3 1 ) with suitable scalars A , B (c f. Fig. 4.40). It is easy to find the values of A and B if such a representation of R exists. For, by assumption, the vectors I and J are orthogonal unit vectors of length I , so that
( 32)
I · I = J · J = 1,
I · J = 0.
Sec. 4.3
Vectors in Two Dimensions 391
Forming the scalar product of Eq. (3 1 ) with I, J respectively we find immediately that A and B must have the values (33)
B = R · J;
A = R · l,
in words, A and B are the (signed) lengths of the projections of the segment representing R in the given directions. The possibility of writing R as a linear combination (3 1 ) of I and J follows from the representation (29) of R in terms of i and j , if we can y
Figure 4.40
show that i a n d j the m se l ve s can be ex p re ss ed i n ter m s However, I = (o:, (J), J = (y, b), can be written as,
(34)
J = yi
I = o: i + (Jj,
Because of (32) the quantities orthogonality relations
o: ,
of I a n d
J.
+ bj.
f), y, b m u s t sati sfy th e so-called
(3 5)
z 2 + (J2 = y 2 + ()2 = I ,
( 36)
(o:b - (Jy)i =
o:y + (Jb
If we m ultiply the first of t h e equations (34) by (1, and subtract we find bl -
(JJ
b,
= 0.
the second one by
392
Applications in Physics and Geometry
Ch. 4
and similarly, ( cxo - fly)j = - yl + cxJ.
(37)
Here for the m utually perpendicular unit vectors I and J cxO - fly = I
(38)
x
J = ±I,
where the upper or lower sign holds depending on the counterclockwise or clockwise sense of the 90° rotation that takes I into J. In either case formulas (36) and (37) express i and j in terms of I and J ; substituting these expressions into (29) j ustifies the representation formula ( 3 1 ) for an arbitrary vector R. Formula (31) also can be interpreted as the representation of the vector R in a new coordinate system with axes pointing respectively in the directions of I and J. The components of a unit vector are at the same time the direction cosines of the direction angle of the vector. Let I and J h ave direction angles 4> and V' respectively. Then (X
= cos rp,
{/ = sin rp,
y = cos V'.
tl
= sin 'I' ·
Here either V' = r/> + �7T or 'I' = 4> - !1r. In the first case (which corresponds to a right-handed system of coordinate vectors I , J ) , we h ave y = {J tl = ex, cxrl - {Jy = + I so that -
( 3 9)
,
I = (cos rp, sin rp),
J = (-
sin ¢, cos rp) .
The formulas (33) giving the components of R referred to coordinate vectors I, J then take the form (40)
A = a cos 4> + h sin ¢,
B = - a sin 4> + h cos rp.
These formulas express the relations between the components of one and the same vector R in two right-handed coordinate systems obtained one from the other by a rotation of axes by the angle <{>. If we assume that the coordinate systems have the same origin 0 and that R is the position vector oP of an arbitrary point P we have in (40) the formulas for changes of coordinate systems already derived on p . 36 1 , Equation ( 1 8). The components a, b and A, B are then respectively the co ordinates of P in the two systems. c. Variable Vectors, Their Derivatives, and Integrals
It i s natural to consider vectors R = (a, b) whose components a, b are functions of a variable t , say a = a(t), b = b(t). For any t we then have a vector R
=
R(t) = (a(t), b(t))
Vectors
Sec. 4.3
R(t) R(t)
in Two Dimensions 393
and we say that is a vector function of t. An example is furnished by the position vector of a point that moves with the time t. We say that has the limit R * = ( * , b * ) for t --+ if has the limit a * and b(t) the limit b * for t --+ t0• In that case the length of tends toward that of * , and in case R * ¥- 0 the direction of tends toward that of * (this means that the direction cosines of tend toward those of * ). The vector R(t) is said to depend contin uously on if lim =
R(t) R(t)
R
t,
R
R
t0 a(t)
a
R
R(l) R( t0), that is, if the components of R are continuous functions of t. The length and, if R(t0) ¥also the di rection of a continuous vector 0,
t -+ to
vary continuously with t. To introduce the derivative of a vector we form for two values t and t + h of the parameter the difference quotient
b( t )] a(t) , b(t [a( t , and define the derivative of R as the limit of the difference quotient for h .--.. 0 : R( t) (dat , db) R ddtR l i m .!. . (d, b) . d dt
.!. [ (t h
R
+ h) _ R( t ) ] =
=
=
,, - o
h
+ h) h
]
[R(t + h) -
+ h) h
=
=
The derivative of a vector is formed by differentiating the components. Derivatives of products of vectors are easily seen to obey the ordinary rules
dR S dR S R dS RS + RS ( RS)" = -dt dt dt dR = R S,. R S)" = -dt •
(R
X
X
+
= - ·
s
.
X
· - =
s +
·
·
X
where for outer products, factors have to be taken in the original order. We define similarly the integral of the vector in terms of the integrals of its components :
R( t)
J:R(t) dt = (J:a (t) dt,J: h(t) dt). The fundamental theorem of calculus implies
-dtd J tR(s) ds = R(l). •
394 Applications in Physics and Geometry
Ch. 4
d. Application to Plane Curves. Direction, Speed, and Acceleration
Velocity Vector
In Section 4. 1 we represented a curve C by two functions x = (t) and y = 1p(t). Each t in the domain of these functions determines a point P = (x, y) on C; here t may be considered as time and P as a moving point whose position at the time t is given by x(t) and y(t). If we identify
x and y with the components of the position vector R = OP of P,
0
Figure
4.41
Derivative of the position vector for a curve.
then C i s described by the end point of the position vector R
=
R(t)
=
(x(l), y(t))
( Fig . 4 . 41 ) . For two points P and P' of C corresponding to the para meter values 1 and 1 + D.t we have i n
W = W - oP = R ( t +
!::>. t ) - R(l)
=
D.R
the vector represented by the directed secant of C with end points P, P'. If here !::>. 1 is positive, that is, if the point P' follows P on C in the direction of increasing 1, then the vector
I
- ( R (t + D.t) - R( r)) D. t
�
R(r) = PP' ; its h as the same direction as the vector R(l + !::> f ) length i s the distance of the points P and P' d ivided by D.t. For D.t tending to zero we obtain in the limit the vector -
R.
=
R(t)
=
(x(t), .li (t)),
Vectors
Sec. 4.3
in Two Dimensions
395
where again the dot is used to denote differentiation with respect to the pammeter t. The direction of R is the limit of the direction of the secants PP' and hence is the direction of the tangent at the point P. M ore precisely R points in that direction on the tangent that corre sponds to increasing t on C. provided R � 0. The direction cosines of R are the quantities C O S IX
=
.j
sin oc =
l'2 + !/
----=----=-c--
!i
-:: , ====
v' .i2 + .1/
introd uced on p. 346 as direction cosines of the tangent. The length of R
.Jx2
+ 1? IRI = can be interpreted as ds/dt, the rate of change of the length s along the curve with respect to the parameter t . I f 1 stands for the time, we have in I R I the speed with wh ich the point travels along the curve. I n mechanics one must consider the velocity of a particle not only as h aving a certain magnitude (the "speed") but also a certain direction . Velocity is then represented by the vector R = (x, .li), whose length is the speed and whose direction is the instantaneous direction of motion, that is, the direction of the tangent i n the sense of increasing t. A cceleration
Similarly the acceleration of the particle is defined as the vector (x, jj). Vanishing acceleration means that i = jj = 0 ; if it = 0 along a whole t-interval, the velocity components have constant val ues x = a, y = b ; the components of the position vector i tself are then linear functions of t : x = at + c, y = bt + d. The particle in this case moves with constant speed along a straigh t line. All our previous results pertaining to curves are easily expressible i n vector notation if the curve i s described by the position vector = R(t) = (x(t), y(t)), with 11. � t � fl. We find for the length [cf. Eq . (8), p. 349] I R I dt, it
=
R
P
J �rR .
while for the signed area enclosed by a curve [cf. Eq . (20), p. 365] A =
X
R dt
(the sign of this quantity depending again on the orientation of the curve) . Finally, we have for the curvature K the formula [cf. Eq . ( 1 5 ), P · 355] R X R K = --- . I R I3
396 Applications in Physics and Geometry
Ch. 4
Tangential and Normal Components of A cceleration These formulas have interesting implications if we interpret t again as the time. Let y be the angle formed by the vector it with the vector R, that is, with the instantaneous direction of m otion. The quantity j lt j cos y represents the proj ection of :it on t o the direction of R ; we call it the tangential component of acceleration. Similarly, j :lt j sin y i s the projection o f :it onto the normal (more precisel y onto that normal
0
Tan gential and normal acceleration .
Figure 4.42
obtained by a 90° counterclockwise rotation from R ) ; this is the normal component of acceleration ( see Fig. 4.42) . By defi nit i o n of inner and outer products
..
R.� . R. ,
) R j cos y = Now
n. R
,., .. ·
where v
=
=
,., .. 1 (n. R
-
·
2
ds/dt
=
)Rj
..
,.,
R .. ) R ) sin y =
+
R n.)
=
.)R.
..
·
·
X
R
--
=
1 d
2 dt
- -
,.,
,.,
(n. n.) ·
=
)R) I
.
dv 2 = 2 dt
- -
dv
r• - ,
dt R is the speed of the point. Hence dv
= v; dt Thus the tangential component of acceleration is identical with the rate of change of speed with respect to time. For the n orma l acceleration the formula for the curvature yields (42) ) :it ) s;n y = K ) R j 2 = Kl'2, (4 1 )
I R I cos y
=
-
that is, the product of the square of the speed with the curvature .
Motion o( a Particle under Given Forces
Sec. 4.4
397
For a particle moving with constant speed along a curve the tan gential acceleration v vanishes. The acceleration vector then is per pendicular to the curve. More precisely it points toward the "inner" side of the curve, the side toward which the curve turns (this is seen, for example, from the fact that sin > 0 when > 0, that is, when the tangent turns counterclockwise). In moving along a curve at constant speed therefore, a point experiences an acceleration toward the inside of the curve which is proportional to the curvature and also to the square of the speed. This fact is of obvious significance because as a result of Newton's law (to be discussed later) a force proportional to the acceleration is needed to hold the point P on the curve. t•
y
4.4
K
Motion of a Particle under Given Forces
The early development of calculus was decisively stimulated not only by geometry but just as much by the concepts of mechanics. Mechanics rests on certain basic principles first lai d down by Newton; the state ment of these principles involves the concept of the deri v ati ve, and their application requires the theory of integration. Wi t hout analyzing Newton's principles in detail, we shall illustrate by some simple examples how calculus is appli e d in mechanics. a. Newton ' s Law of Motion
We shall restrict ourselves to the consideration of a singl e particle, that is, of a point at which a mass m is imagined to be concentrated. We shall further assume that the motion takes place in the .c,y-plane, in which the position of the particle the time t is specified by its coordinates x x(f), y(t), or, equival e ntl y , by its "positi o n vector" R R(t) A dot above a quantity indi c ates diffe renti a ti o n with respect to the time t . The relocity and acceleralion of the parti c l e are then represented by the vectors R (i, y) and it (i, fj). In mechanics one relates the motion of a point to the concept of forces of definite direction and magnitude acting on the point. A force is then also described by a vector F a). The effect of several forces F1, F2 , acting on the same particle is the same as that of a single force F, the resultant force, which is simply the vector sum F = F 1 + F 2 + · · · of the individual forces. Newton's fundamental law states: The mass m mul!iplied by the acceleration is equal to the force acting on the particle, in symbols y =
=
=
=
at
(:r( l), y(f)).
=
=
=
.
��
•
.
mit = �
(p ,
398
Applications in Physics and Geometry
Ch.
4
If we write this vector equation which expresses the fundamental law i n terms of the components of those vectors, we obtain the equivalent pair of equations mi = p, mf) = a. (44)
Since acceleration and force differ only by the positive facto r m, the direction of the acceleration is the same as that of the force. I f no force acts, that is, F = 0 , the acceleration vanishes, the velocity is constant, and x and y become li near functions of t. T h is is Newton's first law : A particle on which no force acts moves with constant velocity along a straight line. N ewton's law mti = F is in the first instance nothing more than a quantitative definition of the concept of force. The left-hand side of this relation can be determ ined by observation of the motion, by means of which we then obtai n the force. H owever, Newton's law has a far deeper meaning, due to the fact that in many cases we can determine the acting force from other physical considerations, without any knowledge of the corresponding motion. This fund mental law is then no longer a definition of force, but it instead is a relation from which we can hope to determine the motion. This decisive turn i n using Newton's law comes into play i n all the numerous i nstances where physical considerations permit us to express the force F or its components p, a in an explicit way as functions of the position and velocity of the particle and of the time 1 . The law of m otion then is not a tautology, but furnishes two equations expressing mi, my i n terms of x, y, x, 1i. and I , the so-called equations of motion . These equations are differential equations, that is, relations between functi ons and their derivatives. Soh•ing these differential equations , that is, finding a l l pairs o f functions x(t), y(t) for which t h e equations of motion are valid, yields all possible motions of a particle under the prescribed force. b. Motion of Falling Bodies
The simplest example of a known force is that of graPity acting on a particle near the surface of the earth. It is known from direct obser vation that (aside from effects of air resistance) every fall ing body has an acceleration which is d irected vertically downward , and wh ich has the same magnitude g for all bodies. Measured in feet per second per second, g has the approx; m ate value 32. 1 6. 1 I f we ch oose an
1 The precise value of g. which also includes in add ition to gravitational attraction , effects of the rotation of the eanh, depends on the location on the ea rth.
Sec.
4.4
Motion of a Particle under Given Forces
399
x y-coord i n ate system in which t h ey a xisItpoi nts vertically upward while t h e x-axis is horizontal, the acceleration (i', f)) h as the components ,
-
w
x = 0,
s
=
ij = -g.
p
Ne ton fundamental law the vector F re rese nt i ng the force of gravity acting on a particle of mass must then be By
'
m
F
(0, - mg).
=
This force vector is likewise directed vertically downward ; i ts magni tude o f the body near the surface of the earth, is When we cancel out the factor the equations of motion of a p a r ti c l e under g a vi ty take the form , the weight
mg.
r
m,
x = 0,
ii =
-
g.
From these equations we can easily obtain descri ption of the most general motion possi ble for a fall i ng body . I ntegrating with respect to yiel d s !i = where and are constants. A fu rther i teg ra tio then shows th at a
i: = a ,
t
a
-
h
x
= at +
.11
c,
=
-
g t + h, n
n
L�t2 + ht + d,
where c and d are c n s tan ts. T h u s the general sol ution of our equa ons of motion depends on fo ur u n-specified constants a, h, c, d. We can i m med iately relate t e val ues these constants for an i nd i vid ual motion to the initial conditions for that moti o n . If the particle the find i n i t ial ti me t = 0 i s at the point ( x11 , y11) , then setting t = 0
o
h
of
ti
we
d = Yo ·
at
velocity R = (i:, .lil = (a, -gt + h) reduces for I = 0 (a, h ) . Th us (c, d) and (a, h) represent r spe t i v ly i n i tial position and i n i tial velocity of the particle. An choice of these i n i tial ndi i on a motion. In case a 7Jf 0, that is, i n case the i n i tial velocity i s not vertical, we can eli minate 1 and obtain a n on parametric representation fo r orbit of the particle. Solving the first qu tion f I and substituting i nto the second yields
The
y
uniquely to
e a
y
=
-
to
e c e
co t s leads the
or
h � -'(x - c) 2 + - ( x - c) + d . 2a 2 a
Hence the pa th i s para bo la For 0 we have co n an t and the whole motion takes pl a e along a vertical st ra i gh t l i e . a
.
a =
c
x
= c
n
=
st
,
400
Ch. 4
Applidations in Physics and Geometry
c. Motion of a Particle Constrained to a Given Curve
In most problems of mechanics the forces acting on a particle depend on the position and velocity of the particle. As a rule, the equations of motion are too complicated to permit us to determine all possible
motions. Considerable. simplification arises if we may consider the cu rve C described by the particle as known and only have to determine the motion of the particle along the curve. In a large class of mechanical problems the particle i s constrained (by means of some mechanical device) to move on a given curve C. The simplest example is the plane pendulum where a mass m is joined by an inextensible string of length L to a point P0 and moves under the influence of gravi ty on a circle of radius L about P0. Along the curve C we use the arc length s as parameter. The cu rve is then given by x = x(s), y = y(s). Finding the motion of the particle along C then amounts to finding s as a function of 1. An equation of motion along the curve is obtained as fol lows. We form the inner product of both sides of Newton's formula m R F with a vector ; : m R · !; = F . !; . I f we take for !; the vector of length I whose directi on is that of the tangent to C in the sense of increasing s, th a t is, !; = dR fds we have in F !; = f the tangential component of the force, or the force acting in the direction of the motion . According to E q u a t ion (4 1 ), p . 396, the tangential component R · !; of the acceleration is j ust dr'fdt = d 2sfdt2, that is, the acceleration of the particle along the curve. Newton's law then yields the formula ms = .r. (45) that is, the mass of the particle multiplied with the acceleration of the =
,
•
particle along its path equals the force acting on the particle in the direction of motion. I n applying this eq uation to a particle constrained to move along C
we assume that the constraints make no contribution to .f.l For a force F = (p, a) we have then by Equation (44), p. 398,
( 46)
f = p ds + a -'ds dx
d?J
-
1 Actually, t he mechanism of constraint has to supply a force that holds the particle on C (in the simple pendulum thi•; is provided by the tension of the string). We assume that this "reaction" force is perpend icular to the curve and thus has no tangential component ; this woulc be the case for frictionless sliding of the particle along a curve.
Sec. 4.4
ll1otion
of a
Particle under Git·en Forces
£ht
40 1
since theyector ,; has the components dx-ds , ....:.ds.. (see p. For a known d11 dx curve C the direction cosi n es - and ....:... of the tangent can be considered ds ds as known functions of s. If likewise the force F = (p, a) depends only on the position of the parti c le, we have in / a known functi o n of s. The motion of the particle along then has to be determined from the relatively simple differential equation m.f = f(s ) . 394 ) .
C
Figure 4.43
Motion on a g iven cu rve under gravit y .
Specifically, for the gravi t ational force ( 46a )
f
F = {0,
d lt = - m g -'- ; ds
- mJ;)
we have
thus the equati o n of moti o n of a particle constrained to move on curve under the influence of gravity becomes C
d2s = dt 2
(4? )
dy
- g ds
·
I f denotes the inclination angl e of the curve, we have dyjds = sin (see Fi g . and the equati o n of moti o n becomes r1.
4.43),
d 2.� dt 2
- =
a
. -g stn
r1. .
L
r1.
For a parti c le constrained to move on circl e of radius about the origin ("simple pendulum") sin 0, = -L cos 0, x = L
y
a
402
Applications in Phy;'ics and Geometry
Ch. 4
0
Figure 4.44
The simple pendu l u m .
where fJ = sf L is the polar angle counted from the down ward direction. Here (see Fig. 4.44) rx = () an d thus
d 2s
dt '
or
4.5
=
-
g
dy d() dO
-;J;
. = - g sm ()
Free Fall of a Body Resisted by Air
We s tar t
with two exam ples of the motion of a part icle a l o n g a We co n s ider o n l y cases w h e re the force acts i n the d i rection of t h e l i ne so that n o mech a n i s m of constraint is n ecessary. T h e pa t h of a body fa l l i n g freely do w nward can be descri bed para m et ri c a ll y by x = c o n st a n t , y = s. If gravity i s the only force a c t i ng , we h ave the eq u a t i o n of motion straight l i ne .
ms = - mg .
For a particle released at the time t = 0 from t h e a l t i t u d e Y o = s0 w ith initial velocity t' o ( co u n te d p o s i t i ve i f u pw ard) , we find then by integration s = - �gl2 + t' 0 / + S0• If we wish to take account of the effect of the .fi-iction or air resistance acting on the particle, we have to consider this as a force whose direction is opposite to the di rection of motion and concerning which we must make definite physical as su mpti o ns . 1 We shall work out the results of 1 These assumptions must be chosen to suit the particular physical system under consideration ; for example, the law of resi stance for low speeds is not the same as that for h igh ones (such as bul let velocities).
Free Fall of a Body Resisted by
Sec. 4.5
A ir
403
different physical assumptions: (a) The resistance is proportional to the velocity, and is given by an expression of the form - rS, where is a positive constant; (b) the resistance is proportional to the square of the velocity, and is of the form - r!/2 for positive s and rs2 for negative s. In accordance with Newton's law we obtain the equations of motion r
ms = - mg - rJ.
(a)
ms = - mg
+ n,'2, where we have assumed in (h) that the body is falling (S < 0) . If we first consider s = u( 1) as the function sought, we have
(b)
' mt
= - mg - n • , mi' = - mg + rr2•
(a ) (b)
Instead of determining as a function of 1 by these equations, we determine 1 as a function of t' , writing our differential equati o ns in the form r
dt du
(a)
g( l + k2v) '
dl dt•
( b)
k.
g( l
-
k 2 l, 2 )
where we have put ,.)r/mg With the help of the methods given in Chapter we can immediatel y carry out the integrations and obtain 1 - --:; log ( I + k2v) + 1 0, ( gk1 - kv I --- + 10. (b) I + kv Solving these equations for we have =
3
a)
I =
1
(a)
V =
I (1
I;"
- -
2 gk
=
l og
r,
- e
--uk' l t� lu l ) .
e- 2 r1ku · t u ) 2 k I + e� ukl t - 1 , 1
l
- -k tanh [gk ( t These equati o ns at once reveal an important property of the motion. The vel o city does not increase with ti m e beyond all bounds, but tends to a definite limit depending on the mass and the constant (which, in turn, depends on the shape of the falling body and the air density).
( b)
l' =
1
I
_
----- =
10)] .
m
r
404 Applications in Physics and Geomerry
For (a)
lim v(t)
t � oo
l i m v(t)
(b)
t � oo
=
=
-
k2
-
Ch. 4
- - mg r
! = - {mg . .j -; k
For the limiting velocities frictional resistance j ust balances gravi tational attraction. A second integration performed on our expressions for v(t) = s, with the help of the methods of Chapter 3, gives the results (which may be verified by differentiation) (a)
s(t)
( b)
.1(1)
=
=
1 ( e-uk, lt-tol - I ) + - I (I - t0 ) - k2 gk4 -
-
2
1
gk
log [cosh gk( t - 10) ] +
c
c,
where c is a constant of integration. Here 10 is the time at which the particle would have had velocity 0 and c its altitude at the time 10. The two constants c and 10 can also be related easily to the velocity and position at any other time 1 1 , if we consider those quantities as initial conditions. 4.6
The Simplest Type of Elastic Vibration-Motion of a Spring
As a second example-of major significance-we consider the m otion of a particle wh ich moves along the x-axis and is pulled back toward the origin by an elastic force. As regards the elastic force we assume that it is always directed toward the origin and that its magni tude is proportional to the d istance from the origin . In other words, we take the force as equal to - kx, where the coefficient k is a measure of the stiffness of the elastic connection. Since k is assumed positive, the force is negative when x is positive and positive when x is negative. Newton's law now tells us that
mi = - kx. (48) This differential equation by itself does not determine the motion completely, but for a given instant of time, say t = 0, we can arbitrarily assign the initial position x( O) = x0 and the initial velocity i(O) = v0 ; that is, in physical language, that we can start off the particle from an arbi trary position with an arbitrary velocity ; thereafter the motion is determined by the differential equation. Mathematically, this is expressed by the fact that the general solution of our differential equation contains two constants of integration , at first undetermined,
Sec. 4 . 7
Motion on
a
Gil'en Curt"e 405
whose values we find by means of the initi l conditions. This fact we shall prove immediately We can easily state such a solution directly. If we put J our differential equati on becomes d2J:jdt2 - w2x. The substitution wt for the independent vari a ble reduces thi s equation to the form , discussed in Chapter 3 , p. 3 1 2 . Thus our differential equation is satisfied by all the fu nctions = c1 co s oJ/ + s i n rot, which may also be verified at once by differentiation (where and denote const nts chosen arbitrarily). In Cha pte r p. 3 1 3 , we saw that there are no other solutions of our differential equa ti on and hence that every such motion under the influence of n elastic force is g i ven by this expression. This can easily be put i n the form = a sin - b) = -a sin cos wl + a cos sin we need only wr i te s i n = and cos = c2 , thus i ntro ducing instead of and the new constants a and M otions of th is type are said to be sinusoidal or simple harmonic. Th ey are periodic; any state [t h at is, position .r(l ) and velocity i(l)] is repeated after the time T = 2n/co, which is called the eriod, since the functions sin o•t and cos wt h a e the period T. The pn um ber a is called the maximum displacement or amplitude of the oscillation. The number 1 / T = w/2n i s called the .frequem:v of the oscil lation ; it measu res the n u m ber of osci llations per u n i t time . We return to the theory of oscillati o ns in Chapter 8 . a
.
rv
=
T = d2x/dT2 =
=
k/m ,
c1
c2
-x
x(t )
c2
a
3,
x(t)
w(t
-a
c1
c2
OJ
o
oJ
a
o
c1
OJ
a
o
il .
(/J
il
OJ/ ;
v
shall
*4.7
Motion o n a Given Curve
a. The Differential Equation and Its Solution
We now turn to the general form of the problem of motion along a given curve under an arbitrary preassigned force mf(s). We shall deter mine the fu ction s(t) as a function of by of the d ifferen tia l equati o n [Eq. (45), p. 400] = f(s), where fts) is a given function.1 This differential equation in s can be solved compl ete ly by the fo ll ow ing device. n
1
s
mean s
Our original equation of motion along a curve was nrJ.' = f"Cs) ; we can , however. always write the function /"(>") in the form m((.l"). obtai ning the simpler form of the equation used here . 1
406
Applications in Physic>' and Geometry
Ch. 4
We consider any pri mitive function F(s) of f(s), s o that F'(s) = f(s), and mu l ti pl y both sides of the equati on s = f(s) = F'(s) by s. We can then write the left-hand side in the form d(S2 /2)/dl, as we see at once by d i fferentiating the expression s2 ; the right-hand side F (s)s however, by the chain rule of differentiation is the derivative of F(s) with respect to the time I, if in F(s) we regard the qu a nt i ty s as a function of 1. Hence we i m mediately have '
or by integration
( ·2)
d ' dt 2
d F( s ) , dt
- - s
= -
�s2
F(s) +
=
,
c,
where c denotes a constant yet to be determined. We have now arrived at an equation which only i nvolves the function s(t) and i t s first derivative. (Later on we shall i nter p ret this equat io n as expressing the conservation of energy during the motion.) Let us write this eq u a t i o n in the form ds/dt = .J2[F(s) + c]. We see that from this we cannot i mm ed iate l y find s as a function of t by i n tegra t i o n However, we arrive at a solution of the problem if we at fi rst content ourselves with finding the inverse function t(s), that is, the time taken by the particle to reach a d e fini te position s . For l(s) we h ave t h e equation dt 1 ' ds .J2[F(s) + c] .
thus the derivative of the function t(s) is known, and we have I =
I)2[F(s) ds
+ c]
+ c ''
where c 1 is another constant of integration . As soon as we have pe r fo rm ed this last i ntegrati o n we have solved the problem, for although we have not determ ined the position s as a function of t, we h ave inversely found the time t as a function of the position s. The fact that the two constants of i n tegrat i o n c and c1 are still ava i la bl e enables us to make the general solution fit special initial conditions. The ge n er a l discussion can be i l l us tra ted by our earlier example of elastic vibrations if we identify :c with s; here f(s) = - oh and corre s p o n d in gl y say, F(s) = § ro2s2• We t h e refo re obtain ,
-
dt ds
Sec.
4.7
Motion 011 a
Git•en Curee 407
and furthermore,
This i n tegral , h owever, ·can easily be eval uated by i ntrod ucing r•>s/ .J2c· as a new variable : we thus obtain
I . WS t = - arc sm -;= + c 1 , .,; 2c
w
or, forming the i nverse function, S
v1 2c . = - Si n (I)
(
OJ I -
c1 ).
We are thus led to exactly the same formula for the solution as before. From this example we also see what the constants of i n tegration mean and how they are to be determined. I f, for example, we req u i re that at the time t = 0 the particle shall be at the point s = 0 and at that i n stant shall have the veloci ty s(O) = I , we obtai n the two equations 0 =
.J2c .
--
(I)
sm
I =
o> c 1 ,
.J"i:: cos wc 1 ,
from w h ich we fi n d that the constants h ave the values c1 = 0, c = A The constants o f i n tegration c and c 1 can be determined i n exactly the same way when the i n i tial position s0 and the i n i tial velocity .50 (at t i me I = 0) are prescri bed arbitrarily. b. Particle Sliding down a CUI·ve
The case of a particle sliding down a frictionless cu rve under the i nfluence of gravi ty can be treated very si mply by the method j ust described . We found al ready on p. 40 1 the equation of motion corre sponding to this case : dy .i' = ' ds
-g
where dots i n d icate d i fferentiation with respect to the time f. The right-hand side of this equation i s a k nown function of s, since we k now the curve and we can therefore regard the q ua n tities x and y as known functions of s. As in the last section, we multiply both sides of this equation by s. The left-hand side then becomes the derivative of frS2 with respect to t. If i n the function y(s) we regard s as a function of I, the right-hand side
408
Ch. 4
Applications in Physics and Geometry
of our equation is the derivative of -gy with respect to 1. On inte grating, we therefore have
ts2
-gy + c,
...
where c is a constant of integration. To find the interpretation of this constant, we suppose that at the time t = 0 our particle is at the point of the curve for which the coordinates are x0 and Yo and that at this instant its velocity is zero, that is, s(O) = 0. Then putting t = 0 we immediately have -gy0 + c = 0, so that Since s2 could never be negative, we see that the altitude y of the particle never exceeds the value y0, and only reaches i t at those i nstants when the velocity of the particle i s zero. The velocity i s larger as the particle is lower. Now i nstead of regarding s as a function of t we shall consider the inverse function t(s) . For this we at once obtain
�
ds
= ± --;.===
which is equivalent to t =
c1
±
f
.J2g( Yo - y) ds
.J 2g( y0 - y)
,
where c 1 is a new constant of integration. As regards the sign of the square root, which i s the same as the sign of s, we notice that if the particle moves along an arc which is l ower than Yo everywhere except at the ends, the sign can not change. For the sign of s can change only where s = 0, that is, where y - y0 = 0. Thus the particle can only "turn back" at points of maximum elevati on y0 on the curve. I nstead of the arc length s the curve can also be referred to any parameter 0, so that X = c/>( 0) , y = 1p( O). Introducing e as independent variable, we obtain I =
r1
±
f
ds
-
dO
dO
.J2 g(y0 - y) .
=
r1
±
f) x'2
+ y'2 dO, 2g( yo - Y) ----
where the functions x' = cj>'(O), y ' = 1p'(O), and y = 1p(O) are known. In order to determine the constant of i ntegration c1 we note that for t = 0 the parameter 0 will have a value 00• This i mmediately gives us our solution i n the form
(49)
t
=
±
io
90
' + '2
J2g(x 2y0 -y y)
dO.
Sec. 4.7
Motion
011 a
Gil·en Curt"e 409
We see that this equation represents the time taken by the particle to move fr6m the parameter value 0 0 to the parameter value 0. The inverse function O(t) of this function t(O) enables us to describe the motion completely ; for at each i nstant t we can determine the point x = cf> [O(t)], y = 1p[O(t)) which the particle is then passing. c. Discussion of the Motion
From the equations just found, even without an explicit expression for the result of the integration we can deduce the genera l nature of the motion by simple i ntuitive reasoning. We suppose that our curve is of y
0
xo
Figure 4.45
XJ
the type shown i n Fig. 4.45, that is, that i t consists of an arc convex downward ; we take s as increasing from left to right. If we initially release the particle at the point A with coordinates .c0 = c/> ( 00), Yo = 1j!(00), corresponding to () = 00, the veloci ty i ncreases, for the acceleration s i s positive. The particle travels from A to the lowest point with ever-increasing velocity. After the lowest point is passed , however, the acceleration is negative, since the right-hand side -g dyfds of the equation of motion is negative . The velocity therefore decreases. From the equation s 2 = 2g(y0 - y) we see at once that the velocity reaches the value zero when the particle reaches the point B whose height is the same as that of the initial position A. Since the acceleration is still negative, the motion of the particle must be reversed at this point, so that the particle will swing back to the point A ; this action will repeat itself indefinitely. (The reader will recall that friction has been disregarded .) [n this oscillatory motion the time which the point takes to return from B to A must clearly be the same as the ti me taken to move from A to B, since at equal heights we have equal values of l si . If
Applications in Physics and Geometry
410
Ch.
4
we denote the time required for a complete journey from A to B and back again by T, the motion will obviously be periodic with period T. If 00 and 01 are the values of the parameter corresponding to the points A and B, respectively, the half-period is given by the expression
T 2
8 l i ' J� I ?lo � I r8· J 4>' 2(0) lp' 2(0) · -)2g ) 8 , I
I )2g
- = -= ( 50)
=
e,
--- dfJ - Y
dfJ + 1p(00) - 1p(O)
If 0 2 is the value of the parameter corresponding to the lowest point of the curve, the time which the particle takes to fal l from A to this l owest point is ( 8 ' x' 2 + y' 2 dO
�I J -}2g Je.
I·
Yo - Y
d. The Ordinary Pendulum
The simplest example is given by the so-called simple pendulum. Here the curve under consideration is a circle of fi xed radius L : x
= L sin 0,
y
= - L eos 0,
where the angle 0 is measured in the positive sense from the position of rest. From the general expression ( 50) we at once obtain using the addition theorem for the cos ine,
J2L�L,(8• Jcos dO
J� L,(8• L
dO ' . Oo . 2 -0 2 Sill - - Sin 2 2 where 00 (0 < 00 < 7T) denotes the amplitude of oscillation of the pendulum, that is, the angular position from which the particle i s released at time t = 0 with velocity zero. 1 B y the substitution T=
(}
- cos 00
sin (()/2) u = --s i n (110/2) '
=
du
dO
J
c o s ({}/2) 2 sin (()0/2)
our expression for the period of oscillation of the pendulum becomes T=
2
J� f
g -1
J( l - (
du
u 2) t -
u 2 s i n2
(�))
during the motion. This excludes tne type of t umbling motion of the pendulum in
' We have assumed here that the velocity does become equal to zero at some time
which () is not periodic and varies monotonical ly for all 1.
Sec. 4.7
Motion
on a
Ciren
Cune
41 1
We h a ve therefore expressed the period of osci llation of the pend ulum by a n elliptic integral (see p. 299). If w e assume t h a t t h e a m p l i t u d e of t h e oscillation is s m a l l , s o that w e m a y w i th sufficient accuracy replace the seco nd factor u nder t h e s q uare root s i g n b y I , w e obtai n the expression
as an a pp roxi m a t io n for the pe rio d of osci l lati o n . We can evaluate t h i s l a s t i n tegral by for m u l a 1 3 i n o u r ta b l e of i ntegral s (p. 263) and obtain the expression 2 .J L/g as an approx i mate value for T. To this order of a pp rox i m at i o n the pe ri od i s i n d epe nde n t of eo. that is, of the a m p l i tu d e of the oscil lation of the pend ul u m . Clearly, the exact period is larger and i nc r e as e s w i t h 00. S i nce in the i n t e r v al of i n te grati o n
1r
- sin 2 0!,
2
we find for t h e period the esti mates
j
21r � < g -
T< - C OS
I
( 00/2 )
=
j
cos 2 0!,
2
,
21r � . g
For angles 00 < 1 0° we h ave I / (c o s 00/2) � sec 5° < 1 .004 ,
t hat the
p er i od will be given by t h e fo r m u l a L/g w i t h a rel ative error of less than ·� " � · For fmer a pp rox i m a t i o n of the e l l i ptic i n teg ral for T see S ec t io n 7 . 6f.
21r.J
so
e. The Cycloidal Pendulum s t r i ct ly
The fact that the p erio d of oscil lation of the ord i n a ry pen d u l u m is not i n d e p end ent of the amplitude of osci llation caused C h ristian H uygens, in his prolonged efforts to construct accurate clocks, to seek a c u rve C for w h ich the period of o s ci l l atio n is i ndependen t of the p osi t ion on C a t which the oscillating particle begins i ts motion.1 Huygens recogn ized that the cycloid i s such a curve. I n order that a particl e may actually be able to oscil late on a cycl oid the cusps of the cycloid must p oi n t in the directi on o pp os i te to that of the force of gravity ; that is, we must rotate the cycloid considered prev i o us ly (p. 3 28) about the x-axis (cf. Fig. 4.2, p. 329). We therefore 1
The oscil lations are then said to be isochronous.
4 1 2 Applications in Physics and Geometry
Ch. 4
write the equations of the cycloid i n the form x = a(O + y
=
rr
+ s in 0),
-a( ! + cos 0),
which also i nvolves a change of the parameter t into 0 +
rr
(Fig. 4.46).
y
0
27l'a
Figure 4.46
Path described by a cycloidal pendul u m .
The time which the particle takes to travel from a point at the height
Yo
=
(0 < 00 < rr}
-a( ! + cos 00)
down to the lowest point, and up again to the height y0, by formula (50) of p. 4 1 0, is
f"" J J 2 2g -Oo Yo -
I
=
I
x' 2 + y'2 d()
y
=
J2ag j'"o
cos (0 /2 ) d&. -·· .)cos (j - cos e.
Using exactly the same substitutions as for the period of the si mple pendulum, we arrive at the integral
I 2
and we therefore obtain
=
2 J� J I .J g -1
du
__
I
- u2
The period of oscillation T, therefore, is indeed independent of t h e amplitude 00. A simple way of actually constraining a particle by a s tr i ng to move on a cycloid will be described on p. 428.
Sec. 4.8
*4.8
Motion in a Gravitational Field
4!3
Mo.tion in a Gravitational Field
As an example of unconstrained motion we consider a particle moving i n the gravitational field of an attracting mass. a. Newton's Universal Law of Gravitation
Kepler's description of the motion of the planets, which was based on the precise observations of Tycho Brahe, led Newton to formulate his general law for the gravitational attraction between any two particles. Let P0 = (.r0, Yol and P = (x, y) be two particles of masses m 0 and m, respectively . Let r = .J ( .r - :r0) 2 + (y - y 0) 2 be the distance between the particles. Then P0 exerts on P a force F which has the direction of � PP0 and the magn itude I F I = ym0m/r2, where y is the " unirersal grm•itational constan t . � ince F can then only differ by a posi tive factor from the vector PP0, which itself has magn itude r, we must have F
=
y m0m
--" r
p po =
�
( y m 0 m ( x0 - x) ym0 m ( y0 - y) ) r
a
•
ra
.
·
This law of attraction refers to particles, that is, to bodies that can be considered to be concentrated i n points, neglecting the actual extent of the bodies (Fig. 4.47). The validity of such an assumption is plausi ble enough for celestial bodies whose mutual distances are tremendous when compared with their diameters. Newton vastly increased the range of application of this law by showing that the same law of attraction also describes the attraction of a body of mass m 0 of considerable extent on a particle of mass m, provided that the body is a sphere of constant density, or, more generally, provided that the body is made up of concentric spherical shells of constant density ; in that case the attrac tion of the body on a particle P located outside the body is the same as i f the total mass m0 of the body were located at its center P0 ( Fig. 4.47). The earth can with fair accuracy be thought of as made up of concentric shells of constant density, so that the attraction of the earth on a particle of mass m on its surface is directed toward the center P0 of the earth (that is, vertically downward for an observer) and has magnitude ym0m/ R 2 , where R is the radius of the earth and m0 its mass. We can identify then ym0 mj R 2 with mg, where g is the gravitational acceleration (see p. 398). I n other words, we have g = ym0/ R 2 • From Newton's fundamental law we find for a particle P of mass m moving under the influence of the attraction of a mass m0 located at P0
414
Applications in Physics and Geometry
Ch. 4
the equations of motion
.. ymo(Yo - y) . y= ,a We now make the further simplifying assumption that m0 is so much larger than m that the effects of the attraction of P on P0 can be neglected m
I
F
= -;:-,--
1 I I I I I I I I
'Y"'o"'
6 mo (a) m
mo (b) Figure 4.47 (a) Newtonian attraction of two particles. tion of the earth.
(b)
G ravitational attrac
and P0 can be considered at rest. This would, for example, be the situation for a pair of bodies like the sun and a planet or the earth and a body on its surface. Taking the origin of coord inates at P0 we then have for P = (x, y) the equations of motion (5 1 )
with
r =
.Jx2 + y2•
Sec.
4.8
Motion in a GraL"ilaliona/ Fit'ld
415
b . Circldar Motion about the Center of Attraction
We shall n ot attempt to fi n d the most general solution of these d i fferential equations (which, as i s well known, would correspond to m otion along a path of the form of a conic section, with one focus at the attracting center). Instead, we shall just consider the simplest types of motion consistent with these eq uations, namely, uniform circular motions abo ut the origin and motions along a rad i us fro m the origin. For u niform circular motion of P along a ci rcle of rad ius a about the origin we have r = a and x
= a cos uJI,
y = a
sin wt,
w here uJ is a constant. The period T of the motion , that is, the time after wh ich P retu rns to the same posi tion, i s T = 21rjw. We fi n d for the velocity components i = - ao)
sin o>t,
so that t he s peed of P i n i ts orbit is .. /-.2-l' = 'o, X + .lj" = (52)
y = aw cos wt
QW
27Ta T
= -
.
The acceleration of P has the components Clearly, the equations of motion (5 1 ) are then sati sfied
if
or
( 53 ) This is just Kepler's third law for the special case of circular motion, according to wh ich the cuhes of the distances of the planets ./fom the sun are proportional to the squares of their periods. We can give some s i m ple i l l ustrations of Kepler' s law for the case where the attracting bo d y is the earth with i ts mass m0 and radius R. Observing that here ym0 = gR2 we have aa =
gR 2
47T2
T2.
For a satellite circling the earth at tree·top level (neglecting, of course, air resistance) we have a = R 3963 miles. We find then from o u r ,.._,
Ch. 4
4 1 6 Applications in Physics and Geometry
formula for the period of the satellite the value T=
21r
and for its velocity in its orbit
( 5 4)
v =
-
21rR T
=
�
1y Rg ,..._,
,..._,
1 . 4 hours
27,000 feet per second .
,
We can compare the value of T for the satellite circli n g the earth with the period of 27.32 days of the moon, that is the time after which the moo n returns to the same positio n among the stars ("sidereal mo n th ) . By Kepler's law the ratio of the distance a of the moon to the radius R of the earth should be given by the i-power of the ratio of the periods. This leads for the d istance of the moon from the center of the earth to the value "
a
=
(
27 32 24 � 0 R · x 1 .4
)
,.._,
60R
,..._,
240,000 miles,
which agree s well with the actual average value of the d istance. c. Radial Motion-Escape Velocity
The second type of motion we shall consider i s that of a p a rt i cle moving from the center of attraction along a ray, say the x-axis. Here y = 0, x = r, so that the equations of motion red uce to ..
ymo
X= -X2 .
Following our general proced ure for equations of the type s u lti p l y both sides of this equation w i th i and obtain
we m
. ..
XX =
or
Thus the expression
i - ym0 -
X2
=
f(s),
Sec. 4.8
Motion in a Gravitational Field 4 1 7
has a constant value h d uring the motion. (Later on we shall recogn ize this fact as an instance of the law of conservation of energy.) If we introduce x instead of t as independent variable, we have then ±
dt 1 = = dx i which by integration leads to ±
t = t0
ix
x0
1
� h =+=( 2= }'1=11:::; x) ' .J;::2:;:: 0/ =
d� . .j2h + (2ym0/�)
We shall not bother to carry out the integration which can be performed easily with the help of the methods developed in Chapter 3. For a particle released at the ti me t0 = 0 at the distance :r0 with i nitial velocity zero we have h = - ym0fx0• The time required for such a particle to fall into tl>� attracting particle (x = 0) is then 1
=
7T d� rxo = J o .j2y m 0( 1 /� - 1 /x0) 2
J Xo3
2ymo .
By Kepler's law this is .J h times the time i t would take the particle to circle the center of attraction at the distance x0 [see Eq . (53), p. 4 1 5] . The relation
� i} 2
-
ym X
o=h
has an interesting consequence when we investigate the circumstances under which a particle can escape to i nfinity. Since !x2 ;;::: 0 we find for x -+ oo that the constant h must be nonnegative, and hence that �i2 - ym0fx ;;::: 0 during the whole motion. I n particular, a particle starting at the distance x = a with velocity v can escape to infinity only if !v 2 - ym0/a ;;::: 0. The lowest possible value of the velocity v which will permit a particle to escape to infinity is then v = .J2ym0/a. This is the escape velocity t' , . For a particle starting at the surface of the earth and escaping to infinity, that is, escaping its gravitational pull, we h ave a = R, ym0 = gR2 so that v.
=
.J2gR
,
,...._,
37,000 feet per second.
Hence [cf. (54), p. 4 1 6] the escape velocity is just .J2 times the velocity needed to maintain a satellite in a circular orbit near the earth. A
418
Ch. 4
Applications in Physics and Geometry
meteor falling from infinity onto the earth also would have velocity v. on impact, if we neglect air resistance and motion of the earth in its orbit. 4.9 Work and Energy a.
Work Done by Forces during
a
Motion
The concept of work throws new light on the considerations of the last section and on many other questions of mechanics and physics . Let us again think of the particle as moving on a curve under the influence of a force acting along the curve, and let us suppose that its position is specified by the length of arc measured from any fixed i nitial point. The force acting in the direction of motion itself will then, as a rule, be a function of s. This function will have positive values where the direction of the force is the same as the direction of i ncreasing values of s and negative values where the direction of the force is opposite to that of increasing values of s. If the magnitude of the force is constant along the path, we mean by the work done by the force the product of the force by the distance (s1 s0} traversed, where s1 denotes the final point and s0 the i nitial point of the motion. If the force is not constant, we define the work by means of a limiting process. We subdivide the interval from s0 to s 1 into n equal or unequal subintervals and notice that if the subintervals are small, the force in each one is nearly constant ; if a, is a point chosen arbitrarily i n the vth subinterval, then throughout this subinterval the force will be approximately f(aJ. If the force throughout the vth sub i nterval were exactly f(a,), the work done by our force would be exactly -
n
I f (a,)
v=l
!:!..s"'
where !:!..s , as usual denotes the length of the vth subinterval. If we now pass to the limit, letting n increase beyond all bounds while the length of the longest subinterval tends to zero, then by the definition of an integral our sum will tend to W=
rf(s) ds,
which we naturally call the work done by the force. If the direction of the force and that of the motion are the same, the work done by the force is positive ; we then say that the force does work. On the other hand, if the directi Jn of the force and that of the
Work and Ene��y 4 1 9
Sec. 4.9
motion are opposed , the work done by the force is nega t i ve ; we then say that work is done against the force . 1 I f we regard t h e coord i nate of pos i t i o n s as a fu nction of the t i me t, so that the force f(s) = p i s also a fu nction of t, then in a plane with rectangular coord i nates s and p we can plot the point w i t h coordinates s = s(t), p = p(t) as a function of the t i me . This po i n t w i l l describe a curve, which may be cal led the work diagram of the motion. I f we are dealing with a periodic motion, as i n any mac h i ne, then after a certain tim e T (one period) the moving point (s(t), p(t)) must return to the same poi n t ; that is, the work d i agram will be a closed curve. In this case the curve may consist s i mply of one and the same a rc, t raversed first forward and then backward ; this happens, for i nstance, in elastic oscil lations. H owever, i t i s also poss i b l e for the curve to be a m ore general closed c u rve, encl osing a n area ; t h i s i s the case, fo r example, with machines i n w h ich the pressure on a piston i s not the same d u ring the forward stroke as d u ring the backward stro k e . The work done in one cycle, that i s , i n t i me T, will t h e n be given s i m ply by the negative of the area of the work d iagram or, i n other words, by the i n tegral
( '"' T
'·
• 'o
p( l ) dt
ds
dt,
where t h e i n terval of t i m e fro m 10 to 10 + T represents exactly one period of the m o t i o n . If the boundary of the area is positively traversed, the w ork done i s negat ive, i f negatively traversed , the work done is pos i t i ve. I f the c u r ve consists of several loops, some traversed positi vely and some traversed negatively, the work done i s given by the sum of the a reas of loops, each with its sign changed . These considerations are i l l ustrated i n practice by the indicator diagram of an old-fashi oned steam engin e . By a suitably designed mechanical device a pencil is made to move over a s heet of paper ; the horizontal motion of the pen c i l relative to the paper i s proportional to the d i stance s of the piston from its extreme pos i t i o n , whereas the vertical m otion i s proporti o n a l to the steam press ure, and hence proportional to the total force p of the steam on the piston. The p i ston therefore describes the work d i agram for the engine on a known scale. The area of t h i s d i agram i s meas ured (usually by m eans of a pla n i m eter), a n d the work done by the steam on the piston is thus found.
Note that here we must carefully characterize the force of which we are speaking. For example, in lift ing a weight the work done by the force of gravity is negative : Work is done against gravity. But from the point of view of the person doing the lift ing the work done is positive, for the person must exert a force opposed to gravity. 1
420
Ch. 4
Applications in Physics and Geometry
on
Here we also see that our convention for the sign of an area, as discussed p. 365 is definitely of practical interest. For it sometimes happens when an engine i s runn ing light, that the highly expanded steam at the end of the stroke has a pressure lower than that required to expel it on the return stroke ; on the diagram this is shown by a positively traversed loop ; the engine i t se l f is drawing energy from the flywheel instead of furni sh ing energy. b. Work and Kinetic Energy.
The law of m oti o n
Conservation of Energy
ms = f
leads to a fundamental relation between the changes i n velocity during the motion of a particle along a curve and the work done by the force f in the d irection of motion. We apply the same device used already several times in the preceding examples and multiply both sides of t he e q uati on of motion by s : ms!i
= f(s)s.
Now ms!i = (d/dt) ! ms2 = (d/dtHmv2, where v(t) s is the v elocity of the particle. I ntegrating both sides of the equation with respect to t be tween the l i mits 10 and 11, we find =
I
I
- m v2(11) - - m v2( 10) =
2
2
.
=
J.''f(s) -ds dt lo
dt
J,,f(s) ds =
W.
The q uan t i ty �mv2 is called the kinetic energy K of the particle . Hence : The change in kinetic energy of a particle during the motion equals the work done by the force acting on the particle in the direction of motion. T h e q uantity / represented the force acting in the direction of motion or the tangential component of force. For a force F = (p, a) the force i n the direction of motion is '•
dR dx d f = F · - = p - + a --'-v . ds ds ds If p and a are known functions of x and y and if the particle i s known to move along a curve x = x(s), y = y(s), t he n .f al s o becomes a known
function of s. Hence i n order to compute the work ( 55 )
W=
J.,,f(s) ds ..
Sec. 4.9
Work and Energy
421
as the particle moves from one pos1t10n (x0, y0) to another (x1 , y1), we have to know in general the p at h along which the particle moves. In an important class of cases the work W depends only on initial and final position and can be expressed i n the form
(56) with a suitable function V(x, y) the potential energy. The formula expressing that the change i n kinetic e n e rgy equals the work done by the force then can also be written in the form
Thus the q uantity K + V, the sum of k i netic and potential mechanical energy, that is, the total e nergy, does not c h ange d u ri n g the motion . T h i s is an instance of the general physical law of consen•ation of A potential energy function V can easily be con structed i n some of the motions d i scussed earlier. Thus for a particle s u bject to gravity we have F = (0, - mg) an d f = - mg(dyjds). The work done by the force of grav i ty as the particle moves fro m a position (:r0, y0) to a position (:rl , 1/ 1) i s then
energy.
W=
J , d·dsij f "' ..
- mg -=- d.�
=
- mg dy = m gy0 - mg!h ·
IIO
qO
We see that W is proportional to the c hange in altit ude between i n itial a n d end position. For the potential energy function V we can ch oose V = mgy (or m o re generall y V = mgy + c, where c is any constant). The l aw of conservati o n of energy then states that the q u a n t i ty
is constant during the motion . We had noticed t h i s fact already i n i nv estig a t i ng the moti o n o f a particle s liding down a curve (p. 408).
c. The Mutual A ttraction of Two Masses Another e xa m ple of a force with which we ca n associate a poten tia l energy function V is furnished by the g ravitat ion al attraction F exerted by a p article P0 = (x0, y0) of mass m0 on a particle P = (x , y) of mass m. Here - Xo) F=
[-f.l(X
r
3
'
-f.l(Y Yo)] r
3
'
422 Applications in Physics and Geometry
Ch. 4
where f! = ym0m and r = J (x - x0)2 + (y - y0)2 • (According to Coulomb's law the same type of formula gives the interaction of two electric charges.) The force i n the direction of mo t io n i s t h en
f![
]
f! dr
d f!
f = - - (x - Xo) - + (y - Yo) - = - -:2 - = - dx
since (x -
Xo)
dx ds
dy
ds
r3
+ (y - Yo)
ds
dy ds
= !2 .:!_
ds
[( x -
r ds
ds r
Xo)z + (y - Yo)z]
1 dr2 dr = -- = r- . ds 2 ds
The work done by the force of attraction when t h e pa r ti cle P moves fr om a position (x1 , y1) t o t h e p o sit i on (x2 , y 2) is then
J(
W = '' .:!_ t. ,,
ds r
)
ds
= .!!:. - .!!:. = r2
x
r1
V(x1 , y1) - V(x 2 , y 2 ) ,
where V(x, y) = - 11/r = {! / J( x0) 2 + (y - y0) 2 is the potential energy. I f we move the particle from the posi tion (x1 , y 1 ) t o i n fi n i ty (corre spo nd i ng to r2 = co), the work done by the force of attraction i s {! / T h e work done b y an opp os i n g force that moves the p artic le to infinity has t he same n u merical value but the opposite s i gn Hence f!/r1 = - V(x1, y 1 ) is the work that has to be done against the force of a tt r ac tio n in order to move the pa r ti cle to i nfin i ty from the position (x" y1). Th i s i mportant expression is called t h e mutual potential of t h e two particles. Therefore here the potential is defined as the work re qu i red to sepa rate the two attracting masses completely, for example, the work requ ired in order to tear an electron completely away from its atom (ionization potential). I f the attracting mass P0 i s considered as fixed, the law of conser vation of en e rgy i mpl i es that the attracted particle P moves in such a way that the exp r e ssio n -
-
r1
-
.
•
l v' - ym o = 2 r
h
(the total energy per unit mass m) has a constant value during the motion. We had derived this fact already for the special case of purely radial motion ; we see now t h a t it holds for any type of motion under
Work and Energy
Sec. 4.9
423
the influence of gravitational attraction . We can conclude again that h ;;::: 0 for a particle escaping to infinity ; i ts orbit is then u nbounded (parabola or hyperbola) instead of bounded (ellipse). The escape velocity v
c
=
J2rmo r
,
which corresponds to h = 0, is the least velocity which enables the particle to escape to i nfinity from a given distance r. It does not depend on the direction in which the particle is released but only on the distance r from the attracting center. d. The Stretching of a Spring
As a third example we consider the work done i n stretching a spring. Under the assumptions on the elastic properties of the spring made on p. 404, t h e force acting is.f = - k.r, w here k i s constant. The work that must be done against this force in order to stretch the spring from the unstretched position x = 0 to the final position x = :r1 is therefore given by the i ntegral I kx dx = - kx/.
i
xl
2
0
*e. The Charging of a Condenser
T h e concept of work in other branches of physics can be treated in a s i m i l a r way. For e x a m p l e , let us consider the charging of a condenser. If we denote the quantity of electricity in the condenser by Q, its capacity by C, and the difference of potential (voltage) across the condenser by V, then we know from physics that Q = C V. Moreover, the work done in moving a charge Q through a difference of potential V is equal to Q V. Since in the charging of the condenser the difference of potential V is n o t constant but i ncreases with Q, we perform a passage to the l i mit exactly analogous to that on p. 4 1 8, and as the expression for the work done i n charging the condenser we obtain
iCJJo V dQ = -C I
Q iolQ dQ = - 1 = - Q, V1 , o
I
2 C
2
1
2
where Q1 is the total quantity of electricity passed into the condenser and V1 is the difference of potential across the condenser at the end of charging process.
424
Applications in Physics and Geometry
Ch. 4
Appendix *A.l
Properties of the Evolute
On p. 359 we defined the evolute E of a curve C as the locus of the centers of curvature of C. Tf C is represented by : x = x(s), y = y(s) , using the arc length s as parameter, then the center of curvature ($, ry) of the point C with parameter s is given by [cf. ( 1 7a), p. 359] TJ = y + pi,
$ = X - py,
(58)
with K
I . .. . = - = xy - yx.
p
.
.
The quantities K and / p / are, respectively, curvature and radius of curvature of C. We can deduce some interesting geometrical properties of the evolute from these formulas. Differentiating the relation i2 + y2 = I leads to, ix + yfj = 0. Since also if} - yx = 1 / p, we have I .
.
1 .. y = - x.
p
p
X = - - y,
(59)
.
Differentiating the formulas (58) with respect to s
�=
x - pfJ -
py =
-
iJ
py,
=
y + pi + pi
=
pi,
and therefore
Since the direction cosines of the normal to the curve are given by x, the normal to the curve C is tangent to the evo!ute E at the center of curvature; or the tangent to the evolute is the normal of the given curve ; or the evolute is the "envelope" of the normals (cf. Fig. A. l ) . If further w e denote the length o f arc o f the evolute, measured from an arbitrary fixed point, by a, we have, using s as parameter, - y and
a• = e:r
=
e
+
ry•.
Since i2 + y 2 = 1 , we obtain from our formulas (59),
a•
=
p•.
426 Applications in Physics and Geometry
Ch.
4
c
Figure A.2
String construction of the i nvolute C of a curve
E:
p1
=
p0
+
-
rr0 •
arc of the evolute E and stretched so that a part of it extends tangentially away from the curve to it ; if in addition the end point Q of this thread lies initially on the origi nal curve C, then as we unwind the thread Q will describe the curve C. This accounts for the name evolute (eL'olvere, to unwind). The curve C is called an involute of the evolute E. On the other hand, we may start with an arbitrary curve E and construct its i nvolute C by this unwinding process. Then conversely E is seen to be the evol ute of C (Fig. A . 2). For the proof we consider the curve E , which i s now the given curve, as given in the form $ = $(a), 'l = r;( a), where the current rectangular coordinates are denoted by $ and r; and the parameter a is length of arc on E. The winding is done as indicated in Fig. A.3 ; when the
E
Figure A.3
Sec. A . l
Properties of the El'olute
421
thread is co mpletely wound on to the evol ute £, its end Q coi ncides with the point A of E corresponding to some arc-length a . If the thread is now unwound until it i s tangent to the evol ute at the poi nt P, corre sponding to the length of arc a > a , the length of the segment PQ will be ( a - a) and i ts d i rection cosines will be - i and - iJ, where the dot now denotes d i fferentiation with respect to a . Thus for the coord inates x, y of the point Q we obtain the expressions
(60)
�
x =
-
(a
-
a)t,
y
=
r1
-
(a
- a)tj,
w h i c h give t h e equations for the i n volute descri bed b y t h e poi nt Q i n terms o f the parameter a . B y differentiation with respect t o a w e obtain x
(6 1 ) Since
y ��
+ i1ii
=
= =
�
ij
t + (a - a)� rj + (a - a) ii
-
= =
(a - a)�. (a - a)ii.
0, we at once find that
which shows that the l i ne PQ is normal to the i nvolute C. We can therefore state that the normals to the curve C are tangPnt to the curve £. Si nce the tangen t to E has d i rection cosines t. 'i we fi nd for the direction cosines of the tangent of C the expressions
-/
(62)
i
i:2 + y2
=
.
1) •
' = -==y=
Jx2 + !?
=
-{
Differentiating the relation {i + ini = 0 with respect to a and sub stituting for i, it . �. ii, th eir ex pressions from the previous equations (6 1 ) , (62) shows that
0
= "X �.= •
+ 1JY .. . +
.. +
�c X
.. 1J. Y
x
·2 +
y· 2
= --a -
a
+
. .. .. . + xy - xy
J
x2 + Y2 .
Hence the radi us of curvature of the curve C corresponding to the point Q = (x, y) turns out to be (see formula ( 1 5) on p. 355) p
1 K
= - =
(x2 + 1i"l% Xfj - yi
= a -
a.
This is also the distance of the point Q from P = (�. 1]). Because P also lies on the normal to C at Q, we have in P the center of curvature of C corresponding to the point Q. Thus every curve E is the evolute
of all its involutes.
428 Applications in Physics and Geometry
Ch. 4
Examples. We consider the evolute of the cycloid y = - 1 - cos t.
x = TT + t + sin t,
By Eq. ( 1 7), p. 359, the center of curvature ( ; 'Y)) for a curve referred to an arbitrary parameter t is ,
x2 + ?l ; = X - y. .. --. -.. , xy - yx
'YJ
. -
. x2 + 1/ = y + X -.-.
.. · .. --
xy - yx
.
A short computation yields then for the evolute of the cycloid
If we put t = ;+
;
= 1r
T - TT,
1r = 1r
+t
then
+
-r
-
sin
+ sin
1,
T,
'YJ =
'YJ
-
I + cos I.
2=
-
I
-
cos
T:
these equations show that the evolute i s itself a cycloid which i s similar to the original curve, and can be obtained from it by translation as indicated in Fig. A.4. y
Figure A.4
The cycloidal pendulum.
This gives us a simple method of constructing a cyc!oidal pendulum (see p. 4 1 2). ! f a mass P is attached by a th read of length 4 to one of the cusps of the evolute, then under tension the thread will partly coincide with the evolute and lie along a tangent to the evolute the rest of the way. The mass P is then forced to lie on the invol ute, that is, on the original cycloid. Under gravity P must describe an isochronous motion over some portion of the cycloid with a period independent of the position at which P begins the motion. (The parameter t to which the cycloid is referred does not correspond to the time in the isochronous motion.)
430
Ch. 4
Applications in Physics and Geometry
This curve is called an astroid. Its graph is given in Fig. A.6. By means of the parametric equations we may readily convince ourselves that the centers of curvature corresponding to the vertices of the ellipse are actually the cusps of the astroid. y
Figure A.6
* A.2
Evolute of the ellipse .
Areas Bounded by Closed Curves.
y y(t),
Indices
In Section 4 . 2 the oriented area bounded by a closed curve x = x(t), oc � t � fi, which nowhere intersects i tself (a so-called simple closed curve), was represented by the integral =
A = -
J:y(t)x(t) dt;
the value obtained is positive or negative depending on whether the sense in which the boundary is described is counterclockwise or clock wise. This formula remains meaningful as a definition of A if we allow self-intersections of curves. It remains to see how A is related to areas
Areas Bounded by Closed Curves. Indices 431
Sec. A . 2
i n such cases. Suppose that the curve C, given by the equation x = x(t), y = y(t), intersects itself in a finite number of points, thus dividing the plane i nto a finite n umber of portions Rto R 2 , Suppose further that the deri vatives are continuous and that :i2 + y 2 ¢. 0, except per haps for a finite nu mber of j ump-discontin u i ties (which may or may not correspond to corners) . Finally, it i s assumed that the curve has •
•
•
•
y
X
0
Figure A.7
I n d ices I' ' of regions
R, fo rmed by orie nted cl osed c u rve. Figure A . B .
a fi n i te n umber of lines of support x = constant, that is, vertical li nes that are either tangent to the curve or pass t h rough a poi n t of self i n tersection of the curve. To each region R , we then assign an integer, the index p,,, defined in the following way : We choose an arbitrary poi n t Q in R;. not lying on any line of support, and erect the half-line extending from Q u pward in the direction of the positive y-ax is. We count the number of ti mes the curve C for increasing t crosses the half-line from right to left, and subtract the n umber of times the curve C crosses from left to right ; the difference is the index p, , . For example, the i nterior of the curve ill ustrated i n Fig. 4. 1 7, p. 343 , has the index p, = + I ; and in Fig. A. 7 the regions R 1 , , R5, R6 have the i ndexes th = - I , p, 2 = - 2, •
•
•
432 Applications in Physics and Geometry
Ch. 4
I and p.6 = 0. This number p.; actually depends on the region R; only and not on the particular point Q chosen i n R;, as we readily see in the following manner. We choose any other point Q' in R;, not on a line of support, and join Q to Q' by a broken line lying entirely in the region R ; (Fig. A.8). As we proceed along this broken line from Q to Q' the number of right-to-left crossings minus the n umber of left-to-right crossings is constant ; for between lines of support the n umber of crossings of either type is unchanged, whereas on crossing a line of support the number of crossings of both types p.3 = - I , p.4 = 0, p.5 =
y
Q -1 - I
I I I I .• - t -
-
' Q Q --� 0�--------------L---�-L--�--------� x Figure A.S
either stay the same or both n umbers i ncrease by one or both decrease by one ; in every case, the difference is unaltered . Here a l i ne of support that meets the curve at several different points, say A, B, . . . , H, is considered as several different lines of support, FA , FB, . . . , FH, where F is a point vertically below all the points A , B, . , H. Our argument then appl ies to each of these lines. Hence the number /1 ; has the same value whether we use Q or Q' in determining it. In particular, if our curve does not intersect itself, the interior of the curve consists of a single region R whose index is + I or - I depending on whether the sense i n which the boundary is described is counter clockwise or clockwise. To see this we draw any vertical l ine (not a line of support) intersecting the curve ; on this line we find the h ighest point of intersection P with the curve, and in R we choose a point Q .
.
Sec. A .2
A reas Bounded by Closed Cun,es. lndius
433
and so near it that no point of intersection lies between P and Then above Q there lies one crossing of the curve, which if the curve is traversed in the counterclockwise sense must be a right-to-left crossing, so that + 1 ; otherwise - 1 . As we have just seen, this same value of holds for every other point of R. For such a curve, and, in fact, for all closed curves, one of the regions, the "out side" of the curve, extends unboundedly in all directions; we see immediately that this region has index 0, and ignore it in what follows. Then the relation between the integral and the areas of the regions R is given by the following theorem: THEOREM. The mlue of the integral -[3yi dt is equal to the sum below P Q.
Jt =
Jt =
Jt
A
1
of the absolute areas of the regions R1, each area R 1 being counted Jt ; times; in symbols
-r yi Lfti ! area R;l· dt =
PROOF. The proof is simple. We assume, as we are entitled to do, that the whole of the curve lies above the x-axis. (Adding a constant to y does not change the value of the integral for a closed curve. The lines of support cut R1 into a finite number of portions; let r be) one of these portions. Then on taking the integral -J yi dt - J y dx for each single-valued branch of the function y y(x) and interpreting it as area between the curve and the x-axis, we find that the absolute area of r is counted + I times for each right-to-left branch above r and - I times for each left-to-right branch above r ; i n all, times. The same is true for every other portion of R1; hence R1 is counted times. Thus the integral round the complete curve has the value ! area R,l, as stated (cf. Fig. This formula agrees with what we have found for simple closed curves, as we recognize from the discussion of the values of for such curves. The definition given for the index has the disadvantage of being stated in terms of a particular coordinate system. As a matter of fact, however, it can be shown that the value of p,1 assigned to a region R1 is independent of the coordin ate system and depends solely on the curve. This can be readily seen by identifying with the total number v of times a point on the curve for t increasing from to f3 runs about any fixed point Q 1 of R1 in the counterclockwise sense, that is with the number of times C winds around Q, . We shall prove the identity of and v1• Let C be given parametrically by x(t), y y(t) where � t � {3. Let Q (�, 'f/) be a point whic h does not lie on a line of support of C. A
=
=
p1
1 p,
p. 1
A.7).
p
p1
Jt 1
1
f1;
x =
=
=
ex
ex
Applications in Physics and Geometry
434
Ch. 4
We take Q as origin of a system of polar coordinates r
2 + ( y - ry ) 2, = .Jcx - � )
r,
() i n which
x- � - TJ . cos e = -- , sin e = Y-r
r
The polar angle e is determined only within whole multiples of 21r ; however, e i s determined uniquely a s a function o f t by its value eo for t = oc if we require e = O( t ) to vary continuously with t al ong the curve C. At t = {3 the angle e will then have a value 0({3) = 00 + 2v1T, where v is an i n tege r . The number v
= - [0({3) - O(oc)] = 1 21T
1 21T
fp •
1 d() - dt = dt 21T
f
G
dO
represents the n u mbe r of times that the oriented curve C winds around Q. The curve C c r o sses the vertical half-line through Q for those values of t for which the expression (1 /27T)[(J(t) - 7T/2] has an i ntegral value n. Consider for a fixed n the t-v a l u es in the parameter interval for which ( 1 /27T)((J - 1rj2) = n . Let a, and T" be the number of such t-values for which dOfdt > 0, re s pectively dOfdt < 0. Obviously, the index a t the point Q is f.1
= I an - L 7n = I C an - T n). n n u
On the other hand, a n - Tn can only have one of the values 1 , 0, - I , for the graph of (J(t) in the e, t-plane must cross the line e = 1rj2 + 2n1T alternately from above or below. Actually, we have a n - T" = sign [0({3) - O(oc)] if 1rj2 + 2n1T lies between O(oc) and 0({3) and an - T, = 0 otherwise. Consequently, f.1 equals the number of values of t he form 1rj2 + 2n1T with an integer n that lie between O(oc) and 0({3) taken with the sign of 0({3) - O(oc) ; that is, f.1 equals the number v. Since e = arc tan [(y - ry)f(x - �)], we have de dt
y( x - � ) - x(y - rJ) (x - � ) 2 + (y - ry}" ·
This yields for the ind e x f.1 of the or i ented closed curve C with r espect to the point (�, ry) the integral representation
1
Jl = 21T
J
P
•
y(x - �) - x( y - TJ) dt 2 (x - �) + ( y - ry)2
Problems
435
which can be simply written (see p. 367) without referring to the param eter t explicitly : ,U _!_ { . (x - �) dy - (y - 'f))0d.r . = 27T J (' (X - �) 2 + (,1/ - 1} ) "
The remarkable feature of these results i s that the integer ,u or v which descri bes a topolog ical relation between the poi nt Q and the curve C can be determi ned analytically. fro m the parameter representation of C, by evaluati ng an i n tegral.
PROBLEMS
SECTION 4. 1 c, page 328
1 . Sketch the hypocycloid for a = 4c (the astroid ) and find its nonpara metric equatio n . 2. Prove t h a t if c/a i s rational t h e general hypocycloid i s closed after the moving c i rcle has rotated an integral n u m ber of times, whereas i f cfa is ·irrational, the curve has i nfi nitely many points where it meets the circum ference of the fixed circle and will not close.
3. Derive the parametric representation ,. = a t - b sin t,
y
= a - b cos t
for ordinary trochoid, t hat is, for t he path of a poin t P attached to a disc of radius a rol l i ng along a line, P having the distance b from the cen ter of the disc (see Fig. 4.7). 4. Find the parametric equations for the cu rve .r3 + y 3 = 3aJ·y (the folium of Descartes), choosing as parameter t the tangent of the angle between the .r -axis and the ray from the origin to the poi n t (.r, y ) .
SECTION 4. l e, page 343 1 . The angle rJ. between two curves at a point of i ntersection is defined to be the angle between t heir tangents at the poin t . Find a formula for cos "- in terms of the parametric representations of the curves. 2. Let .r = f(t) and !I = g(t). Derive formulas for d 2 yjdx 2 and d3yjdx3 in terms of derivatives with respect to the parameter t.
r
3. Fi n d the formula for the angle "- between two curves r = [( 0) and = g( 0) i n polar coordinates.
4. Find the equations of t he curves which everywhere i ntersect the straight l i nes through the origin at the same angle "- · = [( f) and y = g(t) are continuous on the closed i n terval
[a, b] and differentiable on the open i n terval (a, b) with x' 2 + y' 2 > 0, then 5. Prove : if
.r
436 Problems
there is at least one point on the open arc
X = /(1),
y =
g{ l ),
(a < I < b),
where the tangent is parallel to the chord joining the end points. 6. Let P be the point of a ci rcle which traces out a cycloid as the circle rolls on a given line. Let Q be the point of contact of the circle with the line. Prove that at any instant, the normal to the cycloid at P passes through Q. What similar property holds for the tangent at P? 7. Prove that the length of the segment of the tangent to the astroid,
X = 4c COS3 6,
y
= 4c sin3 e.
cut off by the coordinate axes is constant . •s. Show that the two families of ellipses and hyperbolas, (0 < a < b) a• a•
:r2
_
x2 _
;. •
+ b'
-r'
+ '
b
y2
y2
_
_
;.•
=
-r' =
I,
for 0 < ). < a ,
I , for a <
-r
< b,
are confocal (that is, have the same foc i i ) and intersect at right angles. 9. (a) Show for the ellipse that the angle between the two rays from the foci to a poi nt on the curve is bisected by the normal at the point. (b) Show for the hyperbola that the angle is bisected by the tangent. SECTION 4. 1 f, page 348 1 . Prove that the curve defi ned by
{.r'
!I =
sin � ,
0,
{
.<'
.r =
0
has fi nite length, but that the conti nuous curve defined by !/ =
. I .r sm - , 0,
·''
0 <
.r
.r = 0
:5: I
is not rectifiable. 2. Prove that if the function ( is defined and monotone on the closed interval [a, b], then the arc defi n ed by is rectifiable.
y
= [(x),
(a :5:
"'
:5: b),
SECTION 4 . l g, page 352 1 . An elliptic integral of the second kind has the form
i¢
VI
-
k' sin2 0 dO.
(a) Show t hat the arc length of the ellipse
.r =
a cos 0,
Problems !I =
be expressed in terms of a n elliptic i ntegral of the second kind.
437
b s i n () can
(b) Do the same for the trochoid .r =
at
-
b sin t,
!I
=a
-
b cos t.
*(c) Show that the arc length of the hyperbola can be expressed i n terms of e l l i ptic i n tegrals of t he first and second k inds . SECTION 4. l h, page 354
1. Let P be a point of the rol l i n g circle which generates a cycloid and let Q be the l owest point of the circle at any given instant. Show that Q
bisects the segment joining P to t he center of t he osculating circle of t he cycloid at P.
2. Find the center of curvature for !I = .•·2 when .r = 0. Determine the point of intersection of the normal l i nes to the curve when .r = 0 and when .r = • · Calculate the distance of the i ntersection from the center of curvature. Suggest an alternative defi nition for the center of curvature. Prove that this definition is equ i valent to the defi nition given i n the text.
3. Consider the quest ion of whether the oscu lating circle crosses the curve at the point of contact.
*4. Prove that the circle of curvature at a poi n t P of the curve C is the l i m i t of the circles through three points P. P1 , P2 as P1 and P2 tend to P.
5. Let = /( 0) be the eq uation of a curve i n polar coordi nates . Prove r that the curvature i s given by the formula
where
r
' = df do '
r"
=
6. The curve for which the length o f the tangent i ntercepted between the point of contact and the y-axis is always equal to I is called the tractrix. Find its equation. Show that the radi u s of curvature at each point of the curve i s i n versely proportional to the length of t he normal i ntercepted between the poi nt on the curve and the y-axis. Calcu late the length of arc of the tractrix and fi n d the parametric equations i n terms of the length of arc. 7. Let .r = .r(t), y = y(t) be a closed curve . A constant length p is measured off along the n ormal to the curve. The extremity of this segment descri bes a cu rve which is cal led a parallel curve to the original cu rve. Find t he area, the length of arc, and the radius of curvature of the parallel curve. 8. Show that the only curves whose curvature is a fixed constant k are circles of radi u s 1 /k.
*9. I f the cu rvature of a curve i n the "·y-plane i s a monotonic function of the length of arc, prove that the curve i s not closed and that it has no double points.
438
Problems
SECTION 4 . 1 i , page 360
1. Show that the expression for the curvature of a curve
x = x(l), y = y(l)
is unaltered by rotation of axes and also by change of parameter given by 1 = (T), wh ere ¢'(T) > 0. SECTION 4 . 3 d , page 394
1. Prove if the acceleration is always perpendicular to velocity that the speed is consta n t .
2 . T h e velocity vector, considered as a position vector, traces out a curve known as the hodograph . Show whether or not a particle moving on a closed curve may have a st r ai gh t line as its hodograph .
3. Assuming the rolling circle moves at constant speed, fi n d the velocity and acceleration of the point P which generates the cycloid. 4. Let A be a fi xed poi n t of the plane and suppose that the acceleration vector for a moving point P is a lways directed toward A and proportional to the I /IAPI 2 • Prove that the hodograph (cf. Problem 2) is a ci rcle.
5. Let A be a fixed point on a c i rcle. Let P be a poi nt of the circle moving so that the acceleration vector points to A. Prove that the acceleration is proportional to I A P I - 5• 402
1. A particle moves in a straight l i ne s u bject to a resistance producing the retardat i on k u3, where u i s the velocity and k a c o nst a n t . Find expressions for the vel oc i ty (u) and t he t i m e ( 1 ) i n terms of s, the distance from the initial position, and v0 , the i n i t ial veloci ty.
SECTI ON
4 . 5 , page
2. A particle of unit mass moves along the .< -axis and i s acted u pon by a force f(.<) = - s i n .r. (a) Determine the motion of t he point i f at time 1 = 0 i t is at the point ,,. = 0 a n d has velocity L'o = 2 . Show t hat a s I • oc the particle approaches a l i m i t i n g position, and fi n d this l i m i t i n g posi t i o n . ( h ) I f t h e conditions a r e t he same, except th at t0 m a y h a v e a n y value, show that i f L'n > 2 the poi nt moves t o an i n fi n ite distance as 1 - ex' , and that i f r0 < 2 the point osc i l lates about the origin .
3 . Choose axes w i t h their origin a t t h e center of t h e earth, whose rad ius we shall denote by R. Accord ing t o N e w t o n ' s law of gravitation, a particle of unit mass lying o n the y-axis is attracted by the ea rt h with a force _ ,,Mjy2, where ,, is the " gravitational constant" and M is the mass of the earth . (a) Ca lculate t h e motion of the particle after i t is released at the point Yo ( > R) ; that is, if at t i me 1 = 0 i t is at the p o i n t y = !fo and has the velocity v0 = 0. (h) Find the velocity with which the particle i n (a) strikes the earth. (c) Using the result of (h), calculate the velocity of a particle fa lling to the earth from i n fi n i ty . 1
* 4 . A particle perturbed sl ight ly from rest o n t o p of a circle s l i des down ward under the force of gravity. At what point does i t fly u nconstrai ned off the ci rc l e ?
1 This i s the same a s the least veloc i t y w i t h w h i c h i n order that it s h o u l d leave the earth and never
a <'rojec t i l e ret u r n .
w o u l d have t o be fi red
Problems 439
•s. A particle of mass m moves along the elli pse r = k/( 1 - e cos 0). The force on the particle is cmfr2 directed toward the origin. Describe the motion of the particle, find its period, and show that the radius vector to the particle sweeps out equal areas in equal times.
SECTION 4A. I , page 424
I. Show that the evolute of an epicycloid (Example, p. 329) is another epicycloid similar to the first, which can be obtained from the first by rotation and contraction. 2. Show that the evolutc of a hypocycloid (Example, p. 3 3 1 ) is another hypocycloid, which can be obtained from the fi rst by rotation and expansion.
5 Taylor's Expansion
5.1
Introduction : Power Series
It was a great triumph in the early years of Calculus when Newton and others discovered that many known functions could be exp re ss ed as "polynomials of infinite order" or " power series," with coefficients formed by ele gan t t ranspa ren t laws. The geometrical series for 1 /( I - x) or 1 /( 1 + x2) 1
(I)
--
1 - x
= 1 + x + x2 + " · + x " + . .
_1 = 1 - x2 + x • - x" + 1 + x2
(Ia)
_
.
.
.
+
.
( - 1 )" x 2 " + .
.
.
valid for the open i nterval l x l < I , are prototypes (see Chapter I , p . 67). Similar expansions of the form J(x) = a0 + a1x + . . . + a , :r" + . . . =
I a "xv,. "'
\'= 0
with numerical coefficients a,, w i l l be derived i n th is chapter for many other functions. The fol lowing are striking exam ples : x2 x3 xn e• = 1 + x + - + - + · · · + - + · · · ;
3!
2!
. SJn x =
COS X
=
X
_
xs xa -+-
3!
5!
_
1 --+-+ x2
2!
:r4
4!
11 !
+ . . . +
·
·
· +
( - 1) " x2 n 1 1
(2n + 1 ) !
( - 1 ) x2" "
(2n) !
These series expansions are valid for all x . 440
+ . .
+ . . . ; ,
Sec. 5. 1
Introduction : Pmrer Series 441
Newton's General Binomial Theorem . The expansion (I
+
:r
:x ) = I +'
I!
.r
+
i s valid for lxl < 1 and any exponent
:x( :x - 1 ) .r .,· +
---
2!
·
·
·
:x .
To explain the precise meaning of such expansions, we consider the polynomial of order n formed as the sum of the fi rst n + 1 terms of the series, the nth "partia l sum," sll
=
The formula f (x) =
"
� a..,x\1.
v :-. 0
v.
L a ,x',
V = ll
l :rl < a
for
then means : For 11 --+ w the seq uence S., tends to the value of the function f(x) at each point x i n the i nterval lrl < a. The i n finite series is then said to com·erKe to f(x) i n the interval l ·r l < a. The d i fference R ,(x ) = f( x ) - S ,(.r),
the "remainder" of the series, measu res the precision with which .f(x) is approxi m ated by the polynomial S., ( x) at x. For exam ple,
( I I> )
1-
-
1 - x
= 1 + x + x2 +
·
·
·
+ x " + R ,(.r),
where the remai nder R , (x) = �:" 1 1/( 1 - x) tends to zero for l·r l < 1 as
11
increases ; t h us the i nfinite geometric series
L xv u
= 1 /( 1 - x)
results. To find si m ple manageable esti mates for R , in speci fic cases is a task of both theoretical and practical i mportance. I n this chapter we are concerned with such expansions for a wide class of functions, including all the "elementary" transcendental functions. I t i s a stri king fact that in these expansions of transcendental functions the coefficients are elegant expressions in terms of integers. The approach to these expansions will be by Tay lo r s theorem ; later in C hapter 7 we shall discuss a d i fferent approach by a d i rect study of power series. I t should be emphasized that often j ust as for the geometrical series of Eq . ( I a) , the i nfi ni te expansion is not vali d outside some i nterval '
442
Taylor's Expansion
5.2
Expansion of the Logarithm and the Inverse Tangent
x2
Ch. 5
for x-(in the case of the geometrical series, the i nterval < I ) even though the function represented by the series is well defined outside this i nterval.
a. The Logarithm
As simple examples we first derive expansions of the logarithmic and the inverse tangent functions by integration, from the geometric series 1
1- = 1 +
-
- t
t
+ 12 + . . . + rn- l + rJ t)
with r,(t) = t"/( 1 - 1). We substitute this sum for the i ntegrand i n the formula - log ( l - x)
=
ixo
dt
- - I
1
and integrate term by term, obtaining for x < I - l og ( 1 - .r)
=
x
with the remainder R ,.(.r)
x2
xa
x4
2
3
4
+-+-+-+
=
ix 0
r,.
dt
=
i' 0
1
· · ·
--
- t
1"
+ - + R " ( x) , x r1
dt.
11
Hence for any positive i nteger 11 the function - log ( I - x) is approxi mated by the polynomial of nth degree,
and the remainder R , indicates the "error" of this approximation. To appraise the accuracy of this approximation we estimate the remainder Rn. I f we at first suppose that - I � :r � 0, then in the entire i nterval of i ntegration the integrand t"/( 1 - 1) in absolute value, nowhere exceeds [ t " l = ( - 1 )" 1 " . Thus
[ R ,. [
�
I
Jo( t" d t '
I=
[x[" 1 1
11
+ 1
;
hence for every value of x in the closed interval - 1 � x � 0 i ncluding - 1 this remainder can be made as snail as we wish by choosing n
x =
Sec. 5.2
Expansion of the Lr�!{arithm and the Imwse Tangent
443
large enough (cf. p. 6 1 ). For �' > 0 the end point x = I must be omitted ; we have to restrict x to the half-open interval 0 � :r < I ; the i ntegrand d oes not change sign and its absol ute value does not exceed 1 " /( 1 - x) ; we thus obtain for 0 � x < I the estimate x "+1 I t " dt = . I R , I � -I - x o ( I - x)(11 + I )
JJ
H ence again, if x i s fixed, the remainder is arbitrarily small when n i s sufficiently l arge . O f course, t h e estimate h a s n o meaning for x = I . S umming up, :r11 x2 xa l og ( I - x) = - x - - - R" (2) 2 3 11 -
-
-
-
-
·
·
·
where the remainder R, tends to zero as 11 i ncreases, provided that x l ies in the half-open i nterval - I � x < I . I n fact, th is reasoning establishes a "uniform" esti mate for the remainder, i ndependent of x and valid for all values of x in the i nterval - I � x � I - h, where h is any n u m ber such that 0 < h � I ; n a mely, I R , I � 1 / [(11 + l )h]. The fact that the remainder R , tends to zero i n the half-open interval - I � x < I is expressed by saying that in this i nterval the logarithmic fu nction i s given by the i nfinite series'
(3) If
l o g ( I - x)
=
xa .r4 x2 -x - - - - - - 4 3 2
·
·
·
.
we insert the particular value x = - I in this series, we obta in the remarkable formula l og 2 = I - � + ! - ;} + -
(4)
·
·
·
.
This is one of the relati ons w hose d iscovery made a deep i m pression on the early pioneers of the calcu l u s . F o r t h e open i nterval - I < x < I , we h a v e o n l y to write - x i n place of x in (2) i n order to obta i n x2 x 3 x4 x" + ( - 1 )"- 1 - - R ,, ' , (2a) l og ( ! + x) = x - - + - - - + 11 4 3 2 where ·
R '(x) n
=
f.
•
o
x
·
·
t" d t = ( _ 1 )"� 1 {" t " d t Jo I + t .
I - t
·
1 We l eave it as an exercise to the reader to ascertain that for all val ues of x for which /x/ > I the remainder not only fails to approach zero, but, in fact, that / R. / increases beyond all bounds as n increases, so that for such val ues of :r the polynomial is not a good approximation of the l ogarithm and becomes worse with increasing n.
444
Taylor's Expansion
Ch. 5
(1 )
Taking n as even and subtracting (2) from (2a), we have x3 x5 + x 1 - log = ar tanh x = x + - + - + 2 1 - x 3 5 --
where the remainder R, is given by
-
1
Rn = - (R n - R n ) = 2 '
ix 1 0
+
· · ·
x n- l
--
n - 1
+ Rm _
t" dt - 12 '
--
and where ar tanh x is defined according to p. 233. Observing that 1 /( I - 12) ::;:; 1 /( 1 - x2), we find by an elementary estimate of the integral that l x"+' l I Rn l 2 ; n+ - x
::;:; --1 · 1
1
-
thus the remainder Rn tends to zero as n i ncreases, a fact again expressed by writing the expansion as an i nfinite series :
(5 )
1 x3 x5 x' 1 +x = ar tanh x = x + log + + + 2 3 5 7
1-x
-
-
--
-
-
·
·
·
,
for all values of x with l x l < I . Incidentally, this result also could be derived directly by integrating the geometric series for 1 /( 1 - x2). I t is a n advantage of this formula that a s x traverses the i nterval from - I to I , the expression ( I + x)/( I - x) ranges over all positive numbers. Thus, if the value of x is suitably chosen, the series enables us to calculate the value of the logarithm of any positive number, with an error not exceed ing the above estimate for R,. b. The Inverse Tangent
We can treat the inverse . tangent in a way similar to that of the logarithm, starting with the formula
-11 +
where now
f
r.
4 = 1 - 12 + 1 - + 12n
1 + t2
= (- 1)" -
--
.
. . + ( - 1 ) "- 1 1 2 "- 2 +
r
•
•
By integration [see Eq. ( 14), p. 263], we obtain xo xs x2 n-1 + : _ n •-' + R "' arc tan x = x - - + - - + 2n - 1 3 5
Rn
= (- 1 )"
ix r•• 0
1
--
+ 12
·
dt ;
·
·
'
---
Taylor's Theorem
Sec. 5.3
445
we see at once that in the closed interval - I :::;; x :::;; I the remainder R . te nd s t o zero a s n increases, since 1 x 1 2 n+I IR .. I :::;; t 2 " dt = _ o 2n + 1
i lxl
__ .
From the formula for the remainder we can also easily show that for lxl > I the absolute value of the remai nder increases beyond all bounds as n increases. We have accordingly deduced the infinite series 1 x3 x5 x 2 n(6) arc tan x = x - - + - - + + ( - 1 )"- 1 +
3
5
·
.
.
--
valid for the closed interval lxl :::;; 1 . Since arc tan for x = 1 , the Leibnitz-Greg ory series
(7 )
2n - 1 1 = 1Tj4 , we
::: = ! - ! + � - + · · · , 4
an expression as remarkable as that found earlier for log 5.3
obtain
2.
Taylor's Theorem
Newton's pupil Taylor, observed that the elementary expansion of polynomials lends itself to a wide generalization for nonpolynomial functi ons, provided that these functions are sufficiently differentiable and that their domain is suitably restricted . a. Taylor ' s Representation of Polynomials This is an entirely elementary algebraic form ula concerni ng a polynomial i n x of order n, say
f(x) = a0 + a1x + a 2x2 +
· · ·
+ a ,x".
lf we replace x by a + h = b and expand each term in powers of h, there results immediately a representation of the form
(8) Taylor's formula is the relation
(8 a) for the coefficients c. i n terms of f and its derivatives at x = a. To prove this fact we consider the quantity h = b a as the independent -
446
Ch. 5
Taylor's Expansion
variable, and apply the chain rule which shows that differentiation with respect to h is the same as differentiation with respect to h = a + h. Thus successively differentiating the formula (8) with respect to h and each time thereafter, substituting h = 0 yields succe�sively the results c0
=
f(a), c1
=
f'(a), . . . , v ! c, = f'>(a)
and therefore indeed the Taylor formula for polynomials : (9 )
h" h2 f ( a + h) = [(a) + hf'(a) + -f"(a) + . . . + -f 1 ">(a). n! 2 ·
The (n + l )st derivative vanishes for a polynomial of degree n , a nd thus our formula (9) naturally terminates. As stated the formula (9) is nothing but an elementary algebraic rearrangement of a polynomial in powers of a + h, into a polynomial i n powers o f h . b . Taylor's Formula for Nonpolynomial Functions
N ewton and his i mmediate pupils boldly applied formula (9) to nonpolynomial fu nctions for which the expansion does not a uto matically stop at the nth term ; instead they si mply allowed n to increase to infinity, a procedure which for many of the important special functions will be justified l ater on. Assuming the function f differentiable at least n times in an interval containing the points a and a + h we certainly can no longer write for f(a + h) an expression as in (9) of a finite n umber of powers of h, but must account for the discrepancy by an additional "remainder" R,, writing tentatively h" ( 1 0) f( b) = f(a + h) = f(a) + hj'(a) + · · · + - F(a ) + R , ; nl
in fact, ( I 0) is nothing but a definition of the corrective remainder term R , and indicates the expectation that R , might become small and tend to zero for n � oo. If the remainder indeed tends to zero, then the formula ( 10) in the l i m i t n � oo leads to an expansion (11)
f (a + h )
=
f(a) + hf ' (a ) +
·
·
·
+ '!..: f 1 "\ a ) + · · · n!
of f(x) as an infinite power series in h . The crucial problem, far transcending in difficulty that of the alge braic manipulations in Section 5.3a i s then to find estimates for the
Expression and Estimates for the Remainder 441
Sec. 5.4
remainder R, so that the accuracy of Taylor's representation by the finite Taylor polynomial of order n in h ( 1 2)
Tn( h ) =
Io l'\v ! a) h ''
and the passage to the limit for n --+ oo , can be rigorously explored. Taylor's polynomial T,(h) is an approximation to f(a + h) in the sense that at h = 0 the functions Tn and f, as well as their derivatives up to order n coi ncide, so that the difference R , = f - Tn va nishes at x = a together with its first n derivatives. 5.4
Expression and Estimates for the Remainder
a. Cauchy's and Lagrange ' s Expressions
A direct representation of the remainder R , , allowing esti mates of its absolute value I R , I . is the core of Taylor' s theorem . The results are easily obtained on the basis of the mean value theorem of calculus. They are m oreover related to the linear approximation offunctions by dijJerentia/.1· (see p. 1 79). Let us first examine again this approximation. The definition of derivative at the point a states merely that f(a + h) = f(a) + �f'(a) + h <, where • -+ 0 for h -+ 0. We can attain a somewhat sharper approxi mation by ascertaining that • is in fact of order at least as small as h, provided that not only f' but also f" exists and is continuous in our interval J. The estimate i s obtained if we write again a + h = b, introduce a remainder R by ( 1 3)
f(b)
= f(a) + (b - a)j'(a) + R,
and now consider b as fixed and the initial point a as variable ; this equation defines R as a function of a in the interval J; then differen tiation with respect to the variable a yields zero on the left-hand side since f(b) is constant and the rule for differentiating a product shows that 0 = f'(a) - f'(a) + ( b - a)j"(a) + R'(a) and hence (1 4)
- R'(a) = (b - a)j"(a).
Now, for a = b we obviously have R(b) = 0. By the mean value theorem of calculus [R(a) - R(b)]/(b - a) = - R'(�), where � is a not otherwise specified value between a and b ; because of R(b) = 0 we therefore conclude R(a) = - (b - a)R'W = h R ( �) Now by ( 14) R'(�) = - (b - �) J"W and hence I R'W I < h 1 /"(�) 1 since i b - � I S h. -
'
.
448 Taylor's Expansion
Ch. 5
Since l f "(�)l is bounded in an interval around a, we obtain finally an estimate that shows that the remainder or "error" Rn is small of at least second order in h : ( 1 5) I R(a) i < h 2 lf"<m We turn from the special case n = I to that of any order n . The direct characterization of the remainder R n is achieved by the same device as for n = I . We assume that a and b = a + h are points in an interval J in which f(x) is defined and has continuous derivatives up to the order n + I. We consider a as the independent variable and keep the end point b fixed. In formula ( I 0), p. 446, which defines R,(a), we write b - a instead of h. Differentiating and taking into account that f(b) i s constant, we find from the product rule that almost all terms cancel out, and we are left with the formula. ( 1 6)
for every value a in the interval. Since for a = b the remainder R , is zero, this direct expression for its derivative as a function of a completely characterizes ( 1 7)
R,
as the integral R ,(a )
J
L"R,'(!)
dt
=
-
L'R,,'(t)
= " ( h - tl" r + '(t) dt. a
dt or
II !
This is an exact in tegral representation of the remainder. An estimate for R , similar to the one obtained above for n = I follows directly by the mean value theorem of calculus applied to ( 1 6) : R ,( a ) - R n(b) = R ,(a ) = - R,, '(t) = ( b - �)" f"+'(!;) or ( 1 8)
n!
b - a
b - a
R ,(a)
=
( h - a)( h - � ) " ! "+ '( � ).
n!
where � i s a suitable, not specified , i ntermediate value between a and b. The same estimate can also be obtained by applying to the expression ( 1 7) the mean value theorem of integral calculus (Chapter 2, p. 1 4 1 ). Cauchy's Form of the Remainder. If we define � = a + Oh a + O(b - a) we obtai n Cauchy's formula for the remai nder i n Taylor's form ula ( 1 0) =
( 1 9)
Rn(a)
=
hn+l
-
n!
( I - 0) "f"+'1(a
+ (J h ),
where 0 is an unspecified quantity between 0 and I .
Sec. 5.4
Expression and Estimates for the Remainder 449
We can also apply to the integral ( 1 7) for Rn the generalized mean value theorem of integral calculus (see p. 1 42) taking for the "weight function" p(t) the expression p( t ) = (b - t)" which does not change sign throughout the interval of integration. 1 Then (20)
Lagrange 's Form of the Remainder. Setting again yields Lagrange ' s form for the remainder (2 1 )
R"
�=a
+ Oh
h "+l (a) = --f 1 "+ 1 1 ( a + Oh ) (n + 1 ) !
with a suitable quantity 0 satisfying 0 � 0 � I . Lagrange's form i s particularly suggestive, and hence more commonly applied, since it makes the remainder R , i n the formula ( 2 2 ) f ( + h) = f(a ) +
a
.!!._ f' (a) 1!
+ 2h2! f" (a) + +
h" n!
· · ·
l"\a) + R , = P,( h ) + R ,.
look l i ke the term h"+1j! "+1 1(a) /(n + I ) ' that would arise in the expansion (22) to one order higher, only with the argument a replaced by the i ntermediate value a + Oh. For a function f for which /1 "+1 1 is continuous in a closed interval containing the point a, the quantity IJ I "+ 0 ( �) J h a s a fixed bound M . Since then
the Taylor polynomial P,.(h) gives for fixed n an approxi mation to the function f(a + h) with an error of order at least n l in h. Our interest will be directed chiefly toward the question whether the remainder R, tends to zero as n i ncreases ; if this is the case, we say that we have expanded the function in an infinite Taylor series
+
(2 3 )
f" (a) + � j"'( a) + · · · ; f(a + h) = f( a ) + .!!._1 ! f' (a) + !!: 3! 2!
The generalized mean value theorem was proved for the case of a positive p(t), but it applies equally well when p(t) is negative throughout the interval of i ntegration.
1
450
Ch. 5
Taylor's Expansion
in particular, if we first put obtain the "power series" f(x)
=
a =
0 and then write 2
f(O) + !.. f'(O) + ::_ f"(O) + 2! 1!
x
in place of h, we
· · · .
We shall discuss examples i n Section 5.5. For applications the finite Taylor expansion (22) for a fixed n with the remainder term is just as important. If we let h tend to zero in this formula in the terminology of Chapter 3, p. 252, the various terms of the series tend to zero with different orders of magnitude in h . The expression f(a) represents the term of zero order in Taylor's series, the expression hf'(a) the term of first order, the expression h2f"(a)/2 ! the term of second order, etc. We see from the form of the remainder that in expanding a function as far as the term of nth order we make an error which tends to zero of order (n + I ) as h tends to zero. The nearer the point a + h lies to the point a, the better is the representation of the function f(a + h) by the approximating polynomial P,(h) ; in the cases of greatest interest the approximation in the immediate neighborhood of the point x can be improved by increasing the value of n. b. An Alternative Derivation of Taylor's Formula
The integral representation ( I 7) for the remainder term R , i n Taylor's theorem was based on formula ( 1 6) for Rn' (a) . Because of the importance of the theorem we give here a different version of the derivation, which leads directly to the expression for R " by repeated in tegration by parts starting with the formula : (24)
f(b) - f(a)
=
rf'(t)
dt.
T o transform (24) by successive integration b y parts, w e i ntroduce the functions by the relations :
(25)
cfo ,( l ) , q,.(t), . . . ' cfo,(t), cfo ,' ( t )
cPo(t) = l ,
=
and the conditions
(26)
cfo, ( b) =
0,
for
r•
.
.
.
'
cfo ,_ ,( l) z 1
where we consider b as a fixed parameter. Clearly, the conditions (25) and (26) determine successively all cfo ,(t) As is verified i mmediately
Sec. 5.4
Expression and Estimates for the Remainder
451
the ,P,(r) are just the polynomials tj> ,(t)
=
( t - b)' l' !
We note in passing that the functions 1>. originate from each other by successive i n tegration, leaving constants of integration open ; therefore the defining conditions (25) could also be satisfied by functions satisfying other side conditions i nstead of (26) (see p. 1 89).
Since .(a)
=
( - l )'{b -
j ( b } - j ( a) =
a)''j1• ! and tf>Jb)
fh 1
of' d t =
r'u ·
J
0, we obtain
=
'j tj> , ' d t =
tf>t f '
integ ra t i ng the last term again by pa rts, w e fi nd
f
I '· f '·t f" a
-
a
dt ;
f( b ) - f( a ) = (b - a)f'( a ) - cl>2 'f" dt ( b - a ) 2 j"(a) + {'' ¢d"' dt, = ( b - a ) j'(a) + Ja
2!
and r e p ea t i n g the process n t imes,
2 (b f( b ) - f( a ) (b - a )j'(a ) + �1 a) j"(a ) + =
� .
·
·
·
+ c( - 1 ) " ¢,(1)l "+11 ( 1) dr =
(b -
•
a
a )j'( a) + ( h - a / f"( a ) + .
") I - ·
· · ·
where, by the definition of ¢ , ,
f
R , = '! l n i ll ( l ) ( b a
+
+
.
( h - a )" l " \a ) n .,
( h -n . a ) " f " 1( a ) + R , , I
-, I ) " d l .
n.
Thus we have again p ro ved TAYLOR' s THEOREM.
up to the (n
If a function f(t) has continuous derivatives + I )th order on a closed interval containing the two points a and b, then : f( b )
=
f(a ) + ( b -
a)j' ( a) +
· · ·
+
( b - a) n!
"
j1" 1 ( a ) + R n
452 Taylor's Expansion
Ch. 5
with t he remainder R., depending on n, a, and b, given by the expression
(27) By changes of notation we obtain slightly different ex pres si on s of the Taylor formula. Thus, replacing a by x and b by x + h, we have (27a)
f(x + h)
=
f (x) + hf'(x) +
with R,.
or with t
=
x +
(27 b)
If we set x ( 27c)
r,
Rn =
=
I i"'" _.!_ f -
n!
x
· · ·
+
h"
n!
/'" 1 ( x) + R,.,
(x + h - t )"/' "+ 1 1 ( 1 ) dt ;
(h - r) "J' "+11 (x + T) dT.
"
=
n!
o
0 and write x in place of h, we ob t ain 1 f(x)
=
I!
+
with the remainder
R .,
2
f(O) + .:!:_ f'(O) + .::._ /"(0) + x"
n!
I =II I
2!
/ ' " 1 (0) + R
f
o
u
( x - l) "f 1 "+ 1 1 ( 1 )
x
· · ·
dt.
Applying the mean value theorem of integral calculus or its generalized form to the integral leads to the Cauchy formula R
.
=
(I
- , 0)" II .
x n+lf ( n +l i ( (J x)
and respectively. the Lagrange formula
for the re maind e r , as was shown before (p. 448) . Here () is a suitable nonspecified number with 0 � () � I (not the same in bot h formulas) . This special case of the theorem is sometimes without historical justification, called Maclaurin's theorem. Taylor's general theorem was published in 1 7 1 5 ; Maclaurin's special result, i n 1 74 2 . 1
Expansions of the Elementary Functions 453
Sec. 5.5
,.
As an exercise, the reader should construct functions satisfying (25) for which the sii:le conditions (26) are replaced by the relations
for 5.5
v
fv(l)
dt
=
0
� I (see Chapter 8, A ppendi x A). Expansions of the Elementary Functions
The preceding general results pe r mit us to expand the si m ple elemen tary fu nctions in Taylor se r ies. Expansions of other functions will be discussed in C hapter 7 . a . The Exponential Function
First we expand the exponential function, f(x) = ex. I n this case all the derivatives are identical with j(x) and have the value I for x = 0. Lagrange's form for the remainder (p. 449, Equation (2 1 )), yields at once the formula :
e
x
3
xn
x2 x xn+l = I +- + - +- + . . . +- + 2! I! 3! n! (n + l ) ! X
eo x
0 < (} < 1 .
If we n ow let n i ncrease beyond all bounds the remainder R, tends to zero for any fixed value of x. To prove this we note first that e6x � since e" is a monotone increasing function. Let m be any integer greater than 2 1 4 Then for all k � m, l xl/k < �. and ,
el x l
I
xn+ l
---
(n + 1 ) !
so that
I
l xl l x l · · · -l x"' l =-· m! m + 1 n + I --
l x "' l · <- m!
2 xl IR n I < 1 -
"'
m!
I
1 2xl"' I � ·m ! 2" 2 n+l -·m
--
.
el xl
.
..!._ . 2"
-
Since the first two factors on the right are i ndependent of n , whereas l /2" 0 for n oo our statement is proved. -
-
454
Taylor's Expansion
The function
e"
Ch. 5
therefore is represented by the infinite serie s x x2 x3 e" = 1 + - + - + - + · · ·
1!
2!
3!
"' x' =2 v= o v !
- .
This expansion is valid for all values of x. In particular, for x = I we obta i n again t h e infinite series that served to define the number e i n C hapter 1 (cf. p. 77) . Of course, for numerical calculations we must make use of the form of Taylor's theorem with the remainder ; for x = I , for example, (compare with similar computation on p. 78) we have I 1 e9 1 e= 1 + 1 +-++ · · · + - + --- . 2! 3! n ! (n + 1) !
If we wish to calculate e with an error of at most 1 / 1 0,000, we need only choose n so large that the remainder is less than 1 / 1 0,000, and since this remainder is certainly less1 than 3/(n + ! ) ! , it suffices to choose n = 7, since 8 ! > 30,000. We thus obtain the approximate value e = 2. 7 1 825, wi th an error less than 0.000I .
b . Expansion of s i n x , cos x , sinh x , c o sh x For the functions sin x, cos fo rm u l as : f(x) = sin x f'(x) = COS X J"(x) = - sin x
f"'(x) = - cos x J141 (x) = sin x
COS
X
- sin x - cos x sin x COS
X
x,
sinh x , cosh x we find the following
sinh x
cosh x sinh x
cosh x sinh x
cosh x, sinh x, cosh x, sinh x, cosh x.
Thus i n the ap pro x i m a t i ng polynomials in x for sin x and sinh x, th e coefficients of the even powers of x will vanish, whereas in those for cos x and cosh x the coefficients of the odd powers vanish. 1 Here we have made use of the fact that e < 3 . This follows (cf. p. series for e ; for it is always true that 1 /n ! :-:;; 1 /2"- ' , and therefore
e < I
+I +!+t+..
= I
+
I I
_
!
=
3.
78)
from our
Sec. 5.5
Expansions of the Elementary Functions 455
When we use Lagrange's form of the remainder (21 ), p. 449, the Taylor series for our functions take the form :
.
Sin X = X
3!
.
Sinh
X =
;
" 2" x4 (-I) x x2 I --+-- + . + .
.
:____:___ _
2! 4! " ( - J ) i l :r 2 n i 2 C OS (Ox ) + ' ( 2 11 + 2 ) !
X = X
x =
(211 + 1 ) !
( 211 + 3 ) !
x3 xs + - + - + 5' 3!
+ cosh
5!
( - I )"- ilx2 "+3 C OS (Ox)
+
cos
( _ 1 )"x2n+I · · · + .:..._:_ ._ __
xa xs - -+ -+
x 2 " 1 3 cosh (Ox)
----
( 211 + l ) !
.
( 2 11 + 3 ) !
;.r x4 1 + -+ - + 2! 41
x2 n + 1
+
· · ·
(211 ) !
· · ·
+
cosh (():r) + ----'--'-'--' (211 + 2) ! .r 2 "+ 2
x2 u
(211) !
--
where, of course, in each of the four formulas 0 denotes a different number in the interval 0 � 0 � I , a number which in addition depends on 11 and on x. Since in each of these form ulas, the remainder tends to zero as n increases, as can be seen by exactly the same arg ument as in the case of e', we can make the approxi mations as precise as we wish . We thus obtain the four i n fi n ite series, valid for all values x : Sin
X = X
cos
X =
xa
x5
- -+ - - + 5! 3!
.
x2 x4 I - - +-- + . 21
4!
x5
sinh x
=
x + -+ -+ 3! 5!
cosh x
=
1 +-+-+
x3
xz 2!
x4 4!
.
.
.
.
· · · =
· · · =
=
=
I /"f_,
I ( - I r x21'+ 1 ,o o (2Y + I ) ! "
I
( - I Y x2 '
"''
'�"
--- '
(21•) !
2v 1 · 1 x
' �o (2Y + 1 ) ! x2 v
I oc
'
--
,�o ( 2Y) !
'
·
The last two may also be obtained formally from the series for ex in accordance with the definitions of the hyperbolic functions (see p . 228).
Taylor's Expansion
4 56
c.
Ch. 5
The Binomial Series
x)
We pass over the Taylor series for the functions log ( I + and are tan already treated directly in Section 5.2. We shall, however, take up the generalization of the binomial theorem for arbitrary exponents, which was one of the most spectacular of Newton's mathematical dis coveries. We wish to expand the function = (I + in a Taylor series where > - I and Ot is an arbitrary number, positive or negative, + rational or irrational. The function (I is chosen instead of since for the latter at the point = 0 it is not true that all the deriv atives are continuous, except in the trivial case of nonnegative integral values of Ot. We first calculate the derivatives of j(x), obtaining
x
f(x) x)'
x
x
x)"
x•
j'(x) = Ot(J + x)"- 1, j"(x) = Ot(rx - I xr2, j<'1 (x) = 0t(0t 1 ) (Ot + I )( 1 + xr'. In particular, for x = 0 we have +
I )(
-
f"(O) = 0t ( 0t
Taylor's theorem then states (I + ) = I +
x
"
ocx
+
oc
+
( oc
1)
-
-
2!
1)
.
.
'
- v
· · ·
/'(0) = Ot,
!!'1 (0) = oc ( oc
•
· · ·
(oc -
x2 +
·
·
v
1 ), . . .
+ 1).
·
oc(oc - l )(oc - 2) · · · ( oc - n n'
,
+
1)
x"
+
R
n.
Convergence We must yet discuss the remainder. This problem is not very diffi cult, but nonetheless is not quite so simple as the cases previously treated . We shall obtain an estimate for the remainder both directly and also as a special case of a general result of Section A.4. This will permit us to conclude that whenever lxl < I , the remainder R " for the + x)" may binomial expansion tends to zero. Thus the expression (I be expanded in the infinite binomial series
(1
+
�
x)" = 1 + 1 ! x = i (oc)x', v--o
V
+
(
oc oc -
2!
1)
x2 +
Sec. 5.6
Geometrical Applications
()
451
where for brevity we have i ntroduced the general binomial coefficients IX
_
v
IX ( IX - I )
_
. . ( IX -
.
l' !
v
+ I)
(for
v
> 0),
*To prove directly that the remainder R ,. --+ 0 for n ---.. OCJ in the case where - 1 < x < 1 , we make use of Cauchy's form of the remainder ( 1 9), p. 448 :
Rn =
(1 -
,
0
)" x"+y< n H) ( Ox)
n.
(0 ::;;; 0 ::;;; 1 ) . that
Since lxl < I , we have 0 S ( I
1 I Rn l S ( 1 + Ox)"- I lX X I
-
0)/( 1 + Ox) ::;;; 1 so
I ( i) I I ( �) I · · · I ( �) I · 1
-
X
1 -
X
-
1
X
There exists a number q with lxl < q < l . Then obviously also
for all sufficiently large
m,
say for m > N. Thus for n > N
I R n l =:;; ( I + O x)"-1 I IX I ( 1 + I IX I )·'' n-;\'.
a The factor ( I + xOt- 1 is bounded (by 2 a-I if IX � 1, by ( l - q) -I if IX < I) so that clearly R " --+ 0. A slightly m ore general formula gives an expression for (a + b)a. a We only have to factor out a and apply the binomial expansion with x = b/a to obtain for a > 0 and l b l < a q
( �r a• ( ) + �
(a + b)" = a " 1
+
=
IX
( + IX
� :-2 ) ) (�f +
IX IX - 1 ) a- 2 b 2 = a + 01:- a a-l b + ( a + · · · . 1 1 .2 a
5.6
'
'
·)
Geometrical Applications
The behavior of a function f(x) in a neighborhood of the point a, or the behavior of a given curve in a neighborhood of one of
x=
458 Taylor ' s Expansion
Ch. 5
its points, can be described in detail by means of Taylor's theorem, since this theorem perm its us to resolve the increment of the function on passing to a neighboring point x = a + h into a sum of q uanti ties of the first order, second order, etc., in h. a. Contact of Curves
Conta c t of Higher Order
If at a point x = a, two curves y = f(x) and y = g(x) intersect and have a common tangent, we say that the curves touch one a no ther or have contact of the first order. In this case the Taylor expansions of the functions f(a + h) and g(a + h) h ave the same terms of zero order and first order in h. If, in addition, at t h e point x = a the second derivatives of f(x) a n d g (x) are also equal to each other, we say that the curves have contact of the second order. Then the terms of second order in the Taylo r expansions off and g w ill also agree. If we assume that both functions h ave continuous derivatives of at least the third order, then the differem:e D(x) = f(x) - g(x) can be expressed in the form D(a + h) = f(a + h) - g(a + h)
=
� D"'( a + 3!
Oh) =
� F(h), 3!
where the expression F(h) ten ds to f'"(a) - g"'(a) as h ten ds to zero. The difference D(a + h) therefore vanishes to at least the thi rd order w i th h. We can proceed in this way and consider the general case where the Tay lo r series for f(x) and g(x) agree up to terms of the nth o rd er ; that is, f(a) = g(a) , f (a) = g ' (a), . . , f� ''i (a) = g ' " i (a). '
.
We assume that the ( n + l )th derivatives are continuous. Under these conditions the curves defined by our two functions are said to have contact of the nth order at the point x = a. The difference of the two functions is then of the form h n+l D(a + h) = f ( a + h) - g(a + h) = D("+ 1 1(a + Oh) (n + 1) ! h n+l - -- F( h) ' (n + 1) ! --
Sec. 5.6
Geometrical Applications
459
where since 0 � 0 � 1 the q uantity F(h) = n < >•+l l (a + ()h) tends to n p +I l ( a) g < "+ 1 l (a) as h tends to zero. We see from this formula that at the point of contact the difference f(x) g(x) vanishes to at least the (n + 1 )th order. -
-
IIII I
/
I
/
/
/
/
/
/
/
/
/
0
/
Figure 5.1
= f(a) �! a f'(a ) + x
X
Osculating parabolas of e'.
The Taylor polynomials defined by Pn( x)
y
+
.
.
.
+ (x
�! a)n f< n\a)
are characterized geometrically as the "parabolas" of the nth order having contact of the greatest possible order with the graph of the given function at the given point. Hence these parabolas are sometimes called osculating parabolas. (Only for n = 2 are these curves "parab olas" in the ordinary sense.) For the function y = e", Fig. 5.1 shows the first three osculating parabolas at the point x = 0.
460
Taylor's Expansion
Ch. 5
Two curves y = f(x) and y = g(x) that have contact of the nth order at a point x = a, might possibly have contact of an even h igher order, that is, that the equation j<"+1(a) = g<"+l l(a) might also be true. If this is not the case, that is, if J<"+ii(a) ;e g<"+ 1 1(a), we say that the order of contact is exactly n. 1 Contact of Even or Odd Order
From our formulas as well as from i ntuition we can state a remarkable fact often unnoticed by beginners. Let the contact of two curves be exactly of even order ; that is, an even number n of derivatives of the two functions have the same value at the point in question, whereas the (n + I )th derivatives differ. Then the preceding formulas show that the differencef(a + h) - g(a + h) has different signs for small positive values of h and for numerically small negative values of h. The two curves then cross at the point of contact. This occurs, for i nstance, in contact of the second order if the third derivatives have different values. In contrast, contact exactly of an odd order, for example, an ordinary contact of the first order, i mplies that the differencef(a + h) g(a + h) has the same sign for all numerically small values of h, positive or negative ; the two curves therefore do not cross i n a neigh borhood of the point of contact. The simplest example is the contact of a curve with its tangent. The tangent can cross the curve only at points where the contact is at least of second order ; it does actually cross the curve at points where the order of contact is even, for example, an ordinary point of inflection where f"(x) = 0 but f'"(x) ;e 0. At points where the order of contact i s odd the tangent does not cross the curve, as for example, at an ordinary point of the curve where the second derivative is not zero, such as for the curve y = x4 at the origin. We know from Chapter 4, p. 360, that for the circle of curvature at the point x = a given by the function y = g(x) i n a neighborhood of the point x = a, we not only h ave g( a) = f(a) and g (a) = f'(a), but also g"(a) = f"(a). Hence the circle of curvature is at the same time the osculating circle at the point of the curve under discussion ; that is, it is the circle which at that point has contact of the second order with the curve. In the limiting case of a point of inflection, or i n general, of a point at which the curvature is zero and the radius of curvature is infinite, the circle of curvature degenerates into the tangent. In ordinary '
That the order of contact of two curves is a genuine geometrical relation which is unaffected by change of axes is a fact which can be easily confirmed by means of the formulas for change of axes (see Chapter 4, p. 360).
1
Sec. 5.6
Geometrical Applications
46!
cases, when the contact at the point i n question is not of an order higher than the second, the circle of curvature does not merely touch the curve, but also crosses it (cf. Fig. 4.23, p. 359). In conclusion it should be mentioned that sometimes contact of order exactly m is described by saying : the curves have m + I infinitely near points in common ; of course, the precise meaning of such a statement obviously refers to a limiting process. If the curves have, in fact, m + I distinct points P, P1 , P i n common a n d if we let all the points P; tend to P, if necessary modifying one of the curves , then the limiting position might be expected to be that of two curves with a contact of order m . For example, if we draw a circle through three points P, P1 , P2 on a curve C and then let P1 and P2 tend to P, it can be seen that the circle tends to the circle of curvature on C in P. (See Problem 4, p. 437.) •
.
•
m
b. On the Theory of Relative Maxima and Minima
As we have already seen i n Chapter 3, p. 243, a function f(x), whose first derivative vanishes at x = a, has a relative maximum at the point if f"(a) i s negative, a minimum if f"(a) is positive. These conditions, therefore, are sufficient conditions for the occurrence of a maximum or minimum. They are by no means necessary ; for in the case when f"(a) = 0 there are three possibilities open ; at the point i n question the function may have a maximum or a minimum or neither. Examples of the three possibilities are given by the functions y = - x4 , y = x4 , and y = x3 at the point x = 0 . Taylor's theorem at once enables us to make a general statement of sufficient conditions for a maximum or a minimum. We need only to expand the function f(a + h) in powers of h ; the essential point is then to find whether the first nonvanishing term contai ns an even or an odd power of h. I n the first case we have a maxi m u m or a minimum depending on whether the coefficient of h is negative or positive ; i n the second case we have a horizontal i nflecti onal tangent and neither maximum nor minimum. The reader may complete the argument for himself using the formula for the remainder.1 1 The necessary and sufficient condition given previously (p. 242), however, is more general and more convenient in appl ications : provided the first derivative j'(x) vanishes at only a fin ite number of points, a necessary and sufficient con dition for the occurrence of a maximum or minimum at one of these points is that the first derivative J'(x) changes sign as the curve passes through the point.
462 Taylor's Expansion
Ch. 5
Appendix I A.I. l Example of a Function Which Cannot Be Expanded in a Taylor Series
The possibility of expressing a function by means of a Taylor series with remainder of (n + l )th order depends essentially on the con tinuity and differentiability of the function at the point in question. For this reason log x cannot be represented by a Taylor series in powers of x, and the same is true of the function x!-1 whose derivative is infinite at x = 0. In order that a function may be capable of being expanded in an infinite Taylor series, all its derivatives must necessarily exist at the point in question ; however, this condition is by no means sufficient. A function for which all derivatives exist and are continuous throughout an interval still need not be capable of expansion in a Taylor series ; that is, the remainder R. in Taylor's theorem may fail to tend to zero as n increases, no matter how small the i nterval is, in which we want to expand the function. An i mportant simple example of this phenomenon is the function y = f(x) = e-1fx'
x -#
for
0, /(0)
=
0,
which we have already considered in the Appendix to Chapter 3, p. 255. This function and all its derivatives are continuous in every interval, even at x = 0, and as we have seen, at this point all the deriv atives vanish , that is, j!"1(0) = 0 for every value of n. (Geometrically, this means that the line y = 0 has contact of infinite order with the cu rve of the function at the point x = 0). Hence in the Taylor expansion f(O! + .=_ f' ( O) + � f"(O) + · · · 2! 1! 2
all the coefficients o f the approximating polynomials P,.(x) vanish, no matter what value is chosen for n. Thus the remainder remains equal to the function itself, and thus, except for x = 0, can not approach zero as n increases, since the function is positive for every other value of x. Incidentally, this function is useful for the construction of functions exhibiting intuitively unexpected phenomena. For example, g(x) e-1i•'sin (l fx)
=
Zeros and Infinities of Functions 463
Sec. A.J.2
supplemented by g(O) = 0 is again a function with derivatives of all orders, all of which vanish af x = 0 ; the graph of y = g(.r) near x = 0, intersects the x-axis infinitely many times, and oscillates infinitely often .
A.I-2
Zeros and Infinities of Functions
a. Zeros of Order n The Taylor expansion of a function j(x) allows us to characterize the order to which a function vanishes at a point x = a. We say that a function j(x) has an exact n-fold zero at x = a or that it vanishes there exactly of order n, if j(a) = 0, J'(a) = 0, J"(a) = 0, . . . ,fPH l(a) = 0, and j i " l (a) 7Jf 0. We expressly assume that i n the neighborhood the function has conti nuous derivatives at least to the nth order. By our definition we i m ply that the Taylor series for the function in the neighborhood of the poi nt can be written in the form
f(a + h ) =
(28)
h" n!
F( h ) =
h" /1"1 ( a + 0/J), n!
0 < 1,
in which as h tends to zero the factor F(h) = n ! f( + h)/h" tends to a limit differen t fro m zero , namely, the value fi "1(a). Hence f(a + h ) has the same order as h" for h 0 or vanishes to order n i n the sense defined in Chapter 3, p. 252. Similarly, expa n d i ng the derivatives J'(x), j"(x), . . . ,JI'1(x) by Taylor's theorem with the Lagra nge form of the remai nder, we obtain a series of expressions
a
-+
f'(a + h ) = (n h-"- 11 ) ! F1 ( h) = (n h-"-11 ) ! /"1(a + ()!J)
(29 )
f'(a + h ) = ( n h-"_,,v) ! F,(h) = (n h-n-vv) ! /"1(a + () h )
i n all of which the factors () may be different, whereas the factors F1 o F2 , , F,. tend continuously to jl "i(a) a s h -+ 0. Hence f ' vanishes of order n - I, f" of o r de r n - 2, etc. In these form ulas, of course, the assumption i s made that j(x) vanishes of order n ;;;. v. •
•
•
b. Infinity of Order v I f a fu nction cp(x) is defined at all points in a neighborhood of the point x = a, except perhaps at x = a itself, and i f cp(x) = f(x)/g(x), w here at x = a the numerator does not vanish , but the denominator
464 Taylor's Expansion
Ch. 5
possesses a p-fold zero, we say that the function (x) becomes infinite of the Pth order at the point x = a. If at the point x = a the numerator has a ,u-fold zero and if ,u > P, the function has a (,u - P)fold zero there ; if ,u < P, the function has a (P - ,u)fold infinity at the point. These definitions are in agreement with the conventions already laid down (cf. Section 3.7) regarding the behavior of a function. A.I.3 Indeterminate Expressions
We now discuss in a more precise manner, the "indeterminate = j(x)fg(x), in which j(x) and g(x) both vanish at the same point x = a, such as the function (sin x)fx at x = 0. We shall always assign to such functions the value
expressions" of the form cp(x)
( 30)
(a)
=
l im cp(a + h)
h�o
provided this limit exists. These limiting values can be characterized by a simple rule, known as L'Hospital's rule, for which we assume that all derivatives off and g that arise are continuous in an interval containing a. We furthermore assume that the denominator g(x) vanishes at x = a to an order 1• not higher than that of the numerator j(x), so that the function cp(x) does not become infinite at x = a. Then the rule states (3 1 )
J < "( a ) ( a ) = < 'l (a ) . g
By the definition of contin uity, the function cp(x) is then continuous at x = a, and being continuous elsewhere, as long as g(x) ;tf 0, cp is continuous in an i nterval about a. The proof follows immediately from the results of A.2 ; applying Eqs. (28) to both f and g, we find the function cp is, i n a neighborhood of a, given by the relation cp(a + h ) =
f(a + h) g ( a + h)
=
l"( a + (J h ) , g 1 ',(a + e,h)
whence the continu ity of the numerator and denominator, and the nonvanishing of g1•·l(a) yield (3 1 ). We can express the meaning of the last equations in the following way : if the numerator and denominator of a function cp(x) = j(x)jg(x) both vanish at x = a, we can determine the limiting value as x tends to a by differentiating the n umerator and denominator an equal number of times until at least one of the deriv atives is not zero at the point. l f we encounter a nonvanishing derivative in the denominator before one appears in the numerator, the fraction
Sec. A ./.3
Indeterminate Expressions 465
tends to zero. I f a nonvanishing derivative i n the numerator is met before one in the d � nominator, the absolute value of the fraction increases beyond all bounds. We thus have a method of evaluating the so-called "indeterminate expression" 0/0, that is, of determining the limiting value of a quotient in which the numerator and denominator tend to zero. We can arrive at our results in a somewhat different way by basing the proof on the generalized mean value theorem i nstead of on Taylor's theorem (cf. p. 222). Accordingly, if g'(x) ¥- 0 in a neighborhood of the point a, we have f(a + h ) - f(a) r,(a + h ) - g( a )
=
f '(a + O h ) ' g '(a + Oh)
where () is the same i n both numerator and denomi nator. Hence, in particular, when .f(a) 0 = g(a), f (a + h ) g(a + h)
f '( a + O h ) g'(a + O h )
Here 0 i s a value i n the interval 0 < () < I , a n d putting obtain 1 . j '(a + k ) . ( (a + h ) tm I tm · , ,_o
= =
g(a + h )
=
k-o g'(a
k
=
O h , we
+ k)
i t being assumed that the limit on the right exists. If .f'(a) 0 g'(a), we proceed i n the same manner until we reach first index p, for which it is no longer true that sim ultaneously fl''1(a) 0 g <"1(a). Then
=
((a + h ) , _ " g(a + h ) .
Itm ·
= hm
f ' " 1 ( a + I) 1 - o g!i<�(a + /) .
=
'"1(a ) f-' g<" 1 (a )
=
a
an expression i n which we i nclude the case when both sides are i nfinite.
Examples. The following examples which are significant by them selves, illustrate the application of L'Hospital's rule. x-o
lim lim
x-o
sin X
x=
I - cos x X
=
cos 0 I sin 0 1
=
=
1;
O
;
=
e2" - I 2e2" 2 '· = l im log ( I + x) x - o 1 /( 1 + x ) �i� I - 2os �i� �� co
x-o
lim
:
x = s�n x = x
;x= �.
466
Taylor's Expansion
Ch. 5
Other Indeterminate Forms. We further note that other so-called indeterminate forms can also be reduced to the case we have considered ; for example, the limit of sin x
x
as x tends to zero, is the limit of the difference of two expressions both of which become infinite, or is an "indeterminate form" oo - oo. By the transformation x - sin x sm x x x sin x we at once arrive at an expression whose limit as x tends to zero is determined by our rule to be lim
x� o
1 cos x sin x = lim = O. x cos x + sin x . � o 2 cos x x sin x -
-
Derivatives of Indeterminate Forms
The expressions cp(x) = f(x)jg(x) defined at x = a by o u r rule are not only contin uous but also have continuous derivatives provided t h at f and g have continuous derivatives of sufficiently high order. It suffices for us to establish this fact in the case where g vanishes to first order at a , or g(a) = 0, g'(a) :;6 0. For x :;6 a, , g(x)f'(x) - f(x)g'(x) cp (x) = (g(x))'
=
z(x) N(x) '
where again, both numerator and denominator vanish at x = a, since j(a) g(a) = 0. Hence we can determine the l imiting value by applying our rule . z'(x) . . .1.'( I 1m '+' x) = 1 1m x -+ a x -+ 0 N'(x)
=
--
Clearly, d(N(x))j dx = 2g (x)g '(x) , d(z(x))/dx g(x)f"(x) - j(x)g "(x) , both of which again vanish at x = a. Applying L'Hospital's rule once more, as z"(x) lim cp'(x) = lim N"(x) '
=
x -+ a
x ""* a
and noting that N"(x) = 2g(x)g"(x) + 2(g '(x))2, which does not vanish at x = a, we find that g'(a)f"(a) - f'(a)g"(a) . .�,,(. I l ffi 'r X) , 2 2(g'(.1 )) x�a _
Sec. A . /.4
The Convergence of the Taylor Series 467
and this limit is indeed the derivative of cp'(x) at x =
a
(see Chapter
Similar rules for indefinite forms hold for x --+ oo. Thus let f(x) and g(x) be functions for which lim f(x) = lim g(x) = 0 while limf'(x) and 3, p. 26 1 ).
:r: -- oo
x - oo
:r - oo
lim g'(x) exist and are ;C O. Then
f( ) lim f '(x) x ._ = :r- oo g(x) l i m g ' (x ) x_ - , _ _ _
lim
x- oo
The proof follows again from the mean value theorem of differential calculus. * A.I.4 The Convergence of the Taylor Series of a Function with Nonnegative Derivatives of All Orders
We i nsert a general theorem concerning the convergence of Taylor's expansion for functions all of whose derivati ves are nonnegative. Consider the class of functions f(x), differentiable to all orders on the closed interval a � x � b, all of whose derivatives are nonnegative on this interval : v = 1 , 2, . . . . We shall show : For every such function the corresponding Taylor expansion ofj(x + h) in powers of h converges, and the series represents the value of f(x + h) when x and = x + h lie in the open interval (a, h ) and lh l < h - x . For the proof we start with the observation thatf'(x) � 0 by assump tion and hence
;
o
� J(x) - J(q) =
�
frm
J: n;>
a;
a; = f (h) - f(a) = M.
M oreover, for x and ; = x + h in the interval between a and b, we may write f ( " l (x) f(x + h ) - f(x) = hf'( x) + · · · + -- h " + R , .
n!
Assume first that h > 0, or x < ; < b. Then all of the terms on the right-hand side are nonnegative1 and so each is not greater than the This fol lows for R n from the Cauchy or Lagrange formulas and the assu mption � 0.
{'n d ) 1
468 Taylor's Expansion
Ch. 5
value of the left-hand side or than M ; thus
0 For ;
_,.
f 1 "1( x)
�
n !-
�
M
h"
=
b it follows that
M ( ; - x) " ·
M � -- . -n! (b - x) "
J l n l ( x)
(32)
Now, using Cauchy's formula (( 1 9), p. 448) for the remainder, we know there exists some () in the interval 0 S () s I , such that 1 0 � Rn = (
-�
n.
() ) "
h "+ Y i n+l l (x + () h)
S h"+'(n + 1 )( 1 - ())"M (b - X - ()h) n+l
.
Since ; = x + h < b, we may choose a positive n umber p such that O � h s
We then have
or
since
b - x
b - X - ()h � h(1 + p - ())
or
1 + p
+ 1) O � R, S M(n
( I + p - () ) 1 - ()
(
J
1 - () - () + p I
)
"
.
I S M(n + 1 ) p ( I + p)"
<
-1 - < 1 .
1 + pf( l - ()) - I + p
1 - () + p
We know (Chapter I , p. 70) that (n + 1 )/(1 + p)" tends to zero as n increases, so that Rn tends to zero as n increases, when 0 S h < b - x ; thus Taylor's series tends to the function .f for h � 0. For negative h, the fact that R n tends to zero with increasing n follows by using the Lagrange form (2 1 ), p. 449, for Rn : IR I = "
-
1- l h "+l l l f "+� ' (x - () l h l ) l .
(n + l ) !
Now .fl "+2 1 is nonnegative and hence.fl " +ii is monotone nondecreasing. It follows then from the estimate (32) used above that ! n +i l ( x - () h l ) ln l M J '-----'-....l -'-' ...:.. s J +l (x) < ..:.:.:...----,. (n + 1 ) ! - ( b - x)"+l (n + 1 ) !
-- _
Sec.
The Com·e,;rrence of a Taylor Series
A . /.4
Therefore I R ,. I �
and so Rn tends to zero as
n
469
( b l�l J+IM
increases when b - X.
0 < -h <
Thus for any point x with a � x < b, the remainder R, in the Taylor series for f(x + h) in powers of h will tend to zero once l h l < b - x and h > - (x - a). We note that our result is still true if we assume the inequality j< '1(x) � 0 only for all sufficiently large v, say for v > N for some i nteger N, whereas when v � N t he sign ofj< '1(x) may b e a rbi t r ary . To prove this we need only replace the function f in our proof by the function g(x) = f( x) + M(x - a + 1 ) '", for M some positive constant. Then g < •>(x) = j< ''(x) � 0 for v > N, and g < • > ( x) = [< ' 1 (x) + MN(N - I ) · · · ( N - v + l )(x - a + 1 )-'"-• � .f< '1(x) + M for v � N. Thus g < ' 1(x) � 0 for all v if M is chosen su ffi c i en tl y large. This proves that g(x) can be expanded in powers of .r , and the same result follows then for the function (, which differs from g only by a polynomial . The theorem on the binomial series ( p . 456) is an immediate con seq u en c e of this result : We change the notation slightly and c o ns i de r fi rst the function cf>(x) = ( I - x)' in place of ( I + x)". The deri va ti v e s of a re then given by cf> < ' 1 (x) = ( - I )'
(:) rx(
Si nce the binomial coefficients =
rx
( :)
- I)
·
·
o - x)•-' v !
�;
rx
-
1'
+ I)
have alternating signs as soon as rx - v i s negative, we see that either the function cf>(x) or - (:r) belongs to the class o f functions with non negative derivatives from some order on when we l i m i t x to values x < I . Thus for a = - I , b = I , x = 0, a nd I h i < b - x = I o u r general theorem pr oves th a t
�
( I - h ) ' = , 0( - l ) '
( :)
h '.
If here we write x for -h, we obtain the binomial expansion rx ( - I ) x2 rx ( rx - I )(rx - 2 ) 3 x + + x ' = 1 + rx x + rx ( I + x)" = I ·2 3 v�o v I ·2 for any exponent rx and any x with - 1 < x < l .
�. ( rx) ,
·
470 Taylor's Expansion
Ch. 5
Appendix If
*A.II.l
Interpolation
The Problem of Interpolation.
Uniqueness
The Taylor polynomial Pn(x) approximates the function f(x) in such a way that the graphs of f(x) and Pn(x) have contact of order n at a point a, or in such a way that f(x) and P n(x) coincide at n + 1 points "infinitely near" to a. We might "resolve" the point with abscissa a into n + 1 distinct points with abscissas x0, x1 , , xn and seek an approximation to f(x) by a polynomial (x) of degree n which coincides with f(x) at these points. This polynomial, as it turns out, is determined uniquely by a system of linear equations. By a passage to the limit x, x for all i we regain the Taylor polynomials. But "interpolation," that is, the approximation by polynomials coincid ing with f(x) in distinct points is of great importance in many appli cations. The following discussion will give a brief account of the theory of interpolation. We con sider the following problem : Determine a polynomial (x) of nth degree, so that it assumes at n + I given distinct points x0, x1, , xn, the n + I given values /0, /1 , , fn, that is, •
•
•
_...
•
.
•
•
•
•
(xn) = fw
If the numbers /, are the values /, = f(x ,) assumed by a given (possibly less elementary) function f(x) at the points x,, then the polynomial (x) will be named the interpolation polynomial of nth degree of the function f(x) for the points x0 , x1, , x"" There can at most be one such polynomial of nth degree, for if there were two different such polynomials (x) and 1p(x), then their difference D(x) = (x) - v•(x) would be a polynomial of mth degree with 0 ::;; m ::;; n having n + I distinct roots, which is not possible according to elementary algebra.1 We can prove the uniqueness of the interpolation polynomial by yet another method, based on the GENERAL THEOREM OF ROLLE. If a fun c t ion F(x) has continuous derivat ives of order up to n in an interval, and vanishes at least at n + I •
•
•
1 For we would have
C0 ¥- 0 , D(x) = c0(.r - x 1 )(x - x,) . . . (x - xm), since x., . . . , xm are zeros of D(x) ; but then since D(x,) = 0 ,
c 0(x0 - x1)(x0 - x2) (x0 - xm) contrary to the distinctness of :r,, x. , ' xm . • • •
.
•
.
=0
Sec. A .l/. 1
The Problem of Interpolation. Uniquenes.r 471
distinct points x0, x1, , xn of the interval, then there is a point � interior of the interval for which p n>(�) = 0. •
•
•
in the
PROOF. The general theorem follows easily from the special case I which is the Rolle theorem proved on p. 1 75 . Let the numbers X0, x1, , xn be arranged in increasing order. Then by the mean value theorem (or by Rolle's theorem) the first derivative F'(x) must vanish at least once within each of the n subintervals (x;, X ; + l ). This same consideration applied to F'(x), and the intervals between its zeros tells us that F" (x) vanishes at n - l points ; by applying this argument repeatedly, the assertion is proved .
n
=
•
•
•
We now apply this theorem to the difference F(x) = D(x) = (x) - 1f'(x)
which by assumption vanishes at n + l points. We obtain a point � at which the nth derivative vanishes ; D 1 " >(�) = 0. This is, however, n ! d0 , so that d0 = 0 and the difference is a polynomial of at most degree n l , vanishing at n + l points. Again applying the theorem of Rolle, we obtain d, = 0, etc., or D(x) is identically 0 as we asserted . These considerations can be extended to the case where the X; are not all distinct from each other and, perhaps, r of the values x ; agree ; that is, x0 = x1 = · · · = x._,. In the interpolation problem we shall then require that (x) and the derivatives q,'(x), . . . , q,lr-l >(x) should assume preassigned values for x = x0 , and correspondingly for the other points xv. The polynomial D(x) then is of the form c(x - x0)'(x - xr) · · · The general theorem of Rolle and the unique ness theorem , as well as the proofs, hold unchanged in this case. -
.
A.II.2 Construction of the Solution. Newton's Interpolation Formula
We shall now construct an interpolation polynomial (x) of nth degree, such that (x0) = f0, . . . , (x,.) = fn· In order to construct it in a stepwise manner, we shall begin with the constant fo which is a polynomial 0(x) of Oth order which for all x and, in particular, for x = x0 assumes the value A0 f0 . To it we add a polynomial of first order, vanishing for x = x0 and therefore of the form A1(x - x0) ; then we determine A 1 such that the sum has for x = x1 , the correct value f1 . The resulting polynomial of first degree we name 1(x). Now we add to 1(x) a polynomial of second order which vanishes for x = x0 and x = x 1 , and i s thus of the form A 2(x - x0)(x - x1), whose =
472 Taylor's Expansion
Ch. 5
addition thus will not change the behavior at these two points ; the factor A 2 is then determined so that the resulting polynomial of second order, 2(x) , will also take the assigned value, in this case l2 , at x = x2 • This procedure is continued until all points are reached and we obtain the polynomial
( 33) ,P(x) = cPn(x) = A 0 + A 1(x - x0) + A2(x - x0)(x - x1) +
· · ·
+ A n(x - x0) · • (x - xn_1). •
Our method of obtaining the coefficients A ; in the expression for .p is made clear by substituting x = x0 , x = x 1 , , x = xn in order, thus obtaining the system of n + I equations •
(34)
•
•
lo = A o 11 = A 0 + A 1(x1 - x0) 12 = A0 + A1(x2 - x0) + A 2 (x2 - x0)(x2 - x 1)
In = A o + Al(xn - Xo) + · · · + + A n(xn - Xo)(xn - xl ) . . . (xn - Xn-1 ) .
Clearly, we can determine the coefficients A 0, A 1 , , A n successively so as to satisfy these equations, and in this way the interpolation polynomial can be constructed . •
•
•
When the values x. are equidistant, x. = x v- l + h, the result can be written explicitly in a more elegant manner. The equations for the A 1 now become lo = A o
( 35)
11 = A 0 + hA 1 12 = A 0 + 2hA 1 + 2 ! h2 A 2 Is = A 0 + 3hA 1 + 3 · 2h2A2 + 3 ! h3A3 In = A 0 + nhA 1 +
·
·
·
+
n! . i) ! h'A ; + · (n _
·
·
+ n ! h"An.
The solutions may easily be expressed as successive differences of 1: Given any sequence (finite or infinite) of terms lo.f1, f2 , , we call the expressions •
•
•
the first differences of the lk · Applying the differencing process again to the sequence of t!.lk > we obtain the expressions
Sec. A . /1.2
Construction of the Solution. Newton's Interpolation Formula 473
that is, which are the second differences of the fk · The nth difference t:.. y, is defined recursively as :1"-1/k-1 1 - :1"-Yk · When expressed directly in terms of the fk it is given by the formula
which fol lows by a simple inductive argument left to the reader. With this terminology the coefficients A, can be written in t he form (3 7) as can be verified by induction .1
Putting � = (x
=
h(e - r). The expressions (x - x0)(.r - J'1 ) · · · (.r - x, ) assume then the form H� - I ) · · · (e - v)h'+l. Thus we obtain for the polynomials if>(J:)
Newton's Interpolation Formula.
-
x0)/h we have 3:
- x,
from (33), (37), Newton's interpolation formu/a : 2
, If f0,f1,f2, are the values of a function j(x) at the points x0 , x1, x2, where f has continuous derivatives through the nth order, then a'f0/h' is an •
•
•
•
•
•
1 We h ave to verify that the values A, given by (37) satisfy the e q uations (35) ; that is, for any sequence f,, f, , f,, . . . , the identity
i s satisfied . Assuming that this i s true for a certain k , we must show that
f•n = [,
+ (�)
G) + + (;)rt:.r.. , + (�)wr. + a'f,> + + (k + l) (k + l) + . . . k+ (!) (!) 1)/k !
= cr. + t:..f,>
= [n
:1[, +
···
+ a'f,>
D.fo +
I
which is the identity for the case
2 As
D.'f,
2
D. 'fo
I.
on p . 457 we define here the bionomial c oe ffic ients
position in tege rs k by
=
···
$($ - I ) · · · ($ -
k+
for general $ and
474
Taylor's Expansion
Ch. 5
approximation to the derivative f1'1(x0) ; we shall show on p. 476 that 1
lim --; !:>'fo h�o h
(�)
Since also
= J< '>(x0) .
(x xo) '' lim hk = k! 1<�0 k we see that in this case ¢>(x) tends to the Taylor polynomial P11(x) wh en h tends to ze ro . �
We note that the construction of the interpolation polynomial i s possible i n the same m anner, if, perhaps, t h e first r values x0, , xr_1 , f�r- 1 1 are preassigned coincide, and corresponding values /0 , /0 ' , for cfo(x0), f(x0), . . . , q,< r -ll(x0), which coincide with the values •
•
•
•
•
•
f(xo)J ' (xo), . . . Jlr- l l (x.),
for a given function ! For if>(x) we write the form if>(x) = A 0 +
A 1 (:r - x0) +
· · ·
+ A z(x - x0)2 + A .(x - x0)' + A r+ l (x - x0Y(x -
we then determine the A , i n order from the equations f� = A o fo' = A 1 fo" = 2 A ,
xr) +
· ·· ;
f�r- l l
= ( r - 1 ) ! A r- 1 = 1(x + . . . + A r(xr - XoY + A A r Xo) fr o = A o + A 1(xr+ l - Xo) + · · · fr+ l +
Ar(xr+l
- XoY +
ArnCxr+ 1
- Xo)'(x r+l - x r)
A.I1.3 The Estimate of the Remainder
For the foregoing considerations it did not matter how the values
f0, f1 , . . . Jn were originally given . For i nstance, if these values were
obtained from physical observations, the problem of constructing the interpolation polynomial could still be completely solved, giving us then in if>(x) a simple smooth function defined for all x and taking the observed values at the given points, which can be used to "predict" approximate values for f(x) at other x. However, if the function f(x) taking the n + 1 given values fk at the given points xk is defined also for intermediate values x, we have to face the new problem of estimating the difference R(x) = f(x) - if>(x), the error of interpolation. We know at first only that R(x0) = R(x1) = R(x11) = 0. In order to be able to say more, we must make further assumptions on the behavior ·
·
·
Sec. A.I/3
The Estimate of the Remainder
415
of the function f(x), which affect the remainder R (x). We will therefore assume that in the interval under consideration j(x) has continuous derivatives of at least the ( n + l )th order. We note at first that for every choice of the constant c, the function K(x ) = R( x) - c(x - x0)(x - x1) · • ( ;r - x , ) •
, x , . Choose now any value y vanishes at the n + 1 points x0, . . , x , . We can then determine c so that K(y) = 0, x0, distinct from x1 , . that is, R(y) ��� C = ( y - x0)(y - x1 ) · ( y - x , ) •
•
•
--------
-------•
•
Then there are n + 2 points at which K(x) vanishes. We apply the generalized Rolle's theorem used earlier to K(x) ; by this we know there i s a value x = � between the largest and the smallest of the values , x , , y, such that K l » t l l (�) = 0. Since R(x) = f(x) - cf>(x), x0, x 1 , and cf>, as a polynomial of nth order, has an identically vanishing (n + 1 )th derivative, we have p n+ll(�) - c (n + I ) ! = 0, •
•
•
noting that (n + 1 ) ! is the (11 + I )th derivative of (x - x0) · · · (x - x, ) . Thus we have obtained for c, a second expression c = .f' "+1 1 ( �)/(n + I ) ! , containing � and depending in some manner on y. We now use the equation K(y) = 0, i n which y i s completely arbitrary and therefore can be replaced by x, and obtain the representation ( x - x0 )( x - x1 ) · · • ( x - x , ) / l n+ JIW ,
(n + I ) ! � where is some value lying between the smallest and the largest of the points x, x0, x1, . . . , x Thus t h e general problem of i nterpolation for a given function f(x) is completely solved. We have for f(x) the representation
(38)
R (x) =
•.
(39) f ( x) = A0 + A 1 ( x - x0) + A 2(x - x0)(x - x 1 ) + · · · + A ,( x - x0)(x - X1) • • (x - x n_1) + R n , where the coefficients A0, A 1 , , An can be found successively from the values off at the points x0, x1 , , xn by the recursion formulas (34) on p. 472 and where the remainder R n is of the form •
•
•
•
•
(40)
Rn
=
•
•
( x - Xo) (x - xl) . . . (x - x n ) t
(n + l ) !
with a suitable number � between the largest and smallest of the values
476
by
Taylor's Expansion
Ch. 5
If we take the corresponding formula (39) for f(x) with n replaced n - I and subtract, we obtain A n(x - x0)(x - x1) • · • (x - Xn_1) + Rn - Rn_ 1 = 0 .
For x = xn we have R" = 0, and hence for the coefficient A n (using (40) with n replaced by n - I ) the representation n fl l A n = -m n .,
where � lies between the smallest and largest of the values x0, x1 0 • • • , x ". Similar representations exist for A n-1, An_2 , • • • , A0• Thus we recognize that if the points x0, x1, • • • , xn are tending together to one and the same point , perhaps the origin , then our interpolation formula (39) goes term for term into the Taylor formula (27a), p. 452, with the Lagrange form (2 1 ), p. 449, of the remainder. The Taylor formula can thus be considered a limiting case of the Newton i nterpolation formu l a. This formula enables us to give precise m eaning to an expression commonly used in geometry. The osculating parabola which meets a given curve at a point, of nth order, is said to have "(n + I) consec utive points in common" with the given curve at the point. Actually, we obtain this osculating parabola if we find a parabola having n + 1 points in common with the curve, and then draw these points together. Analytically, this just corresponds to the transition from the inter polating to the Taylor polynomial. In the same fashion we can characterize the osculation of arbitrary curves. For example, the circle of curvature is that circle which has three consecutive points in common with the given curve. The interpolation formula can be expected to give the values of a function whose values at some definite points are known, with a high degree of accuracy between these points (both I JI '+ 1 1(�) 1 and the l x - x,l are then bounded). If the value x lies outside the intervals of the points x0, x1, , x " , we speak of extrapolation. By means of such an extrapolation we shall obtain good agreement provided the point x is sufficient ly near the given points. The Taylor formula corresponds i n a sense to complete extrapolation ; in general, it is suitable for use only in a neighborhood of a point. •
•
•
A.JI.4 The Lagrange Interpolation Formula
In closing, we solve the interpolation problem by a somewhat differ ent formula, due to Lagrange, and diffc.-ring from Newton's inter polation formula insofar as each individual term contains only one of
Problems 477
the given values of the function. Moreover, the formula gives cP(x) quite explicitly, not requtring any solution of recursive formulas. For brev ity we introduce the polynomial of (n + l )th degree 1p(x)
=
(x - x0)(x - x1 ) • • • (x - xn),
corresponding to the given points x, . Differentiating by the product r u l e and substituting then successively for x the values x0, • • • , x,. , we obtain the relations
1p'(x .) = (x n - x0)(x. - x1)
•
•
·
(x. - x,_1).
We n ote that 1p ( x )
(x - x.)1p'(x, )
( x - x0) · • · ( x - x,_1 ) ( x - x,+1 ) • • · ( x - x,.) (x,. - x0) • • • ( x , - x,_1)(x , - x ,. , 1) · · · ( x , - x,. )
is a polynomial of nth degree, having at the point x = x,, the val ue I , and at the rem a i n i ng points x,, the value 0 ; then it i s i m m ed i ate l y clear that the expression ( 4 1 ) rp( x)
=
[
1p( x)
f1 fo + + . . . + fn ( x - x ,.)1p' (x n) (x - x1 )1p'( x1) ( x - x0) 1p' (x0 )
J
is the desired interpolation polynomial. This is the interpolation formula of Lagrange. PROBLEMS SECTION 5 .4b, page 540 1. G ive the complete formal derivation of the remai nder formula (27), p. 45 2, using mathematical induction.
2. (A Variant of Proof of Tay lor's Theorem) (a) If g(h) has continuous derivatives through the (n + l )th order for 0 :S; h :S: A , and if g(O) = g'(O) = · · · = g C n l(O) = 0, while lgCnHI(h)l :S; M on [0, A ], for M a constant, show that lgCnl(h)l :S: Mh, lgCn-ll(h)l :S: Mh2/2 ! , . . . , !gcn-il(h)! :s; Mhifi ! , . . . , lg(h)l :S: Mh"/n !, for all h in the
interval.
478
Taylor's Expansion
Ch. 5
(h) Let .f(x) be a sufficien t ly differentiable function o n a ::;; x ::;; h, and Tn(h) be the Taylor polynom i a l for .f(x) at x = a . Appl y the result of (a) to the function g(h) = R, = .f(a + h) - Tn(h) to obtain Taylor's formula with a ro u gh esti mate for the remainder.
have a continuous derivative in t he interval a ::;; x ::;; b, and for every value of x. Then if ; is any poi n t in the i nterval, the curve now h e re falls below its tangent at the p oi n t x = �. y = [(n. (Use the Taylor expansion to three terms.) 4. Deduce th e i n te g ra l formula for t h e remainder Rn by applying inte gra t io n by pa r t s to
let ("("·) ;:>: 0
3.
Let f(-")
f(x +
h) -f(:t:) =
5. Integrate by parts the formula R,
and s o obtain
=
nI 1.
1" o
(h
R , = {(.•· + h) - .f(x)
*6. Su pp ose t h at obtained, namely
J:'
f'(J +
T) dT.
- T)''j'lnHI(x + - hf'(x)
-
i n some way a s e ri es
· · ·
T) dT ,
hTI
- -. f'"1(J·). n.
for the function f(a') has been
f(.r) = Go + ae: + az:t·2 + . . . + anx" + R,(x) ,
where
a0,
a1 ,
•
•
•
,
a, are constants,
t h at
is, t hat
the
R"(.r) is
n
as :r � 0. S h o w that series is a Taylor series.
and R"(:1·)jx" -- 0
=
(f'(O)jk 1) (k
ti mes cont inuous! y di fferen t ia bl e ,
ak
=
0, . . . , n),
SECTION 5 . 5 , page 453 1. Find the fi rs t four nonvanishing terms of the Ta y l o r fo l lo w in g functions in the neigh borhood of r = 0 : (a) x cot x
series
for the
.
'\/�
(b) --,; ;r
2. Find the Ta y l o r series for arc sin u si n g ( c ) sec
.r
arc sin x =
Compar e Sect ion 3.2, Problem 2.
((J log s i n x - log :t'. :r in t he ne i g h bor h o od
lx 0
of .r =
0 by
dt
_ ;- .
v I - t2
*3. Find the first three non vanishing terms of the Taylor series for s i n2 x in the neighborhood of x = 0 by multiplying the Taylor series for s i n x by itse lf. Justify this procedure. *4. Find the first three nonvanishing terms of the Taylor series for tan x in the neighborhood of .r = 0 by u s i ng the relation ta n :r = sin xfco s x, and justify the procedure.
Problems 4 79 • s . Find the first three non vanishing terms of the Taylor series for --.1 cos x i n the neighborhood of x = 0 by applying the binomial theorem to the Taylor series for cos x , and j ustify the procedure.
Find the Taylor series for (arc sin x)2• Compare Section 3 . 2, Problem 2.
*6.
of
7. Find the Taylor series for the following functions in the neighborhood
x=
(a)
( lx
0:
sinh-1 x .
c)
0
sin t dt . t
-
•s. Estimate the error involved in using the first n terms in the series i n Problem 7.
9. The elliptic function s(u) has been defined (Section 3 . 1 4a) as the inverse of the elliptic integral u(s)
=l'
0 ,i ( I
-
x
dx
2J ( J
-
�:
k 2 2)
.
Find the Taylor expansion of s(u) to the term of degree 5. 1 0. Evaluate the following l i mits :
}��x[(' �r J (b) }� {� x {( U - e]}. � [( U ( 1)1 +
+ a·
* ( c)
1 �x .
1 +
-e
! +
(d) lim
x -o
(e) lim z - oo
- e log l +
( )·J rx' ( x)lrx' · sm x
x
,
,
sin .X
* 1 1 . Find the first three terms of the Taylor series for [I + ( 1 /x)]x in powers of J f.r.
Two oppositely charged particles + e, -e situated at a small distance
d apart form an electric di pole with moment M = ed. Show that the potential * 1 2.
energy (a) At a point situated on the axis of the dipole at a distance r from the center of the dipole is (M/r2)( 1 + <), where is approximately equal to •
d2/4r2. (b) At a point situated on the perpendicular bisector of the dipole is 0.
(c) At a point with polar coordinates r, 0 relative to the center and axis of the dipole is [M cos (Ofr2)]( 1 + • ) , where is approximately equal to (d2/8r2)(5 cos2 0 - 3). •
(The potential energy of a single charge q at a point at a distance r from the charge is q /r ; the potential energy of several charges is the sum of the potential energies of the separate charges.) SECTION 5.6, page 4 57 1 . Prove if j(a) = 0 and j(x) has sufficiently many derivatives at x = a that j(xt has at least an (n - l )th order contact with the x-axis.
Ch. 5
480 Taylor's Expansion
2. The curve y = f(x) passes through the origin 0 and touches the x-axis at 0. Show that the radius of curvature of the curve at 0 is given by
xz
p = lim - · z-o 2Y
*3. K is a circle which touches a given curve at a point P and passes through a neighboring point Q of the curve. Show that the limit of the circle K as Q P is the ci rcle of curvature of the curve at P. *4. Show that the order of contact of a curve and its osculating circle is at least three at points where the radius of curvature is a maximum or minimum. *5. Show that the osculating circle at a point where the radius of curvature is a maximum or minimum does not cross the curve unless the contact is of higher than third order. *6. Find the maxi ma and minima of the following functions : �
(a) cos
cosh x
(b) X + COS X
*7. Determine the maxima and minima of the function y ;e
p. 242) .
=
e- 1 1•' (see
S ECTION A . 3 , page 464 1. Prove i f.f is continuous on the interval [0, I ) that lim "'
x-o
2. Prove that the function
y
i' x
=
· 2 dz = f(O). z ((z)
(x2)x , y(O)
=
I is continuous at x
=
0.
6 Numerical Methods
The task of solving an analytical problem al ways remains uncom pleted . The proof of the existence and of some basic properties of the solution is usually considered satisfactory, but relevant questions always remain to be answered. Thus, when the solution is defined by a limit process, for example by an integral, the problem ari ses of actually finding approximations to this limit and of estimating the accuracy of these approximations. Not only are such questions of basic importance theoretically but they are also i nevitable, if we wish to apply analysis to the description and control of natural phenomena which i n principle can be described only in an approx imate manner. Accord ingly it is a great challenge to carry the sol ution to the point where numerical answers and estimates of their accuracy come into reach . Recently, with the advent of high-speed automatic computi ng machines, theoretical and practical aspects of "numerical analysis" have received a great sti mulus ; they are presented in a variety of textbooks. 1 For centuries, however, many of the foremost mathematicians, such as Newton, Euler, and, i n particular, Gauss, have greatly contributed to numerical methods. I n this volume we cannot present n umerical analysis i n a com prehensive way, but at least we shall discuss some of the simple classical results. 1 See for example, Hildebrand, Introduction to Numerical A nalysis, McGraw-Hill Book Co., 1 95 6 ; Householder, Principles of Numerical A nalysis, McGraw-Hill Book Co., 1 953 ; and Whittaker and Robinson, The Calculus of Observations, Blackie and Sons, Ltd., 1 929. 48 1
482 Numerical Methods 6.1
Ch. 6
Computation of Integrals
Although the existence of the integral of a (continuous) function i s assured b y the theory of Chapter 2, the evaluation of such an i ntegral or "quadrature" 1 cannot be effected by elementary functions except i n relatively rare cases. W e must therefore devise methods for numerical i ntegration and for estimating the accuracy of the numerical approxi mation . To compute approximately the i ntegral J
(1)
f!(x) dx x
=
with a < b, we subdivide the interval a � � b into n equal parts, each of length h = (b - a)/n by means of the n + I points
(2) Then
x,
= a + vh,
nh
=
J
where
J
( 3)
,
=
b - a,
v
=
0, 1 ,
.
.
.
, n.
n
= 'J ,J• . V=l
r�l f(x) dx;
the problem of computing the i ntegral J is reduced to that of obtaining good approximations for the areas J, of strips of width h into which we have dissected the enti re area, represented by J. a. Approximation by Rectangles
The most direct approximation, paraphrasing the original definition of the integral, yields the relation J
n
=
v=l
'J ,J.
where for abbreviation we set
f, f(x,). =
1 The word "quadrature" indicates the process of "squaring", that is, of measuring an area inside a curve by finding a square having tne same area (as i n the problem of "squaring the circle").
Sec. 6.1
Computation of Integrals 483
Here (and throughout this chapter) the symbol ""' means "approxi mately equal. " To estimate t h e accuracy or "error" of t h i s approximation, w e assume that f( x) is continuous with a u niformly bounded derivative on the interval a � x � h : lf'(x) l � M1 . Then it can be proved easily (see Problem 4 p. 507, 6. 1 ) that
I - \�/ I
(4)
hf, l �
II,
or therefore
J
� n
�
M h2
�
M h2
,
-
(5 ) = � M 1 (b a)h . Thus the accuracy of the approxi mation of the i ntegral by the finite sum is of the order h of the "mesh width" i n the terminology of Chapter 3, p. 252. h. Refined Approximations-Simpson 's Rule A better approximation i s obtained with hardly more effort if we approximate the areas J, not by rectangular strips but by the slender trapezoids, as in Fig. 6. l a. The approximation form ula (trapezoid form ula) is then
(6)
J
R:< =
ih(fo + J1 ) + � h(fl + f2 ) + · · · + �hUn -! + f, ) h h (fl + J2 + · · · + fn - 1) + 2 Uo + f,),
since every function value except the first and the last appears twice. An approximation which is general ly slightly more precise than that of the trapezoid formula i s that in which the vth strip is approximated by a trapezoi d bounded above by the tangent to the curve at the m i dpoint x 1 + h/2 of the i nterval x,_ 1 � x � x,. The area of this trapezoid is simply
,_
hfv- 1 /2
=
(
hf x,_l +
�)
and we obtain by addition the tangent formula, J
,
h (f1 12 + /s/2 + · · · + fr2n-11 / 2). As we shall see on p . 486, the accu racy of this approximation i s of order h 2 when the second derivative ofjis continuous in the i nterval a � x � h and IJ"(x) l < M2 , with some constant bound M2 • Finally, we mention the famous approximation of Simpson, which with little additional effort yields a much more accurate approximation (7)
R:<
484 Numerical Methods
Ch. 6
y
0
0
.to
.tv- 1
Xv
%()
Xv - 1
x,.
(a)
(b )
Xn-1
Xn
Xn - 1
Xn
Figure 6.1 (a) The trapezoid formula. (b) The tangent formula.
Sec. 6. 1
Computation of Integrals 485 y
x� x.---.-1----� a �--� x.� + -l ------------� x
Simpson's rule.
Figure 6.2
if the fourth derivative of f exists and is uniformly bounded in the interval : l j<4>(x) l s M4 , with M4 a constant. Simpson's formula for n = 2 m is (8)
J
�
4h
3 U1 + fs + /s +
· · ·
+ /2 m-1l
2h
+ 3 U2 + /4 + fo +
· · ·
h
+ /2m- 2 ) + 3 Uo + /2rnl ·
The formula is easily obtained if we approximate the region composed of the vth and (v + I )th strips by a stri p of width 2h bounded above by the parabola which agrees with f at the three abscissae x,_ 1 , x, = x ,_ 1 + h, and x v + 1 = x,_1 + 2 h (see Fig. 6.2). Newton's inter polation formula (p. 473) yields the equation of this parabola : Y
=
/,v-1 + ( x
_
fv Xv- 1 ) . J, - -1 h
+
( x - x,_1 )( x - x , _1 - h ) /,+1 - 2/, + /,_1 .
hence we have the approximation J,
+ 1,+1
�
= =
{"V+t
J
,,_,
Y
dx
h2
2
=
{"v- t+ 2 h
),,_ ,
Y
'
dx
� h - 2h 2hfv-1 + 2h (f, - fv-1) + -- (/,+1 - 2/, + fv-1) 2 h 3 Uv-1 + 4/, + /,+ 1 ) .
486 Numerical Methods
Ch. 6
The formula is now obtained for even n = 2m by adding all these approxi mate values for v = I , 3, 4, . , 2m - I or all the areas of the pairs of strips. .
.
*Accuracy It is not difficult to estimate the accuracy of our approximations. Each quadrature proceeds by approximating the function f(x) i n an interval by an easily i ntegrated function c/>(x) (a polynomial). An estimate for the error i n the integration formula can thus be obtained by estimating lf(x) - cp(x) j . In the tangent formula (p. 483) we replaced f(x) in the i nterval [x,_ " x,] by its tangent at the midpoint x, - (h/2), that is, by c/>(x)
=
f
( �) ( +
x, -
x - x,
+
�) ( �) r x, -
.
By Taylor's theorem with Lagrange's form of the remainder j (x)
=
c/>(x) +
where � lies between x and x, one strip is estimated by IJ' - hf, _ ! 'i '
=
�
I r�, r ·· J xv·- l
-
'
2(I X - X, + 1)h ' 2 rm, h/2. H ence the error corresponding to
[f( x) - .fo(x )] d x
I
l f( x) - .fo(x) j dx � M 2
- -M
h3
24
x r , J x v- h 2 '
�(X -
x,
+
�r dx 2,
2•
For the total error in the tangent formula contributed by the various intervals 1 we find then the upper bound h3
n -
24
M2
h2
= -
24
M2 (b - a).
1 This is the total error inherent in using the approximating formula, the so-called truncation error ; in practice, additional error ar'ses because of round off in the computation. The total effect of round-off errors increases most likely with the number of steps taken, that is, with decreasing h, whereas the truncation error
decreases,
Sec. 6.1
Computation of Integrals 487
We use this derivation as a model for estimating the error in the other quadrature formulas. I n the trapezoidal rule (6) w e approximate f(x) in the i nterval [x, _ 1, x.J by the linear int e r p o lati o n polynom ial
From the error esti mate for the remainder i n the i nterpolatio n formula [ se e p . 475, Eq . (40)] fo r n = I , we find
f (x ) - ( x)
=
Hx - x , _1 ) (x - x ,.)f"(�).
where � lies between x , _ 1 and x, . H ence the absolute value of the error in the computation of J, i s at most
and the total error i s then at most
n times this quantity : 2 h - M2( b - a ) .
12
The same tech n i q ue can be applied to S i m pson' s rule (8), tak ing for (x) the quadratic polynomial agreeing with f in the points x,_1, x , , x,.n leadi ng to an error i n J,, + 1,+ 1 of the order h4• Actually, however, the error estimate can be i mproved by one order of magnitude by using a c ubic polynom ial (x) that gives a better appro x i mation to f i n the i n terval [x,,_ 1 , x,.+ d than the q uadratic one, and still has the same i ntegral , thus l ead ing to the same approximation form ula (9) for the i ntegral J . We simply use the interpolation polynom ial which agrees with j(.r) at the poi nts x,,_ 1 , x,, x,+ 1 and for which '(x.) = f'(x,.) ; it has the form
(:r)
=
A 0 + A1( x - x, _1) + A2(x - x,_1 )( x - x ,)
+ A , ( x - x ,_1 )( x - x, ) ( x - x ,+ 1 ) .
Here t h e first t h ree terms represent t h e q uadratic interpolation poly nomial agreeing with f at the three points x , _1, x,, x,+ 1 . The constant A 3 has to be determined from the condition '(x,) = f'(x ,) . The last term
A 3(x - x, + h ) ( x - x ,)( x - x , - h )
=
A 3[( x - x ,)2 - h2] · [x - x.J
o b v ious ly is an odd function of x x, and therefore does not con tribute to the i ntegral between the lim i ts x , - h and x, + h . For the -
488 Numerical Methods
Ch. 6
error in the approximation to f we then have the estimate [cf. (40), p. 475, with n 3 and with two of the interpolation points coincident at x,]. f - 4> = - (x - x,_1)(x - x,)2 (x - x,+l )f 141 ( ; ). =
1
4!
This yields for the error i n the computation o f J, + 1,+1 the estimate h6
-
90 M
,,
and hence for the total error the estimate n
h6
2 90 M,
=
h4
1 80
( b - a ) M, .
Naturally, we may attain h igher accuracy by approximation of the functionf(x) i n a strip by a polynomial of a still higher order.
Examples. We apply these methods to the calculation of log, 2 =
1
2 dx
I
-
X
.
Dividing the interval I � x � 2 into ten parts of length h = using the trapezoidal rule (6), we obtain
= 1 .1 = 1 .2 x3 = 1 .3 x, = 1 .4 x6 = 1 .5 x. = 1 .6 x 1 = 1 .7 x8 = 1 .8 X9 = 1 . 9 x1
x2
fl
1Jc0 ,
and
0.90909 0.83333 = 0.76923 J. = 0.7 1 429 !6 = 0.66667 J. = 0.62500 !1 = 0.58824 fs = 0.55556 fg = 0.52632 !2 !3
=
=
Sum 6. 1 8773 x0 = X 1o =
Since the graph of to wa rds the x-axis,
1 .0 2.0
t/0 = 0 . 5 lf10 = 0.25
6. 93773 · lo
log, 2 "" 0.69377.
the integrand function has its convex side turned this value is too large.
Computation
Sec. 6. 1
Using the tangent r ule ( 7 ) we have
x0 x1 x2 x3 x4 ·�'s
X6
.r7
X8
Xg
+ + + + + + + + + +
�lz = 1 .05 � lz = 1 . 1 5 � h = 1 . 25 �h = 1 . 35 �lz = 1 .45 �h = 1 . 55 � lz = 1 . 65 � h = 1 . 75 � h = 1.!! 5 �lz = 1 .95
/1 ' 2 = /3;2 = /5 2 = /7 12 = /9/2 = /1 1 ;2 = ./;3/2 = ./;5;2 = j;, 2 = /19 2 =
�
489
0.95238 0. 86957 0.80000 0. 74074 0.68966 0.645 1 6 0.60606 0.57 1 43 0.54054 0. 5 1 2S2 6.92S36 ·
l og,. 2
of Integrals
0.692S4,
t 'u
w h ich, owing to the convexity of the curve, i s too s m a l l . For t h e same subdivision we obtain a much m o re precise res u l t using Si mpson's rule (S) . We have
X! = J . l x3 = 1 . 3 x5 = 1 . 5 x, = 1 . 7 Xg = 1 .9
fl /3 Is f, 19
= = = = =
0.90909 0. 76923 0.66667 0.58824 0.52632
Sum 3 .45955 4 ·
1 3 . 83820
x2 x4 x6 x8
= = = =
1 .2 1 .4 1 .6 1 .8
/2 f, Is /8
= = = =
0.83333 0.7 1 429 0. 62500 0.55556
Sum 2.728 1 8 2 ·
x0 = 1 .0 X10 = 2.0 In
5 . 45636 1 3. 83 820 .fo = 1 .0 .f; o = 0 . 5
20. 79456 · 3\,
l og, """ 0.693 1 5 . reality
loge 2 = 0.693 1 47 . . . .
490 Numerical Methods
6.2
Ch. 6
Other Examples of Numerical Methods
a. The "Calculus of Errors"
The "calculus of errors" is simply a numerical application of the basic fact of differential calculus : a functionj(x) which is differentiable a sufficient number of times can be represented in the neighborhood of a point by a linear function with an error of higher than the first order, by a quadratic function with an error of higher than the second order, and so on. Consider the linear approximation to a function y = f(x) . If y + f:).y = f(x + /:).x) = f(x + h), we have by Taylor's theorem /:).y = hf'(x) +
� f"W, 2
where ; = x + ()h (0 < () < 1 ) is an intermediate value which need not be more precisely known. If h = /:).x is small, we obtain the practical approximation /:).y ""' hf '(x).
Thus we replace the difference quotient by the derivative to which it is approximately equal, and the i ncrement of y by the approximately equal linear expression in h . This simple fact i s used for numerical purposes in the following way. Suppose two physical quantities x and y are related by y = f(x) . We then ask what effect an i naccuracy in the measurement of x has on the determi nation of y. If i nstead of the "true" value x we use the in accurate value x + h, then the corresponding value of y differs from the true value y = j(x) by the amount /:).y = j(x + h) - j(x). The error is therefore given approximately by the above relation. We illustrate the usefulness of such linear approximations by examples. Examples. (a) In a triangle ABC (cf. Fig. 6.3) suppose that the sides b and c are measured accurately, whereas the angle tx = x is only measured to within an error J /:).xl < tl . What is the corresponding error in the value of the third side y = a = ,j b2 + c2 - 2bc cos tx ? We have /).a ""' (be sin ex /:).ot)/a ; the percentage error is therefore
1 00 /).a a
100 be a2
. .._ -- ""' -- Sill ex ucx.
Sec. 6.2
Other Examples of Numerical Methods
I n the special case when b = 400 meters, c we have y = a = 458.2576 meters, so that
.. ua
�
200000 458 . 2576
1 X !
"
=
we find that at worst
X
,
IX =
60°,
,-3 ..
If �oc can be measured to within I 0 seconds � oc = 1 0 " = 4846
500 meters and
49 1
u cx .
of arc,
that is, if
I 0-8 radians,
�a """' 1 . 83 em ;
thus the error i s at most about 0.004 � � · B'
b' = b I
I 6
I
/
b a
Figure 6.3
(b) The following example illustrates the usefulness of the lineari zation for physical problems. It i s known experimentally that i f a metal rod has length /0 at tem perature t0, then at temperature t its length will be I = /0( 1 + oc(t t0)), where IX depends only on t0 and the material of which the rod is com posed . If now a pendulum clock keeps correct time at temperature 10, how many seconds will it Jose per day if the temperature rises to t1 ? For the period T(l) of oscillation we have (see p. 4 1 1 ) -
T(l)
= 2Tr
hence dT
dl
=
A
g
;
Tr
.Jig .
If the change of length is �/, the corresponding change in the period of oscillation is
492
Numerical Methods
Ch. 6
where /1 = /0( 1 + cx(/1 - /0)) and 11/ = cx/0(11 - 10). This is the time lost per oscillation. The time lost per second is tl Tj T ""' M/210 ; hence i n one day the clock loses 43,200 /1/j/0 = 43,200 cx(/1 - 10) seconds. I n this case and i n many other cases where the function under con sideration is a product of several factors, we can simplify the calculation by taking the logarithms of both sides before differentiating. In this example we have log T = log 2 7T - ! log g + -� log / ; l
differentiating, we have
T
l dT dl 21
- -
Replacing dTjdl b y 11 Tj/11 gives 11 T
111 21 in agreement with the preced ing result. T
*b. Calculation of
1T
A different example, using special artificial devices, is classical, although perhaps made obsolete by modern computers . Leibnitz's series "/4 = I - � + � - �- + · · · [Eq . (7), Section 5.2, p. 445], using the series for the inverse tangent, is not sui table for the calculation of " • because of the extreme s lowness of i ts convergence. We may, however, calculate " with comparative ease by the fol l owing artifice . If, in the addition theorem for the tangent, tan ( "' + {3)
=
I
tan " + tan {i -
tan o: tan (i '
we introduce the i nverse functions o: = arc tan 11, {3 = arc tan the fo rm u l a 11 + v . . arc tan 11 + arc tan v = arc tan I - uv
(
,
)
1'.
we obtain
11 and v so that (11 + v)/( 1 - 11v) = I , we obtain the value "/4 on the right-hand side, and if u and v are small numbers we can easily calculate the left-hand side by means of known series. If, for example, we put 11 = !, v = !. as Eu ler did, we obtai n
No w cho os ing
(9)
�=
arc
l + .t
If we further notice that 1
_
:
21
tan ! + arc t2.n ! .
=2 I
,
Ser. 6.2
493
Other Examples of Numeriral Methods
we have arc tan ! arc tan k + arc tan } that by (9), 4 2 arc tan � + arc tan � . Usi n this mu l Ve calculated the number " to pl ces e u a i n (g + i )/( 1 By ea :\,. we arc t n! = tan 1 + tan � or �· + tan �· + arc 4 =2 " by means of the series This ex nsi n extremely useful for for if we su st u e for value arc tan x = x Xlf3 + �. �. or �. w e obt i e r e of accuracy, since the terms diminish rapidly. e artificial these The reader who is ot especiall y te e te of the py nc p e . m i ht be satisfied with an =
, so
=
1T
- •'0 ) =
ga a, for n s of the q t o
g m
a
1T
pa
arc
1 40 a . further obtain
arc
2 arc tan � .
tan
arc
calculating b it t x5f5 - · · · ; a n with b u t few terms a high d g e
is
o
-
in r s d in
n
manipulations,
t
skilfu l ,
understanding
g
'" the
ri
i l
*c. Calculation of Logarithms
For the n umerical calculation of logarithms we t ransform the logarithmic series (Eq . (5), p. 444 ]
where 0 <
I
"-' I og -1 - .r
:r
i nto the series =
X
= .r
.rs :r3 + - + - + . . ·
< I , by the substitution
l + .r = _L__ I
log p
+
- X
p" - I
.
3
5
'2 p " - I
.r = ---
� l og (p - I ) + � l og ( p + I ) +
'2 p 2
I
_
+
1
I + . . . 3(2p 2 - 1 )3
•
where 2p2 - I > 1 or p2 > I . I f p is an i nteger and p + I can be resolved i nto smaller integral factors (for example, if p + 1 is even), this last series expresses the logarithm of p by the logarit h m s of smaller integers plus a series whose terms d i m i nish very rapidly and whose sum can therefore be calculated acc urately enough by use of only a few terms. From this series we can therefore calculate successively the l ogarithms of any prime number, and hence of any n u m ber, provided we have already calculated the val ue of l og 2 (for example, by its i ntegral representation, as on p. 489).
494
Numerical Methods
Ch. 6
The accuracy of this determination of log p can be estimated more easily by means of the geometric series than from the general form ula for the remainder. For the remainder R. of the series, that is, the sum of all the terms following the term l /n(2p2 - 1 ) ", we have Rn <
I
(n + 2)( 2p" o
I
-
I )"+2
(1 +
(n + 2)(2p2 - 1 )" (2p" .
- I
I
( 2 p- - I ) 4
I )"
,, +
( 2 p2 1 1 )2
o
+ ···
).
'
1
and this formula immediately gives the required esti mate of the error. Let us for example calculate log, 7 (under the assumption that log 2 and log 3 have already been found numerically), using the first four terms of the series . We have p = 7, 2p2 I = 97, log 7 = 2 log 2 + ·� log 3
0\ 2
� 0 . 0 1 030928,
3
-
+ 9? +
.
I 97
I
3
�
� log 3
log 2 "' 1 . 3 8629436,
hence
3
.
I 97
,
+
0. 000000 3 7 ,
�
0. 549306 1 4 ;
log, 7 "' 1 . 9459 1 0 1 5 . Estimation o f the error gives
R 11 <
I 5
·
9 7' X 9 72
-
I
I
<
I 36 X 1 09 •
only given to within an error of 5 x t o-•, so that the last place in the com puted value of log 7 might be wrong by 2. As a matter of fact, however, the last place is also correct.
However, we note that each of the four numbers which we have added is
6.3
Numerica l Solution of Equations
We add some remarks about the numerical solution of the equation j(x) = 0, where f(x) need not be a polynomial.1 We start with some tentative fi rst value x0 of one of the roots and then improve this approxi mation. How the fi rst approximation for the root is chosen and how good that approximation is may be left open. We may, for example, take a rough guess, or better, obtain a first approximation from the 1
We a re, of course, concerned only with the determination of real roots of ((x) = 0.
Sec. 6.3
Numerical Solution of Equations 495
graph of the funct.ion y = f(x), whose intersection with the x-axis indicates the required root. Then we try to improve the approximation by a process or mapping which takes the value x0 into a "second approximation," and repeat this process. Solving the equationf(x) = 0 n umerically consists in carrying out such successive approximations repeatedly (or as one says "iter ating" the process) with the expectation that the iterated values xh x2 , . . . , x, converge satisfactorily to the root �. We shall consider various such proced ures and briefly discuss their accuracy. a.
Newton's Method
Description of Method. Newton's iterative proced ure is based on the fundamental pri nciple of the differential calcul us-the replacing of a curve by a tangent in the i mmediate neigh borhood of the point of con tact. Starting from a first approximate value x0 for a root � of the equation j(x) = 0 we consider the point on the graph of the function y = j(x) whose coordi nates are x = x0, y = j(x0). To fi nd a better approxi mation for the i n tersection � of the curve with the x-axis we determine the point x1 w here the tangent at the point x = x0 , y = f(x0) i ntersects the x-axis. The abscissa x1 of this intersection represents a new and, under certain circumstances, a better approximation than x0 to the required root � of the equation. Figure 6.4 at once gives
hence the new approximation ( 1 0)
x1
=
x 0 - f(.x.,) .- . f'(Xo)
Starting with x1 as an approx imation, we repeat the process to find x2 = x1 - f(x1)/f '(x1 ) and so on . The usefulness of this process depends essentially on the nature of the curve y = f(x). In the sit uation indicated i n Fig. 6.4 the successive approxi mations x n converge with increasing accuracy to the required root �. However, Fig. 6.5 shows that with a plausible choice of the original value x0, our construction need not converge to the required root at all. It is therefore necessary to examine i n general the circum stances under which Newton's method furnishes useful approximations to the solution of the equation.
496
Numerical Methods
Ch. 6
y
Figure 6.4
Newton's method of app roximation.
y
Figure 6.5
Quadratic Convergence of Newton's Method Assuming that in a sufficiently wide interval about the root � the second derivative f"(x) is not "too large" and the first derivative f'(x) not "to small", the main fact concerning Newton's approximation is that the successive "errors"
Sec. 6.3
Numerical Solution of Equations
497
converge to zero quapratically i n the sense that l hn+l l � p.h"2 with a fixed constant p.. This indicates an extremely rapid rate of convergence ; if we write the i nequality in the form l hn+IP. I � lhnp. l 2 it implies, for example, that when lhnfl l < w- m we have lhn + l!'- 1 < J 0-2m, that is, the number of "significant digits" in p.x, is doubled at each step. The proof of the quadratic convergence is i mmediate. From the relations x ,. + 1 = xn - j(xn)/ j '(xn ) and /(�) = 0 we find that h n+ l
=
� - Xn+ l = �
By Taylor's formula /(�) - f(x,)
=
-
f xn X n - rm - ( ) ·
(� - x , )j '(x 11) + H �
where 'I) lies between � and x"' H ence ( 1 1)
f ' ( x n) -
·
x ,,)2j"('l)),
h n +l = - f"('ry) h 2 2f '(x,.) ,.
To establish convergence we assume that x, b elongs already to a fixed i nterval � - 6 < x < $ + 6 i n which 1 /" 1 has the maximum value M2 , I J' I the positive minimum value m1, and for which 6 is so small that � bM /m1 < I . Putti ng p. = � M /m1 we have p.IJ < I and 2 2 This inequality shows first of all that x,H 1 belongs again to the same 6-neighborhood of � so that the argument can be repeated. Thus, i f only x0 lies i n the b -neighborhood of �. all subseq uent x, will do the same. From l h r�+tl � f.!b l h , l it follows then that lh, 1 1 1 � (f.lr5)"+ 1 l h 0 l , which impl ies that h11 --+ 0 or that x, --+ � ; m oreover, the q uadratic law of decrease l hn+tl � f.l l h n l 2 will hold for the errors. lt i s clear then that Newton's method will provide us with a sequence x, which certainly converges toward the solution � provided !' and r e x is t , and are con tinuous near �. that J 'W ;t. 0, and that x0 is already sufficiently close to r The q uadratic character of the approxi mation is often a decided advantage of Newton's method over others (see p. 503). *b. The Rule of False Position
Newton's method is the limiting case of an older method, the "rule of false position," in which the secant appears i n place of the tangent. Let us assume that we know two points (x0, y0 ) and (xi> y 1) in the neighborhood of the required intersection with the x-ax is. If we replace the curve by the secant joining these two points, the intersection of this
498 Numerical Methods
Ch. 6
y
Figure 6.6
The rule of false position.
secant with the x-axis can be an improved approximation to the re quired root1 of the equation. For the abscissa � of the point of inter section, we have (Fig. 6.6)
(12 ) which leads to
or
� - Xo = � - XI f(xo) f( x l)
x0j( x 1) - x,f(x0) �= f(xl) - f( xo) xof(xl) - xof( xo) + Xof(xo) - xtf(xo) f( x l) - f( xo)
( 1 3) This formula, which determines the further approximation $ from x0 and x1, constitutes the rule of false position. It is useful if one value of the function is positive and the other negative, say as in Fig. 6.6, where y0 > 0 and y 1 < 0. 1
This amounts essentially to linear interpolation applied to the inverse function.
Numerical Solution of Equations 499
Sec. 6.3
The approximation formula of Newton results as a limiting case for x 1 ---+ x0, for the deno minator of the second term o n the right-hand side of formula { 1 3) tends to J'(x0) as x1 tends to x0• Although the rule of false position may be considered more elemen tary than Newton's method, the latter has the great convenience of requiri ng only one value of x as initial approximation instead of two values. c. The Method of Iteration
The Iteration Scheme. We now turn to a far-reaching scheme for solving equations written i n the form x = cp(x),
where cp is a continuous function with a continuous derivative. The solution of equations of the form j(x) = 0 can be reduced to that of x = cp(x) if we put cp(x) = x - c(x)j(x) where c(x) is any function different from zero. 1 In the particularly suggestive method of iteration we begin again with a suitably chosen i nitial approximating val ue x0 and then determine a sequence x1, x2, x3, of values by the conditions •
xn+ l
•
•
=
cp(X11),
n =
0, I, 2, . . . .
If this "iteration" sequence X11 converges to a limit �. then � = cp(�) is a solution of our equation, since then lim X11 +1 = � and l i m cp(x11) = cp (�) n-+ n-� oo because of the continuity of the function cp. ao
Convergence. The sequence of values x" in the iteration process con verges to a solution u nder a very general assumption : I f the first ap proximatio n x0 lies in an interval 2 J about the solution �. in which
[ cp'(x) [ < q
with a constant q < I , then X11 converges to � For supposing that x0 lies i n J, we have
x1 - � = cp(x0) - cp( �) .
Sometimes called the method of successive approximation . The method is used in many different mathematical contexts for solving equations of one kind or other. ' A lthough ; is unknown, we can very often determine such an interval a priori. 1
500 Numerical Methods
Ch. 6
By the mean value theorem, the right-hand side of this equation equals (x0 - �)4/(i), where i lies in J. Thus by our assumption lx1 - �� � q lxo - �1 .
so that x1 belongs to J, and then also In general, we obtain
since q " -->- 0 as n --.. CIJ , our assertion is proved. We see, moreover, from the preceding, that the iteration sequence xn does not converge when c/)(x) > I in an i nterval about � ; if lc/>'WI = I we cannot make a general statement. Attrac ting and Repelling Fixed Points It is useful to consider the iteration process in terms of a mapping or transformation. The function y = cp(x) represents a transformation which maps a point x on the number axis into an image point y of this number axis (see p. 20). The solution � is then a point not changed by the transformation cp, a so-called fixed point, and the problem is thus one of finding a fixed point of the mapping ; this problem is solvable by iteration when 1 4/(�)1 � q < I as we have seen. The mapping y = cp(x) of the neighborhood of the root or fixed point � has, for lf(x)l < q < 1 , the property of being contracting, that is, diminishing the distance of the original from the fixed point. Such fixed points of contracting mappings are called attracting fixed points. Their construction by iteration converges as the terms of a geometric series with the quotient q. If the root �. or the corresponding fixed point of our transformation is in an interval in which lcp'(x)l > r, where r is a constant larger than 1 , the transformation is expanding, the iteration process diverges, and the fixed point is called repelling. If at the fixed point we have l cp ' (�)l = 1 , no general statements con cerning the convergence of the iterations can be made ; such fixed points are sometimes called indifferent. The following observation should be stressed : a fixed point � of the mapping cp is automatically also a fixed point for 1p, the inverse mapping : � = 1p{g} . I f WWI > I in a neighborhood of a root � and x = 1p(y) is the inverse function of cp, then l'f/J'WI < I . Thus � is an attracting fixed point for this i nverse mapping and it is possible to replace the originally divergent iteration scheme by a convergent one ,
Sec. 6.3
Numerical Solution of Equations 501 y
X
Figure
6.7 Intersectio n (;, ;) of the curves
y =
tan
x
and
y = x.
for the inverse mapping. As an example we consider the equation x = tan x.
I t is clear from the graphs of the functions y = x and y = tan x that these i ntersect somewhere i n the i nterval 7T < x < ·� 7T and that our equation will have a root � in that i nterval (Fig. 6. 7). Since
d tan x = _1_ > 1 , dx cos 2 x the iteration procedure with any point x0 i n the i nterval does not con verge. H o wever, we obtain a convergent iteration sequence if we write the eq uation in the i nverse form (using the notation arc tan x for the principal branch), x = arc tan x + 7T . Since here d 1 - arc tan x = < 1, 1 + x2 dx ---
the sequence defined by converges to �.
xn+l
= arc tan xn +
7T
and, say, x0 =
7T,
502
Numerical Methods
Ch. 6
d. Iterations and Newton's Procedure
As mentioned before the solution of an equation of the form = 0 can be reduced to that of the form x = ,P(x) if we choose for any expression of the form
f(x)
,P(x) = x - c(x)f(x) where c(x) is a nonvanishing function. If we want to solve the resulting equation x = ,P(x) by iteration we have to make sure by a suitable choice of c(x) that the fixed point � of the mapping is "attractive", that is, that I¢'(�) ! < I . Now for the solution � off(�) = 0 we have '(�) = I -
c
'
WJW - c(�)j'(�) = 1 - c(�)j'(�).
The simplest choice is to take for c(x) the expression 1 /f'{x). Then certainly WWI = 0 < I . This choice of c(x) leads to the iteration sequence X
A-(xn) - x n - f( xn ) n+l - '!' J' (xn ) '
which is j ust the sequence of approximations ( 1 0), p. 495, in Newton's method. For the error x, - � = h,. we have the estimate where q is the maximum of 1'(x)l in the interval with end points � and
x n . Since here
x x ' ( x) = f( )j"( ) / ' "(x )
and j(x) = j(x) - JW = f '(YJ) (x - �) . we see that q itself is of the order of h n , and thus confirm again the quadratic character of the approximation in Newton's method. Another sim ple choice for c(x) is to take the constant value 1 /f'(x0), leading to the recursion formula
x xn+l = ,P( x,) = xn - f( ,) . f ' ( x0)
Here ¢'(�) = I - f'(�)ff'(x0) . If f' is continuous and different from zero, we will have an attractive fixed point � if our initial approximation x0 is already so close to the solution � that
Numerical Solut ion of Equations 503
Sec. 6.3
This iteration sequence is somewhat simpler than the one used i n · Newton's method ; however, convergence will b e much slower, like that for a geometric progression, as is the case with most iteration schemes.
Examples. As an example we consider the cubic equation
f(x)
=
x3 - 2x - 5
=
0.
Since /(2) = - 1 < 0, /(3) = 1 6 > 0, a root ; certainly exists in the interval 2 < x < 3. Since, moreover, f'(x) = 3x2 - 2 > 3{2)2 -2 > 0, the interval contains only one root. By Newton's method we find starting with the approximation x0 = 2 successively
-I fx = 2.1 f (xl) = 0.061 x - ( o) = 2 0 f'(x0) 3(2)2 - 2 ' 0 · 061 = fx) X = X1 - ( 1 = 2. 1 2.094 568. 2 3 ( 2. 1 )2 - 2 f ' ( x1) Since /(2. 1 ) > O , f(2) < 0, the root ; lies between 2 and 2. 1 . In the i nterval 1 .9 < x < 2.2, and a fortiori then in the interval ; - 0. 1 < x < ; + 0. 1 , we have the estimates ! f" (x) ! = ! 6x ! < 6{2.2) = 1 3 . 2, j'(x) = 3x2 -2 > 3(1 .9)2 - 2 = 8.83. I t follows [see ( I I ) , p. 497 ] that x1
=
< 1 ; - X n+ll -
� l xn - ; 1 2 2(10. 83)
<
0 . 7 5 1 xn - ; 1 2
provided l x,. - ;1 < 0. 1 . Since l x0 - ; 1 = 1 ; - 2 1 < 0 . 1 , we find suc cessively l x1 - ;1 < (0. 75)(0. 1 )2 = 0.0061 l x2 - ; 1 < (0.61 )(0.006 1)2 < 0.000042.
If this degree of approximation is not sufficient, we obtain a further approximation x3 with an error < {0. 75)(0.000042)2 < 0.000 000 001 3. All x11 after x0 must be larger than ; as is obvious from the fact that f' and f" are positive, which implies that h n + l = -J"(YJ )hn2/2J' (x n) < 0 . Applying instead the rule of false position [( 1 3), p. 498] to the values x0, x1 we find for the i ntersection ; with the x-axis of the secant joining the points (x0, f(x0)) and (x1,f(x1))
;
=
x x X o _ f( o)( l Xo) f(x 1 ) - f(x o)
=
2 .09 425 . . . .
Numerical Methods
504
Ch. 6
Since the curve is convex in the interval in question, the secant lies above the curve and the approximation ¢ must be less than the root ¢. As a second example, let us solve the equation f(x) = x log 1 0 x - 2 = 0.
,
We have /(3) = -0.6 and f(4) = + 0 . 4 and therefore use x0 = 3.5 as a first approximation Using ten-digit logarithmic tables we obtain the successive approxi m ations
.
x0
=
3.5 ,
x1 = 3.598,
x 2 = 3.5972849,
x3 = 3.5972850235.
Appendix
* A.l
Stirling's Formula
In many applications, particularly in statistics and the theory of probability, we find it necessary to have a simple approximation to n ! as an elementary functi on of n . Such an expression is given by the follow ing theorem, which bears the name of its discoverer, Stirling (see also Chapter 8, p . 630) . As n
---+ oo ,
( 1 4) more exactly,
( 14a)
.J2.,. n "+ll 2e
__
,
<
n!
(
< .J2; n "+112e- n 1 +
�) .
4
In other words, the expressions n ! and .J2.,. n"+1 12e-" differ only by a small percentage when the value of n is large-as we say, the two expressions are asymptotically equa l-and at the same time the factor 1 + 1 /4n gives us an estimate of the degree of accuracy of the approxi mation. We are led to this remarkable formula if we attempt to evaluate the area under the curve y = log x. 1 By integration (p. 276) we find that A n , the exact area under this curve between the ordinates x = 1 and 8, p.
1 The method used here is a special instance of the Euler MacLaurin formula which
will
be discussed in Chapter
624.
Sec. A . J
x
= n,
(15 )
Stirling's Formula
505
is given by An =
i"
log x dx
=
x log x - x
��
= n
log
n
- n + 1.
If, however, we estimate the area by the trapezoid rule, erecting ordinates at x = I , x = 2, . . . , x = n as i n Fig. 6.8, we obtain an approxi mate value Tn for the area [cf. (6), p . 483] ( 1 6)
T, = log 2 + log 3 + · = log 11 ! - ! log 11.
· ·
+ log (n - I ) + ! log n
If we make the reasonable assumption that A, and Tn are of the same order of magnitude, we find at once that 11 ! and 11"+ll 2e- " are of the y
Figure 6.8
same order of magnitude, which is essentially what is stated in Stirling's formula. To make this argument precise, we first show that the difference an = A n - T" is bounded, from which it will immediately follow that Tn = A n(! - an/A 11) is of the same order of magnitude as A n. The difference ak+1 - ak is the difference between the area under the curve and the area under the secant in the strip k < x < k + 1 . Since the curve is concave and lies above the secant, ak+ I - ak is positive, and an (an - a11_1) + (an-l - an_2) + + (a2 - a1) + a1 is mono tonic increasing. Moreover, the difference ak+ 1 - ak is clearly less (cf. Fig. 6.9) than the difference between the area under the tangent =
.
.
.
506
Numerical Methods
Ch. 6 y = log x
k
k
k
+ 1/2
Figure 6.9
+
1
at x = k + ! and the area under the secant ; hence we have the i n equality aw t log (k + 1 ) ak < log (k + ! l i log k -
=
( ..!..) ( _l__) -
t log t
+
< � J og 1 + "
2k
2k
[ -1-tlJ [ -1- ] .
-
-
-
t Jog 1 +
2(k +
l log t +
2(k + 1)
Adding these i nequalities for k = I, 2, ,n ! we find that all the terms on the right-hand side except two will cancel out, and (since a1 = 0), we have a n < l log t
-
.
.
.
( J
} log 1 +
1
2
-
,
< i log l
Since a. i s bounded, and in addition monotonic i ncreasi ng it tends to a limit a as n ...... oo . Our i nequality for a k+ l a k now gives us a
-
an
=
I (ak+1 - ak) < ! log
k� n
Since by definition A, log n ! or, writing
G( n
=
= el-an,
t +
2n
T. = a., we have from ( 1 5), ( 1 6),
-
I
( ..!..)' . -
-
a, + (n + D log n
-
n,
The sequence rx n i s monotonic decreasing and tends to the limit rx = e 1-a ; hence 1 <�= G(
=
ea-an
J1 2n
<
n e lli 2 1 Iog IH1/2 )
1 4n
1 + - < 1 +- .
Problems 501
Hence we have
( I 41J +
cx. n n+;/2e-" < 11 ! < cx.n n+li 2e-"
It only remains for us to find the actual value of the limit we make use of the formula (80) of Chapter 3, p. 282 : . (n ! ) 22 2 n J7T = i. 1 m ( 2 ' n ) . Jn n - oo _
a.
Here
.
Replacing n ! by cx. ,. n " H 1 2 r" and (2n) ! by cx. 2 ,.22"+1/2 n2"+1/ 2 e-2 ", we i mmediately obtain (J. n 2 J7T = lim -_ n --+ oc_, CX. 2 n J2 cx. 2
rxJ 2 '
from which ex. = J27T . The proof of Stirling's formula i s thus complete. In addition to i ts theoretical interest, Stirling's formula is a very useful tool for the n umerical calculation of n ! when n is large. I nstead of m ultiplying together a large number of i ntegers, we have merely to calculate Stirling's expression by means of logarithms which involves far fewer operations. Thus for n = 1 0 we obtain the value 3 598696 for Stirling's expression (using seven-figure tables), whereas the exact value of I 0! is 3628800 . The percentage error i s barely i !;; . PROBLEMS SECTION 6. 1 , page 482
1. Prove if f"(x) ::?, 0, that the trapezoid rule yields a greater value and the tangent rule a lesser value than the exact integral of f. 2. Estimate the value h = (b - a)/n needed for a calculation by Simpson's rule accurate to p decimal places of
(a) log 2 =
1
2 1
1 X
I, b) rr = 4
d.r,
il
1
--
o l + :r2
dx.
3. Estimate in terms of k and s (k < I and s < 1 ), the number of points needed to calculate within an error • the elliptic integral
u(s) =
a
(
Jo
'
v{I
-
dx x2)(! - k2x2) .
4. Let f(x) be a continuous function on the interval :x s x s :x + h, with uniformly bounded derivative : l['(x) l s M 1 for M 1 a constant. Prove
SOB
Numerical Methods
that for any fixed point $, "' :5: ; ::::;; "' + h , the estimate
5. Calculate
f."'
I r+h
f(:r) dx - hf
:5:
I
Ch. 6
M�h·
e_,, d.r n umerically to within 1 / 1 00 .
SECTION 6.2, page 490 1 . The period of a pendulum is given by
where I is the length of the pendulum. If the pendulum drives a clock which gains a minute per day determine the necessary correction in /. 2. To measure the height of a hill, a tower 100 meters high on top of the hill is observed from the plai n. The angle of elevation of the base of the tower is 42" and the tower itself subtends an angle of 6'. What are the limits of error in the determination of the height i f the angle 42' is subject to an error of I o ? SECTION 6.3, page 494 1. (a) To solve the equation "' = j(x), show how best to choose the constant a so that the iteration scheme ,,.,,+ 1
=
+ a [x,
x,.
-
j(xc ) ]
converges as rapidly as possi ble in the neighborhood of the solution. (b) Apply this method to solve the equation for VA, :r
=
A -
a:
.
(c) Show if A z I that the number of accurate decimal places is at least doubled at each step of the iteration scheme obtained in (b). 2. (a) Show how best to choose a polynomial
g (;c)
=
so that the iteration scheme for '1A, ;r,_ �1
=
;r,
a
+
+ g(->:,)
b.t·2
(r, - �)
converges most rapidly in the neighborhood o f the solution. (b) Estimate the rapidity of convergence. (c) Show how to further improve the convergence by suitable choices of polynomials g (.r) which are of higher degree.
Problems 3. Investigate suitable schemes of the type of Problems calculation of ,· A.
509
and 2 for the
S ECTION A. l , page 504
.
1 . Prove that hm 11 - co
\Vii! I e n n +l/ 2
l
-- =
*2. By considering
1/ 2
-
log ( !X + x) dx,
O".
> 0, show that
!1.(!1. + I ) · · · (!X + n) = ann ! na, where an is bounded below by a positive number. Show that a. is mono· tonically decreasing for sufficiently large values of n. [The limit of an as n oo is 1 / l'(!X) .] nl ! n 2 ! · · · n z ! . . . , wh ere n 1 + n 2 + 3 F m d an approx1mate expresswn for I og n ., , = n. +n 1 n is 4. Show that the coefficient of x i n the binomial expansion of __,.
·
·
·
•
asymptotically given by
I V1rn
--=
VI -
.
X
7 Infinite Sums and Products
The geometric series, Taylor's series, and a number of examples previously discussed in this book, suggest that we may well study those limiting processes of analysis which i nvolve the summation of i'!,finite series from a more general point of view. In principle, any limiting value S = Jim
s,
can be written as an infinite series ; we need only p ut for n > 1 and a, = s1 to obtain
a = s" ,
- s n-I
and the value S thus appears as the limit of s n o the sum of n terms, as n i ncreases. We express this fact by saying that S is the "sum of the infinite series" a1
+ a2 + as + · · · .
Such an "infinite sum" i s simply a way of representing a limit where each successive approximation is found from the preceding by adding one more term . Thus the expression of a number as a decimal is in principle merely the representation of a number a in the form of an infinite series a = a 1 + a 2 + a s + · · · where, if 0 � a � I , the term a n is replaced by oc , x 1 0-" and oc, is an integer between 0 and 9 inclusive. Since every limiting value can be written in the form of an infinite series, a special study of series may seem superfluous. However, very often it happens that limiting values occur naturally in the form of such infinite series which exhibit particularly simple laws of formation. ,
510
Sec. 7. 1
The Concepts of Convergence and Divergence 5 1 1
Not every series has an easily recognizable law of formation. For ' example, the number 7T can certainly be represented as a decimal (which i s a series I:c. I O- •), yet we know no simple law enabling us to state the value of an arbitrary digit, say the 7000t h, of this decimal . If, however, we consider the Leibnitz-Gregory series for ?T/4 i nstead, we have an expression with a perfectly clear general law of formation [see (7), p. 445] . Analogous to infinite series, i n which the approximations to the limit are formed by repeated addition of new terms, are infinite products, in which the approximations to the limit arise from repeated m ulti plication by new factors. We shall not go deeply into the general theory of i nfinite products, however ; the principal subject of this chapter and of Chapter 8 will be infinite series. 7.1
The Concepts of Convergence and Divergence
a. Basic Concepts
Cauchy's Convergence Criterion. We consider an infinite series with the "general term" an ; the series 1 i s then of the form a 1 + a2 + · · ·
= �;a .. 00
V=l
T h e symbol on t h e right with t h e summation sign i s merely an abbre viated way of writing the expression on the left. If as n i ncreases, the nth partial sum n
'a s = a1 + a2 + · · · + an = k v
approaches a limit
n
V=l
S = l i m Sn
we say that the series is convergent ; otherwise we say that it is divergent. I n the first case we call S the sum of the series. We have already encountered many examples of convergent series ; for i nstance, the geometric series I + q + q2 + · · · , which converges to the sum 1 /( 1 q) when l q l < I , the series for log 2, the series for e, and others. In the language of infinite series, Cauchy's convergence test (cf. Chapter I , p. 75 ) is expressed as follows : -
1 For formal reasons we include the possibility that certain of the n umbers a. may be zero. If all terms from an index N onward (that is, when n > N) vanish, we speak of a terminating series.
512
Infinite Sums and Products
Ch.
7
A necessary and sufficient condition for the convergence of a series is that the number (1)
(m > n), becomes arbitrarily small if m and n are chosen sufficiently large. In other words : A series converges if, and only if, the following con dition is fulfilled: for a given positive number e, it is possible to choose a n index N = N(e), in such a way that the above expression Ism - s,l is less than e , provided only that m > N and n > N. q
We can i llustrate the convergence test by the geometric series for !- If we choose • = f0 , we need only take N = 4. For
=
is m
-
snl
=
2n + · · · +
I
l
�
=2
-1
G
+
2 m- 1
� + · · · + z;-n) <
2n- l <
and
I
if n
I
lO
I
2n-1
> 4.
If we choose • equal to Ito• it is sufficient t o take 7 a s the corresponding value of N, as may easily be verified.
Obviously, it is a necessary condition for the convergence of a series that lim a n = 0. Otherwise, the convergence criterion certainly cannot be fulfilled for = n + 1 . But this necessary condition is by no means sufficient for convergence ; on the contrary, it is easy to find infinite series whose general term an approaches 0 .as n increases, but whose sum does not exist, since the partia l sum s,. i ncreases without limit as n increases.
m
Examples. An example is the series
1 l 1 1 + 7+ ... + -+ ... ;- + ; v2
vn
,3
,
the general term of which is l j ,J � . We immediately see that s,
>
�- +
vn
.
.
.
+
�-
vn
=
n /_
' n
=
,jrl.
The nth partial sum i ncreases beyond a!! bounds as therefore the series diverges.
n
increases, and
Sec. 7. 1
The Concepts of
Convergence and Divergence 5 1 3
T h e same is true fqr t h e classic example o f t h e harmonic series ! + - + - + - + · · · . 2 4 3
1
Here
a n-tl
+ . . . +
I a 2 n = -n+ I
I
1
+ . . . + -
I I >2n 2n
+ . . . + -
I I = 2n 2 -
.
Since n and m = 2n can be chosen to be as large as we please, the series diverges, for Cauchy's test i s not fulfilled ; i n fact, the nth partial sum obviously tends to infinity, since all the terms are positive. On the other hand, the series formed from the same numbers with alternating signs, I I I I ( - 1 )"-1 1 - - + - - - + - - + . . . + --- + . . . . n 2 3 4 5 converges [cf. (4) Chapter 5, p. 443], and has the sum log 2. l t is b y n o means true that i n every divergent series s , tends to + oo or - oo. Thus i n the series 1 - I + I - 1 + 1 +
we see that the partial sum s" has the values I and 0 alternately, and on acco unt of this oscillation backward and forward, neither approaches a definite l i m i t nor increases n umerically beyond all bounds. The following fact, although it is self-evident, is very important and should be noted . The convergence or divergence of a series is not changed by inserting a finite number of terms or by removing a finite number of terms. As far as convergence or divergence i s concerned, it does not matter in the least whether we begi n the series at the term a0, or a 1 , or a6, or any other term chosen arbitrarily. b. Absolute Convergence and Conditional Convergence
The harmonic series I + ! + l + ! · · · diverges, but if we change the sign of every other term the resulting series for log 2 converges. On the other hand, the geometric series I - q + q 2 - q3 + - · · · converges and has the sum I /(I + q), provided that 0 � q < I , and o n making all the signs plus we obtain the series 1 +
q
+
q2
+
qa
+ . . .
,
which is also convergent, having the sum 1 /( 1 - q).
S!4
Ch.
Infinite Sums and Products
7
Here there appears a distinction which we must examine. With a series whose terms are all positive there are Qnly two possible cases ; either it converges or the partial sum i ncreases beyond all bounds as n i ncreases. For the partial sums, be in g a monotonic i ncreasing seq uence, must converge if they remain bounded. Convergence occurs if the individual terms approach zero rapidly enough as n i ncreases ; on the other hand, divergence occurs if the terms do not approach zero at all or if they approach zero too sl owly . However, in series some terms of which are pos i t i ve and some n e gative it may be that the changes of sign bring about convergence, when too great an i ncrease in t h e pa r ti a l sums, due to the positive terms, is compen sa te d by the negative terms, so that as the fi nal result a definite l i m i t is approached. To understand the po s s ibi li ti es better we consider a series 2 av having ,
oo
v=l
positive and negative terms and form for comparison the series which has the same terms all with p o s i t ive signs, that is, 00
la t l + la , l + · · · = 2 l a v l · V=l
If this series converges, then for sufficiently large val ues of n a nd m > n, t h e expression l a nHI + l a n + 2 1 + · · · + l a ml w i l l certainly b e as small as we please ; because of t h e relation
l a n+l + · · · + a mi � l a n+d + · · · + l a ,l t h e expression o n t h e l e ft i s also arbitrarily s mal l a n d s o b y the Cauchy 00
,
test the original series 2 av converges. I n this case the original series is v=l
said to be absolutely convergent. J ts convergence is due to the absolute sma l l ne ss of its terms and does not depend on t h e changes in sign. I f, on the other hand, the series with the terms la,l diverges and the origi nal series still converges, we say that the original series is con ditionally convergent. Conditional convergence results from the terms of opposite signs compensating one another.
Leibnitz's Test. For cond itional convergence Leibnitz's conuergence test is frequently useful : If the terms of a series are of alternating sign and in addition their absolute values l an l tend monotonically to 0 (so that l an+ I I <. la, l ), the series 2 a, converges. [Example : Leibn;tz's series, (7), p. 445.] 00
\1=1
Sec. 7. 1
The Concepts of Conve�l{ence and Divergence 5 1 5
For t h e proof w e .assume that a 1 > 0 , which does not limit the generality of the argument, and write our series in the form
where all the terms b, are now positive, bn tends to zero, and the condition b, n � b, is satisfied. I f we bracket the terms together i n the two different ways b 1 - (b2 - b a) - (b4 - h5) - • • •
(b 1 - b2) + (b a - b , ) + { b 5 - b 6) + · · ·
and
"
we see at once that the partial sums s, = L a, satisfy the fol lowing 1 two relations s , � Sa � Ss � • • • � s2n+l � . . . s2 � s4 � s6 � • • • � s2 n � • • • •
'
On the other hand, s2, � s2n+l � s, and s2,+1 � s2n � s 2 . The odd partial sums s1, sa , . . . therefore form a monotonic decreasing sequence, Ss
Figure 7.1
Ss
s
Convergence of an alternating series.
which in no case falls below the val ue s2 ; hence this seq uence possesses a l i mit L (p. 7 3 ) . The even partial sums s2, s4, li kewise form a m onotonic increasing sequence whose terms in no case exceed the fixed number s1 , and therefore this sequence m ust h ave a limiting val ue L ' . Since t h e numbers s2 , a n d s2 ,+1 differ from one another only by the number b2,+1 wh ich approaches 0 as n increases, the limiting val ues L and L' are equal to one another. That is, the even and the odd partial sums approach the same limit, wh ich we now denote by S (cf. Fig. 7 . 1 ) . This, h owever, implies that our series i s convergent, as was asserted ; its sum i s S. •
.
•
*Abel's Test A test for conditional convergence that incl udes the Leibnitz test as a special case is Abel's convergence test. Let a 1 + a2 + be an infinite series whose partial sums sn = a 1 + · · · + a, are bounded independently of n. Let p1, p2, be a sequence of positive numbers decreasing monotonically to the value zero . Then the infinite series ·
•
(2)
•
•
·
·
Ch. 7
5 1 6 Infinite Sums and Products
converges. (For the special series a1 + a2 + · · · = + I - I + I - I + - · · we find that p1 - p2 + p3 converges, which is Leibnitz' test.) The proof follows if we apply Cauchy's test using "summation by parts" to estimate •
= =
•
·
•
IPn+tan+l + Pn+20n+2 + · · · + PmOmi
IPn+t(sn+l - Sn) + Pn+2(sn+2 - SnH) + · · · + Pm(Sm - Sm_, )J 1 -Pn+tSn + PmSm + (Pn+l - Pn+2JsnH + (Pn+ 2 - Pn1 s)SnH + · · · + (p ,._, - Pmlsm-t l
:::;; p,.+tM + PmM + (Pn+t - Pn+2 + Pn+2 - Pn+S + - · · · + Pm-t - p,. ) M
=
2p.HM,
where M is a bound for t he l sil ; since Pn+I ...... 0 the convergence of the series (2) follows by Cauchy's test.
*In conclusion, we make another general remark about the funda mental difference between absolute convergence and conditional con"'
v= l
vergence. We consider a convergent series 2 a,. We denote the positive terms of the series by p1, p2, p3,
- q2 , - q3,
•
•
•
•
•
•
•
, and the negative terms by - q1,
If we form the nth partial sum s.
n
=
2 a, of the given v=l
series, a certain n umber, say n', of positive terms and a certain number, say n", of negative terms m ust appear, where n' + n" = n . Fur thermore, if the number of positive terms as well as the number of negative terms in the series is infinite, then the two numbers n' and n" will i ncrease beyond all bounds as n does. We see im mediately n' that the partial sum s. is simply equal to the partial sum 2 P, of 'TI N
v=l
the positive terms of the series pi us the partial sum - 2 q, of the nega"'
v=t
tive terms. If the given series converges absolutely, then the series ofpositive terms
2 p,
v=l
and the series of absolute values of the negative
terms 2 q, certainly both converge. For as m i ncreases, the partial sums oo
2 p , and 2 q, are monotonic nondecreasing sequences with the upper m
v=l
v=l
oo
m.
v=l
bound 2 la, J . v=l
The sum of an absolutely convergent series is then simply equal to the sum of the series consisting of the positive terms only, plus the sum of the series consisting of the negative terms only, or, in other words, is equal to the difference of the two series with po>itive terms.
Sec.
The Concepts of Convergence and Diuergence 5 1 7
7.1
For
L n
v=l
a,
=
'
L n
v=l
p , .:..._
L n
v=l
q, ;
as
n
i ncreases n' and n " also increase
beyond all bounds, and the l imit of the left-hand side must therefore be equal to the difference of the two sums on the right. I f the series con tains only a finite number of terms of one particular sign, the facts are correspondingly simplified . I f, on the other hand, the series does not converge absolutely, but does converge conditionally, then the series
L p , and L q, must both be divergent. For if both were convergent the 00
00
v=l
v=I
series would converge absolutely, contrary to our hypothesis. If only one diverged , say
L 00
v=l
p,,
and the other converged, then separation i nto
positive and negative parts, s,
=
L p, n
'
v=l
-
L q shows that the series 11 "
v = l,
could not converge ; for as n increases n' and all bounds, whereas the term
n"
L
v=l
q,
L "
,,
p , would increase
v .,..- 1
beyond
would approach a defin ite lim it, so
that the partial sum s, would increase beyond all bounds. We see, therefore, that a conditionally com•ergent series cannot he though t of as the differ ence of t 11·o convergent series, the one consisting of its positit•e terms and the other consisting of the absolute values of its negative terms. Closely connected with this fact is another difference between abso l utely and conditionally convergent series which we shall now briefly m ention . * c. Rearrangement of Terms It is a property of fi nite sums that we can change the order of the terms or, as we say, rearrange the terms at will without changing the value of the sum. The question arises : what is the exact meaning of a change of the order of terms i n an infinite series, and does such a rearrangement leave the value of the sum unchanged ? Although in fi n i te sums there is n o difficul ty, for example, in adding the terms in reverse order, in i nfinite series such a pos sibi l i ty does not exist ; there is no last term with which to begi n . Now a change of order in an i nfinite series can only mean this : we say that a series a 1 + a 2 + a,1 + · · · is transformed by rearrangement into a series b1 + b2 + b3 + · · · provided that every term a, of the first series occurs exactly once in the second and conversely. For example, the amount by which a, is displaced may increase beyond a l l bounds as n does ; the only point is that a, must appear somewhere in the new series. If some of the terms are moved to later positions in the series, other terms must, of course, be moved to ,
518
Infinite Sums and Products
Ch. 7
earl ier positions. For example, the series
1 + q + q2 + q4 + q2 + t + q1 + q6 + rf + ql6 + I + q + q2
.
.
.
is a rearrangement' of the geometric series + · · ·. With regard to change of order there is a fundamental distinction between absolutely convergent series and conditionally convergent series .
In absolutely convergent series rearrangement of the terms does not affect the convergence, and the value of the sum of the series is unchanged, exactly as infinite sums. In conditionally convergent series, on the other hand, the value of the sum of the series can be changed at will by suitable rearrangement of the series, and the series can even be made to diverge if desired. The first of these facts, referring to absolutely convergent series, is easily established . Let us assume initially, that our series has positive terms only,
I a,. All the terms of this partial sum occur i n the mth partial sum t'" = I b, of the rearranged series, provided and consider the nth partial sum sn =
v=l
v=l
only that m is chosen large enough . Hence
t, � s, . On the other hand, we
can determine an i ndex n' so large that the partial sum s, . =
f a,
of the first
series contains all the terms bv b2 , . . . , bm. It then follows that t m � sn. � A, where A is the sum of the first series. Thus for all sufficiently large values of m we have sn � tm � A, and since sn can be made to differ from A by an arbitrarily small amount, it fol lows that the rearranged series also converges and, in fact, to the same limit A as the original series. If the absolutely convergent series has both positive and negative terms, we may, in fact, regard it as the difference of two series each of which has positive terms only. Si nce in the rearrangement of the original series each of these two series merely undergoes rearrangement and therefore converges to the same value as before, the same is true of the original series when rearranged. For by the case just considered the new series is absolutely convergent and is therefore the difference of the two rearranged series of posi tive terms. To the beginner the fact just proved may seem a triviality. That i t really does require proof, and that i n this proof the absolute convergence is essential, can be shown by an example of the opposite behavior of conditionally convergent series. We take the familiar series for log 2, below which we write the result of multiplication by the factor �. v=l
-!
+
}
-
i
+
!
-
i
+
}
-
� +
k
+
-
·
·
·
·
·
·
= log
2,
= } log 2,
1 For each n > 0 the terms q• with 2• < k � 2•+• are written in reverse order.
The Concepts of Convergence and Dive �r:ence 5 1 9
Sec. 7. 1
a n d add, combining the terms placed i n vertical columns.1 W e thus obtain . I
+
}
-
!
+ + !
t
-
i
+
�
+
ir
-
t
++
- · · · =
! log 2 .
This last series can obviously be obtained by rearranging the original series, and yet the value of the sum of the series has been multiplied by the factor ·�. It is easy to imagine the effect that the discovery of this apparent paradox must have had on the mathematicians of the eighteenth century, who were accustomed to operate with infinite series without regard to their convergence. •we shall give the proof of the above theorem concerning the change in the sum of a conditionally convergent series E an which arises from change of order of the terms, although we shall have no occasion to make use of the result. Let p1, p 2 , be the positive terms and -q�> -q2 , the negative terms of the series . Since the absolute value l an l tends to 0 as n i ncreases, the numbers Pn and qn must also tend to 0 as n i ncreases. As we have already •
•
•
•
•
•
seen, moreover, the sum L p, must diverge, and the same is true of L q,. �
00
1
1
Now we can easily find a rearrangement of the original series wh ich has an arbitrary number a as sum. Suppose, to be specific, that a is positive. We then add together the first n1 positive terms, just enough to bring about that 11]
1'1 1
1
1
the sum L p, is greater than a. Since the sum L p, increases with
n 1 beyond
all bounds, it is always possible by using enough terms to make the partial sum greater than a. The sum will then differ from the exact value a by Pn , • at most. We now add j ust enough negative terms sum
n1
m1
1
1
L P v - ! q,
is less than
a;
-I1 q, to ensure that the
this is also possible, as fol l ows from the
oc
divergence of the series L q,. The difference between this sum and a is now
q, 1
1
at most . We now add j ust enough other positive terms
f p,
11 1
to make
i-1
t h e partial sum again greater than a, a s i s again possi ble, since t h e series of positive terms diverges. The difference between the partial sum and a is now
p,. , at most. We again add j ust enough nega tive terms
m,
m 1+1
- L q,, beginning
next after the l ast one previously used, to make the sum once more less than a, and continue in the same way. The val ues of the sums thus obtained w i l l oscil late about t h e number a, and w h e n the process is carried far enough the oscil lation wil l only take place between arbitrari ly narrow bounds ; for, since the terms p, and q, themselves tend to 0 when v is sufficiently l arge, the length of the interval i n which the oscillation takes place wil l also tend to 0. The t heorem is thus proved .
1 For the addition of series see Sect ion 7. 1 d.
520 Infinite Sums and Products
Ch. 7
In the same way we can rearrange the series in such a way as to make it diverge : we have only to choose such large numbers of the positive terms as compared with the negative that compensation no longer takes place.
d. Operations with Infinite Series
It is clear that two convergent infinite series a1 + a2 + · · · = S and b 1 + b2 + · · · = T can be added term by term, that is, that the series formed from the terms en = a n + b n converges and has the value S + T for its sum. 1 For n n 11 ! c, = ! a, + ! b, -+ S + T. v=l
v=l
v=l
It is also clear that if we m ultiply each term of a convergent infinite series by the same factor, the series remains convergent, its sum being multiplied by the same factor. For these operations it is immaterial whether the convergence is absolute or conditional. On the other hand, further study shows that multiplication of two infinite series by the method used in m ul tiplying finite sums does not necessarily lead to a convergent series for the value of the prod uct, unless at least one of the two series i s absol utely con vergent (cf. A p pe ndi x , p . 555). 7.2
Tests for Absolute Convergence and Divergence
I n Section 7 . I b we have already encountered Leibnitz' useful test for the conditional convergence of series. In t h e following pages we shall o n ly consider criteria referring to absolute convergence. a. The Comparison Test. Majorants
All such considerations of convergence depend on the comparison of the series i n question with a second series ; this second series is chosen in such a way that its convergence can readily be tested . The general comparison test may be stated as follows :
If the numbers b" b2, and if
•
•
•
are all positive and the series ! b, converges, "'
la, l � bn
v- 1
00
for all values of n, then the series ! an is absolutely convergent. r1 = l
1 This theorem is really nothing more than another statement of the fact (cf. Chapter I , p. 72) that the limit of the sum of two terms is the sum of their limits.
Sec.
Tests for Absolute Convergence and Divergence
7.2
521
By Cauchy's test the proof becomes almost trivial. For if m � we have i a n + · · · + a m i � l a ,. l + · · · + l a m l � b n + · · · + b ,,. .
n,
·
00
Since the series � b,. converges, the right-hand side is arbitrarily small, n=l
provided that n and m are s ufficiently large. It follows that for such val ues of n and m the left-hand side is also a rbitrarily small, so that by Cauchy's test the given series converges. The convergence is absolute, since o ur argument applies equally well to the convergence of the series of absolute values l a,,l. The a nalogous proof for the following fact can be left to the reader. If 00
00
and the series � bn diverges, then the series � an is certainly not abson=I
n=t
lutely convergent. Someti mes the above series with the positive terms b ,. are called majorant and minorant series, respectively, for the one with terms a 11 • b. Convergence Tested by Comparison with the Geometric Series
In applications of the test the comparison series most frequently used as a m ajorant is the geometric series. We at once obtain the following theorem . THEOREM.
00
The series � a,. is absolutely convergent iffrom a certain 11 '--'
1
term on ward a relation of the form
( 3)
l a , l < cq" holds, where c is a positive number independent of n and q is any fixed positive number less than I . Ratio and Root Tests.
This test is usually expressed in one of the 00
fol lowing weaker form s : the series � a,. converges absol utely, i f n =l
from a certain term onward a relation of the form (4 a )
I a;:I I <
q
holds, where q is again a positive n umber less than I and independent of n, or : if from a certain term onward a relation of the form
(4b)
522
Infinite Sums and Products
Ch. 7
holds, where q is a positive number less than 1 . In particular, the conditions of these tests are satisfied if a relation of the form
1 an 1
=
k< 1
lim V:Ta:"l
=
k<
lim a n +l
(Sa)
n -+ 00
or
(Sb)
1
is true. These statements are easily established in the following way. Let us suppose that the criterion (4a), the ratio test, is satisfied from the suffix n 0 onward, that is, when n > n0 • For brevity we put an,+m+l = bm and find that
and so on ; hence
and then for
n
>
110,
and
c =
q-n,- l l bol
= cq "
which establishes our statement. For the criterion (4b) , the root test, we at o nce have Ia" I < q" , and our statement follows immediately. Fi nally, i n order to prove the c r i teria (5), we consider an ar bi trary number q s u c h that k < q < 1.. T h en from a certain 11 0 onw ard , that is, a ' < q and V1iG,;] < q respec when 11 > ll o , Eqs. 4a, b imply that
I ;: I
I l
tively, si nce from a certain term onwards the values of an+l or of v a,
iG,;j
differ from k by less than (q k). The statement is then establi shed on the ba sis of the results already proved . We stress the point that the fou r tests 4a, b, Sa, b, derived from the original criterion l an l < cq" are not eq·u ivalent to one another or to the original, that is, that they cannot be derived from one another in both directions. We shall soon see from examples that if a series satisfies one of the conditions, it need not satisfy all the others. -
Sec. 7.2
Tests for Absolute Convergence and Divergence
523
For completen�ss it may be pointed out that a series certainly diverges if from a certain term onward l an l > c
for some positive number
or
c,
if fr om a
V" l a,.l > I ,
1 1
l i m � = k, a ?/
or i f
n -+ oo
li m
or
n 4 00
certain term onward
v1TO:J
= k,
where k is a number greater than I . For, a s w e i m medi a tely recog n i ze in such a series the terms cannot tend to zero as n increases ; the series must therefore di verge . ( I n these circumstances the series cannot even be conditionally convergent.) Our tests furnish sufficien t conditions for the absolute convergence of a series ; that is, when they are satisfied we can conclude that the series converges absolutely. They are definitely not necessary conditions, h owever ; that is, absolutely con v ergent series can be formed which do not satisfy the cond i tions. Thus the kno wledge that ,
l 1 a
lim u-� �
" '1 = 1 Qn
lim V"iilJ = l
or
11 - 0C·
does not imply anythi n g about the convergence of the series. Such a series may converge or diverge. For example, the series
2 -n · "' I
for which l i m
v
u --" ]
I I
f a , I = I and l i m a ,, ' = I , is d i vergent, as we saw on a ,, 1
p. 5 1 3 . On the other hand, as we shall soon see, the series satisfies the same relations, is convergent. n - � oo
1/
... cr.,
2, FI 11
2 C/)
which
As an example of the application of our tests we first consider the series For this series
q + 2q2 + 3q3 +
· · ·
+ nq" +
· · · .
l i m \' I a n i = lql · lim \' � = l ql ,
lim
Ia
"+! I
-
an
71 -+ 00
= lql · l im
n - .. oo
n
+ I
--
n
= lql .
That the series converges if lql < 1 fol lows from the ratio test and from the root test also, even in the weaker form (5). n - C�J
524
Infinite Sums and Products
1 2q q2 zqa 12q2n+ll 7 2 l ql
q2n 2q2n+l
If, on the other hand, we consider the series +
+
+
+ . . . +
+
+ . . .,
we can no longer prove convergence by the ratio test when l � for then lim n � oo
-i'llanl
� I.
=
=
lql
Ch.
7
<
I;
But the root test immediately gives us
lql, and shows that the series converges provided that lql
<
I,
which, of course, we could also have observed directly. c.
Comparison with an Integra/ 1
We now proceed to discuss quite a different method of studying convergence. We shall explain it for the typi c al particularly si mple and i mportant case of the series ,
00 1
1
1
+-+···, ! -=! +2" 3"
"� 1 n"
where the general term a" i s l fn•, r:1. being a positive n u mb e r I n o rd e r to investigate the convergence or d i vergence of this series, we consi der the graph of the func t i o n y = I fx• and mark off o n the x-axis t h e in tegral absci ssae :r = 1 , x = 2, . . . . We first construct the rectangle of height 1 fn• over the i nterval n - 1 � x � n of the x a x i s (n > 1 ) , an d compare i t w i th the area of the regi on bounded by the same i nterval of the x-axis, th e o r di n at e s at the e n d s , and the curve y = 1 fx• (th is region i s shown shaded in Fig. 7.2). Secondly, we constr uct the .
-
y
Figure 7.2
Comparison of se r ie s with an integ ral .
1 I n this connection see also the A ppendix t o Chapter 5 , p .
505 .
Tests for Absolute Convergence and Divergence 525
Sec. 7.2
rectangle of height. l /n" lying above the interval n � x � n + ! , and similarly compare it with the area of the region lying a bove the same i nterval and below the curve (this region is cross-hatched in Fig. 7.2). In the first case the area u nder the curve is obvi ously greater than the area of the rectangle ; in the second case it is less than the area of the rectangle. In other words,
in+ l n
n
1 - <- <
dx
xa.
na.
i"
dx
n- 1
-. xa.
Writing down t hese inequal ities for n = I , 2, 3 , . . . , m, respectively = 2, 3, . . . , m , and summing. we obtain the following estimate for the mth partial sum s,. =
(6 )
Now as
m
i m+l
]
L -;; : m
n=l n
1
dx
- < xa.
< 1 +
sm
i ncreases the i ntegral
i"' 1
dx
-. x"
[ '" � dx tends to a finite limit or
J . xa
i ncreases without l i m it depending on whether rx > I or rx � 1 . Consequently, the monotonic seq uence of n u m bers S111 is bounded or increases beyond all bounds depending on whether rx > I or rx � I , and we thus have the following theorem . THEOREM.
The series of reciprocal po wers en
J
]
1
1
L - =-+-+-+ ... 1a 2" 3"
u � l 11°
is com•ergen t if and only if rx > 1 .
For rx = I the divergence of the harmonic series, which we prev iously p r oved in a different way, i s an i mmediate consequence ; l i kewise the senes
I I converge while the series -r + 1- + vi ,2 oo I The convergent series L � for rx > V series i n investigations of convergence.
v=l
r
I
v3
+ · · · diverges.
1 frequently serve as comparison
For example, we see at once that
526 Infinite Sums and Products
Ch. 7
I � converges absolutely if the absolute values Jc. l v=l V of the coefficients remain less than a fixed bound independent of v.
for ex > 1 the series
Euler's Constant. From the estimate (6) for ex = I it follows at once I I 1 that the sequence of numbers en = 1 + - + - + . . . + - - log 11 =
sn - log n > log (n + 1 )
-
2
3
n
log n > 0 is bounded below. Since from n+l dx I 1 = log (n + 1 ) - log n = + the i nequality < n x n + I n + l J en - cn +l• we see that the sequence is monotonic decreasing, it must approach a limit
--
lim en = lim
n - oo
n-400
(
) +
-
�+!
2
3
--
)
+ . . . + l - log n = c . n
..
The number C whose value is 0. 5772 . , is called Euler's constant. In contrast to the other important special numbers of analysis, such as 71' and e, no other expression with a simple law of formation has been found for Euler's constant. Whether C is rational or irrational is not known to this day. 7.3
Sequences of Functions
As emphasized frequently before, the limit process serves not only to represent known numbers approximately by other, simpler ones, but it also serves to extend the set of known numbers i nto a wider one. It is of decisive i mportance in analysis to study limits not only for sequences-or infinite series-of constant numbers, but similarly for sequences of functions, or series whose terms are functions of a variable x, as, for example the Taylor series or power series in general . Not only the approximation of given functions by simpler ones requires such limiting processes but also the definition and analytic description of new functions must frequently be based on the concept of limit of sequences of functions : .f(x) = l i m .fn(x) for n --+ w . Equivalently, we may consider .f(x) as the sum and the .fn(x) as the partial sums of an i nfinite series .f(x) = I gr(x) of fun•;tions g n(x) where g,(x) =
r�l .fn(x) - .fn_1(x) for n > I and g1(x) = .ft(x). oc
We shall now discuss precise definitions and geometrical inter pretations.
Sequences of Functions 527
Sec. 7.3
a. Limiting Processes with Functions and Curves
Definition. The sequence f1 (x), f2(x), . . . converges in the interval :$; x :$; b to the limit function f(x), if at each point x of the i nterval the valuesfn (x) converge in the usual sense to the value f(x) . In this case we write limfn(x) = j(x) . According to Cauchy's test (cf. p. 75) a
n - � oo
w e can express the convergence o f the sequence with out referring to the l i mi t function j(x) : The sequence of functions converges to a limit functio n if and only i f at each point x i n our i nterval and for every positive number < , the quantity lf,.(x) - f,(x) l is less than <, pro vided that n and m are chosen large enough, that is, larger than a certain number. This number N = N(<, x) usually depends o n < and x and i ncreases beyond all bounds as e tends to zero. We have frequently met with cases of l i mits of sequences of functions. We mention only the definition of the power x� for i rrational values of !Y. by the equation where r1, r2, , r . . . i s a sequence of rational numbers tending to !Y.; or the equation x " ex = l i m 1 + - , n •
•
•
"'
n -�' rx:
( )
where the approximating functi ons f,.(x) on the right are polymomials of degree n . T h e graphical representati on of functions by means of curves suggests that we can also speak of li mits of seq uences of c urves, saying, for example, that the graphs of the preceding l i m i t functions x� and e' are to be regarded as the l imit curves of the graphs of the functions xrn and
(]
+ ':
)
11 ,
"
respectively.
There is, however, a fine d istinction between passages to the limit with functions and with curves, not clearly observed until the m iddle of the n ineteenth century. We shall illustrate this point by an example and then discuss i t systematically i n the next section. We consider the functions J,(x) = x",
n = I , 2, . . .
i n the interval 0 :$; x s 1 . All these functions are continuous, and the limit functio n l i m f,(x) = j(x) exists. But this limit function is not
528 Infinite Sums and Products
Ch. 7
continuous. On the contrary, since for all values of n the value of the function fn( l ) = I, the limit f( l ) = I ; while, on the other hand, for 0 � x < I , the limitf(x) = lim fn(x)
=
0,
as we saw in Chapter I , p. 65. The function f(x) is therefore a dis continuous function which at x = I has the value 1 while for all other values of x in the interval has the value 0. n-- co
y
I
C
1
_ ____ _, /
Figure 7.3
/
/
1
Limit
Iii
I f· ', I I
/ if :
I
/ I lc : I Cz; I / / /. I I
//
/
I
1··1
I
C4 ,
.
'
,
/ c,:/ /.·
curve
j
'
I
I I !
1
and limit function .
X
This discontinuity is geometrically illustrated by the graphs C,. of the functions y = fnCx). These (cf. Fig. 1 .44, p. 66) are continuous curves, all of which pass through the origin and the point x = 1 , y = I , and which draw in closer and closer to the x-axis as n increases. The curves do possess a limit curve C which is not discontinuous at all, but consists (cf. Fig. 7.3) of the portion of the x-axis between x = 0 and x = I , and the portion of the line x = I between y = 0 and y = I . The curves therefore converge to a continuous limit curve with a vertical portion, whereas the functions converge to a discontinuous limit function . We thus recognize that this discontinuity of the limit function expresses itself by the occurrence in the limit curve of a portion perpendicular to the x-axis. This limit curve is not the graph of the limit functio n ; for corresponding to the value of x at which the vertical portion occurs the curve gives an infinite number of values of y and the function only one.
Un iform and Nonun iform Convergmce
Sec. 7.4
529
Hence the limit of the graphs of the functions fn(x) is not the same as the graph of the li m it j(x) of these function. Corresponding statements, of course, hold for i nfinite series as well. 7.4
Uniform and Nonuniform Convergence
a. General Remarks and Definitions
The distinction between the concept of the convergence of functions and that of the convergence of curves i s a phenomenon which the y
a--------------------� b--------� x o L---�
Figure 7.4
To illustrate uniform convergence.
student should clearly grasp. This i nvolves the so-cal led nonun iform c onvergence of sequences or i nfinite series of functions which we shall discuss in some detail. That a function f(x) is the limit of a sequence f1(x), j;(x), in an interval a � x � b means by definition merely that the usual limit relationship f(x) = lim [" (x) holds at each point x of the interval. Such convergence is a local property of the sequence at the point x. I t is, however, natural to require somewhat more than the mere local convergence of our approximations : that if we assign an arbitrary measure of accuracy £, then from a certain i ndex N onward all the func tions f,.(x) should l ie between f(x) £ and f(x) + e , for all values of x, so that their graphs y = J,(x) lie entirely in the strip shown in Fig. 7.4. If the accuracy of the approximation can be made at least equal to a preassigned positive number e, everywhere in the interval at the same .
-
.
.
530 Infinite Sums and Products
Ch. 7
time, that is, by everywhere choosing the same number N(E) i ndependent of x, we say that the approximation is uniform.1 If l i m J,,(x) = j(x)
a � x � b, there exists for every E > 0 a corresponding number N = N(E) such that lf(x) - f.(x) l < E for all n > N a n d al l x in the interval. Many people were quite surprised when i n the middle of the ni neteenth century it was noticed by Seidel and others that convergence of functions need not at all be uniform as had been naively assumed.
uniformly for
Examp les of Nonuniform Convergence. The concept of uniform convergence is illumi nated by examples of nonuniform convergence. (a) The first example occurs for the sequence of fu nctions j ust considered, [n(x) = xn ; in the interval 0 ::;; x ::;; I this sequence converges to the limit function f(x) = 0 for 0 ::;; x < I , [( I ) = I . Convergence occurs at every point in the i nterval ; that is, if • is any positive number, and if we select any definite fixed value x = ;, the i nequality W - [WI < e certainly hol ds i f n is sufficiently large. Yet this approx imation is not u n i form . For, if we choose < = �. then no matter how large the n umber n is chosen , we can find a point x = "' F- I at which l r1 n - [("1)1 = r1 n > ! ; this is, i n fact, true for all points x = "' where I > "' > ''l It is therefore impossi ble to choose the number n so large that the difference between f(x) and fn (x) is less than l
throughout the whole interval.
This behavior becomes i ntelligible if we refer to the graphs of these functions (Fig. 7.3). We see that no matter how large a value of n we choose, for values of ; only a l i ttle less than I the value of the function fn(;) will be very n ear I, and therefore cannot be a good approximation to f( ;) , which is 0. Similar behavior is exhibited by the functions
f,, (x)
=
in the neighborhood of the points x established . Here [(�) = I for lxl < for l x l > I .
I I + x 2n =
I and x = - I ; this can easily be = ! for lxl = I and f(x) = 0
l , f(x)
(b) In the above two examples the non uniformity of the convergence is connected with the fact that the limit function is discontinuous. Yet it is also easy to construct a sequence of continuous functions which do converge to a continuous limit function, but not uniformly. We restrict our attention to
1 Compare with the analogous definition at uniform continuity, p. 41, where we can choose the same number !I( e) independent of x.
Sec. 7.4
t h e interva l 0 :::;;
Uniform and Nonumform Con!'el;t;ence
x
:::;; I a n d make t h e following defini tions for 1 1 � 2 :
f,,(.r)
=
xn'
I
0 :::;; .l' :::;; - '
for
II
I 2 - ::>: x _ <-, 11
for
j;,(.t)
=
fo r
0
531
: s
"
11
11
::>: I ,
.r
where to beg i n w i t h we can choose any val u e for
-x ,
but m u s t then keep this
value of "' fixed for all terms of the sequence . G raph ica l l y , our functions are
y
To illustrate non uniform convergence. 2/n
lfn
Figure 7 . 5
0 s
:::;;
of the .r-axis, w h ereas from :r =
represented by a roof-sh a ped figure made of two l i n e segments lying o ve r
the i n terval
If
.r
2/11
graph i s the :r-axis i tse l f (cf. F i g . "' <
2/11
onward t h e
7.5).
I , t h e a l t i t u d e of t h e h i ghest point o f t h e graph, which h a s i n
general the value
n'· l , wi l l tend to zero a s n i n c reases ; the cu rves w i l l then tend toward the .1·-axis, a n d t h e fu nctions j;,(.r) w i l l con verge u n i fo r m l y to the l i mi t fu nction f(:r) = 0.
I f "' = I , the peak o f t h e graph w i l l h a v e the heigh t I for every v a l u e of 11 . If "' > I , the height of the peak w i l l i ncrease beyond al l bounds as 11 i n crease s .
= 0. For, if
. . i s p o s i t i v e , we have
H owever, n o matter how "' is c h o s e n , the seg uence f/rJ, f2(.r) , . . . a l ways ten d s to t h e l i m i t function of the graph a n d j;,(.r ) =
f(.r)
for a l l sufficie n t l y l arge val ues of equal to
0, so that
11
in e i t her c a se l i m
0;
for
:r
.fn(.r)
i s less t h a n
n - 'l�
i every where
so l a rge that the e x p re ss i o n
i n the i n terval .
11
:r,
= 0 a l l the fu nctional values .f,(.t') are = 0.
j;, (.r)
The con vergence is c e r ta i n ly n o n u n i form, however, i f
pl a inly i m po s s i bl e to choose
2/11 <
so t h a t .r is not u n der t h e roof-shaped pa r t .r
"' � I l f(:r)
; for i t is -
.fn (.1')1
532 Infinite
Sums and Products
Ch. 7
(c) Exactly similar behavior is exhi bited by the sequence of functions where, i n contrast with the preceding case, each function of the sequence is represented by a single ana lytical exp ression . H ere agai n the equation lim [n(x) = 0 holds for every positive value of x, since as n increases the
u- oc,
y
I : I
-'"\11 = 4
..
\
.
: l '-\ I ·,';' \.
: /.,
-- ·
.... - - �
II' 1 l,q/'\,
''
.
I II I
Figure 7.6
·\, \'"' . . ..... ' '
'- .. ·-...._
' -.. ..._
-..... . - -.:.::-.. --:::-::.._
......._ _ _. ......., ___ - - -
Nonuniform convergence of the sequence /n(x) 2
X
=
n'xe- nx.
function e- nx tends to zero to a higher order than any power of 1 /n (cf. Section 3.7b, p. 250). For x = 0, we have al ways fn(:r) = 0, and thus
= lim [n(x) = n -.. c:o
f(x)
0
for every value of x• in the interval 0 :0: .r :s; a, where a is an arbitrary positive number. But here agai n the convergence to the l imit function is not uniform . For at the point ,,. = 1 /11 [where fn(x) has its m a x imum ] we have
[n(x)
( 1) 1 •-1
= · {n -11
=e
•
and we thus recognize that if rt. ;:: I , the convergence is n o n u n i form , for every curve y = fn (x), no matter how l arge 11 is chosen, w i l l contain points (namely, the poi n t x = 1 /n, which varies with n) at wh ich fn(.r) - f(x ) � fn(x) > 1 /2e (cf. Fig. 7.6).
(d) The concepts of u n i form and nonuniform convergence may, of course, be extended to an infin ite series. We say that a series gl(x)
+ g2(x·)
+ . . .
Sec.
Uniform and Nonuniform Conce�l{ence
7.4
533
is u n i formly convergent, or not, according to the behavior of its partial sums
fn(.r). A very s i mp l e example of a non u n iformly c o n ve rge n t series is gi ve n by .?'2 ;t'2 .1.2 /(.!') = .r2 + -- + --- + --- + . . . 1 + ,1.2 ( I + r2) 2 ( I + .r2)3 For x = 0 e ve ry partial sum f,,(:r) = :r2 + · · · + .1:2j( I + .>· 2 )"-1 h a s the value •
.
0; therefore f(O) = 0 .
For
,, . "" 0 t he series is s i m p l y a geometric series y
\ \
Figure 7.7
\
\
\
I
\
\
\
1 \
', '
\. :I
' '- \'.
?
H''J(X) / : ftn (x) /
I/
: .
!I /
I
/
I
" ./ ...-
X
Con vergence to fu nct i o n w i t h remova b l e j u m p d i scont i n ui t y .
w i t h t h e p os i t i v e rat i o I /( I + .1·2) <
I ; we
elementary rules a n d t h u s obta i n for every
The
I f1 (x)
l i m i t function f(.r)
expression f(.r) =
I +
.•.
can therefore s u m i t by the
"" 0 the s u m
:-----,-,-:-:- ----;;- = I + r2 I - 1 /( 1 + .r2) .
is
t h u s gi v e n
.r2, w h ereas /( 0 1
•
except at .r = 0 by the i t t h ere fo re h a s a removable
e v e rywhe re
=
0;
d i scont i n u i ty a t t h e origi n .
For the d i fference .f'(.r) - j;,(.<·)
=
r,(.r) is always 0 for : r = 0, 1 /( l + .r2 )"- 1 for a l l other
H ere aga i n w e have non u n i form convergence i n every i n terval con ta i n i ng t h e origi n .
val ues of :r , as t h e reader may verify for h i ms e l f. If we r e q u i re t h i s e x pre s s i o n to be less t h a n , say 1. t h e n fo r e ac h fi xed value of 1· t h i s can be a t ta i n e d by choosing 11 large enough . But we ca n fi n d no v a l u e of 11 s uffi ci e n t l y l a rge to e n s u re that r,(.r) is e v e ry w here less t h a n � ; fo r if we choose any value of n, no matter how l a rge, we can m a k e r,(.r) g rea t e r than ·� by t a k i n g :r n e a r enough to 0. A u n i form a pprox i m a t i o n to wi th i n ! is therefore i m possible. The matter becomes clear i f we consider the approximating cu rves (cf. Fig. 7.7). These c u rves, except near .r = 0, l i e nearer a n d nearer to the para b o l a whereas it i s given by the express i o n r,/<') =
534
Infinite
Sums and Products
Ch. 7
y = I + x2 as n increases ; near x = 0, however, the curves send down a narrower and narrower extension to the origin, and as n increases this extension draws in closer and closer to a certain straight line, a portion of the y-axis, so that for the limiting curve we have the parabola plus a linear exten sion reaching vertically down to the origin. As a further example of nonuniform convergence we mention the series "' z g,(x), where g,(x) = x' - x'-1 for v � I , g0(x) = I , defined in the
v=O
interval 0 s x s I . The partial sums of this series are the functions x' already considered in Example (a), p . 5 30. b. A Test of Uniform Convergence
The preceding considerations show us that the uniform convergence of a sequence or series is a speci al property not possessed by all sequences and series . We now repeat the definition of uniform con vergence as it applies to i nfinite series : the series g1 (x) + g 2(x) +
· · ·
is uniformly convergent to a function f(x) i n an i nterval if f(x) can be approximated to within a margi n of approximation £ (where £ is an arbitrarily small positive number) by the sum of a fixed and sufficiently large n umber of terms g ,(x) + · · · + gs(x) = fv(x), i ndependent of x i n the i nterval . We again have a test (Cauchy's test) for uniform convergence that does not require knowledge of the limit function J(x) : the series con verges uniformly (or equivalently, the sequence of functions J,(x) converges uniformly) if and only if the difference 1 /,(x) - fm(:r) l can be made less than an arbitrary quantity E everywhere i n the i nterval by choosing n and m larger than a number N i ndependent of x. For, first, if the convergence i s uniform, we can make 1 /,(x) - f(x) l and lf,,(x) - J(x) l both less than £/2 by choosi ng n and m greater than a number N i ndependent of x, fr om which it follows that 1 /,(x) - fm(x) l < £ ; and secondly, if 1/,(x) - .fm(x) l < £ for all values of x whenever n and m a r e greater than N, then on choosing any fixed val ue of n > N an d Jetting m i ncrease beyond all bounds we have the relation l fn (x) - f( x) l = lim l f,( x) - fm( x) l �
m 4 a:_,
E,
for every value o f x, s o that t h e convergence is uniform. As we shall see i t i s j ust this condition cf uniform convergence that makes i nfinite series and other li miting processes with functions i nto convenient and usefu l tools of analysis. Fortunately, i n the l i m i ting
Sec. 7.4
Umform and Nonuniform Convergence 535
processes usually encountered in analysis and its applications, non uniform convergence occurs only at isolated exceptional points and will scarcely trouble us for the present . Usually, the uniformity of convergence of a series i s established by means of the fol lowing criterion (comparing the series with a majorant of constant terms) : If th e terms of the series _L g,(x) satisfy the condition l g,(x) l :5: a,, "'
v=l
where the numbers a , are positil'e constants which form a convergent ro
ro
v=l
v=l
series _L a , then the series _L g.(x) conrerges uniformly (and absolutely).
For we then have
and since by Cauchy's test the sum
i a, can be made arbitrarily small
by choosing n and m > n large enough , this expresses exactly the necessary and sufficient cond ition for uniform convergence. A first example is offered by the geometric series 1 + x + x 2 + · · · where x is restricted to the interval j.cj .::;; q, q being any positive number less ,
than I . The terms of the series are then numerically Jess than or equal to the terms of the convergent geometric series ':£q'. A further example is given by the "trigonometric series" c1
sin (x
)2
-
r\)
provided that l cnl < then we have
gn(x)
=
c,
+
c2 sin (:r
22
-
<12 )
+
c3
sin (x
32
-
03 )
+ . . .,
where c is a positive constant independent of n. For
Cn
sin (x - on) n
2
so that
l gn(x) j
<
";(i. . C
Hence the uniform and absolute convergence of the trigonometric series
follows from the convergence of the series _L 2 . 00
c
11 = 1 n
c. Continuity of the Sum of a Uniformly Con vergent Series of Continuous Functions
The significance of uniform convergence lies i n the fact that a uniformly convergent series i n many respects behaves exactly like the sum of a finite n umber of functions. Thus, for example, the sum of a
536 Infinite Sums and Products
Ch. 7
finite number of continuous functions i s itself continuous, and cor respondingly we have the fo llo wing theorem. THEOREM. If a series of continuous terms converges uniformly in an interval, its sum is also a continuous function. PROOF. The proof i s quite simple. We subdivide the series f(x) = g1 (x) + gi x) + · · ·
into the nth partial sum fn (x) plus the remai nder RJx). As usual, fn(x) = g1(x) + + g n (x) . If now any positive n umber E is assigned, we can in virtue of the uniform convergence choose the number n so large that the remainder is less than £/4 throughout the whole interval, and hen ce ·
·
·
I Rn(x + h)
-
R n(x) l < _: 2
for every pair of numbers x and x + h i n the interval. The parti a l sum fn(x) consists of t h e sum of a finite number of continuous functions and i s therefore conti nuous ; for each point x i n the interval, therefore, we can choose a positive b so small that l fn( x + h ) - fn(x) J <
�
provided I h i < b and the points x and x + h lie i n the i nterval. It then follows that IJ(x +h)
-
f(x) l = lfn(x + h) - fn(x) + Rn(x + h) f,(x)l + l Rn(x + h) - Rn{x) l <
S lfn(x + h)
-
-
Rn(x) l
E,
which expresses the conti nuity of our function. The i m portance of this theorem becomes clear when we recal l that the sums of nonuniformly convergent series of continuous functions are not necessarily conti nuous from our previous examples. From the pre ceding theorem we may conclude : if the sum of a convergent series of continuous functi ons has a point of d i scontinuity, then in every neighborhood of this point the convergence is nonuniform . Hence every representation of discontinuous functions by series of continuous functions must be based on the use of nonuniformly convergent limiting processes. d. Integration of Uniformly Convergent Series
A sum o f a finite number of continuous functions can be integrated "term by term" ; that is, the integral of the sum obtai ned by i ntegrating
Sec.
Uniform and
7.4
Nonuniform Conver�ence
537
each term separately and adding the i ntegrals. In a convergent infinite series of continuo us functions the same procedure i s permissible, provided that the series converges uniformly in the interval of i ntegra tion.
2; 00
A series
v=l
gJx)
=
f(x) ll'hich conrerges uniformly in an in term! can be
integrated term by term in that interml; or, more precisely, if a and x are t wo numbers in the interml of uniform com•ergence, the series
,� lx
g ,(I)
x,
dt converges, a:d, in fact, conrerges uniformly ll'ith respect to
its sum being equal to
l )
f ( t dt.1
To prove this we write as before j(x)
=
L gJx) ""
V=l
