Integration for Engineers and Scientists (Modern analytic and computational methods in science and mathematics)

Modern Analytic and Computational MetlhodS in Science and Mathennallfs RICHARD h EIJ.MAN. editor

Integration for Engineers and Scientists William Squire West Virginia University Morgantown, West Virginia

American Elsevier Publishing Company, Inc. NEW YORK 1970

AMERICAN FLSEVIER PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue, New York, N.Y. 10017

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Copyright © 1970 by American Elsevier Publishing Company. Inc.

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Preface HIS book is primarily intended for students and practicing scientists or engineers faced with the problem of evaluating integrals

beyond those treated in elementary textbooks or listed in the standard tables. It should also be useful to teachers. Although a search of the library

will turn up a number of advanced works dealing with the fundamental theory of the integral, these are of little help in evaluating specific integrals. Actually, for this aspect of the subject, the older problem-oriented advanced

calculus texts are generally superior to their modern counterparts. The general neglect of Liouville's work on integration in finite terms is difficult

to explain. Apparently no American nor European calculus text makes more than a brief reference to it, though there are two excellent presentations (Hardy and Ritt) available in English. In regard to numerical methods the situation is better. The advent of the high-speed computer has brought about a revival of interest, and a number of excellent treatments are available, particularly the monographs by Davis and Rabinowitz, Nikolsky, and Krylov. I believe, however, that the present treatment fills a definite need for an extensive presentation on a somewhat more elementary level. The following quotation by J. L. Synge [Quart. App!. Math., 9 (1951), p 113-114] is apropos: "The modern meaning of the words 'mathematical proof' is well known: they imply a faultless logical chain

which starts from undefined elements and axioms, no loop-holes or exceptions permitted. Those subjects which, like topology, have been developed in that spirit of rigor, present an almost impenetrable front to applied mathematicians or engineers. The extraction of one needed result, and an

understanding of what it means, demand a long and careful study in an uncongenial atmosphere, unrelieved by interpretations in terms of natural phenomena. It is obvious that this state of affairs cannot persist. Each mathematical

subject must be treated on several levels, varying from that of extreme rigor down to simple intuitive descriptions with no proof at all. We get this variety of treatment in older branches of mathematics, partly because the

creators had not got the modern standards of rigor, and partly because these matters have been looked at so long by so many people and from so many different angles.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

Much of the argument that passes for proof in physics and engineering is

not proof in the mathematical sense, and it is unlikely that the practical subjects will ever submit to the strict mathematical discipline. There seems to be an incompatibility between those minds which excel in logic and those

which are capable of dealing successfully with problems suggested by nature. The word `proof' has such a general usage that it is inconceivable that it should be employed only in its strictest mathematical meaning. Physicists and engineers will continue to use it in a looser sense, and will regard as proved any proposition with regard to which they can assemble sufficient evidence to convince them of its truth. As Descartes pointed out long ago,

to `prove' something by a series of logical steps, and to `see' or 'understand' it are not the same thing; and what the physicist or engineer needs is the `seeing' and the `understanding'." The present treatment assumes no background beyond the usual undergraduate course in differential equations. This has resulted in the omission of some powerful methods, such as contour integration and the method of steepest descents, that depend on a knowledge of the theory of functions of a complex variable, but it is believed that this is not serious because of the availa-

bility of adequate treatments. While we have tried to take the impact of the computer into account, no specific knowledge of computer programming

is assumed in the text. However, a number of illustrative FORTRAN programs are included as appendices and many of the numerical problems are intended to be worked with a computer rather than by hand. The first chapter of the book covers background material and tries to

relate the practical aspects to the basic theory. The second chapter is a review of the methods for evaluating indefinite integrals covered in calculus

courses unified by Liouville's theory. Use is made of devices, such as differentiation with respect to a parameter, that are not used in the elementary

courses because of the conventional sequence of topics, which postpones partial differentiation to a later stage. The third chapter covers exact and approximate methods for evaluating definite integrals. I have found that engineering students can reach graduate school without learning that there are methods for evaluating a definite integral other than substituting the limits in the indefinite integral.

The fourth and fifth chapters treat numerical methods for evaluating integrals. The methods in the fourth chapter use only values of the integrand, whereas those in the fifth chapter also use the values of the derivatives of the

integrand. While the presentation is not as rigorous or detailed as those available elsewhere, it is probably the most extensive single compilation of methods presently available. At this time when the impact of the computer is revolutionizing computations, it does not appear possible to make a definite

PREFACE

judgment regarding the best methods. The objective is therefore to present a wide range of methods so that the reader can either pick one adapted to his particular problem or see how to devise a modification suited to his particular circumstances. The last chapter gives a brief account of the theory of integral equations and shows how the quadrature methods in Chapter 4 can be used to develop methods for numerical solution. Although the book has not been designed for any of the usual courses in American universities, the last three chapters could be used as the basis for a specialized second course in numerical methods. The first three chapters would be a useful supplement for honors sections in calculus.

I have tried to present integration not as a completed logical structure but as a field with its roots in the past yet still offering ample scope for further development. An attempt has been made to cite references of varying

degrees of difficulty, particularly for important topics. In order to show current developments, it has sometimes been necessary to cite reports of limited availability. On the other hand, secondary sources have often been cited. To some extent this has been done on the basis of my convenience, and apologies are tendered to those deprived of their due. An attempt has been made to provide interesting and meaningful exercises. Correction of errors and suggestions would be appreciated. The most

interesting applications occur en passant in papers on other subjects and are difficult to locate through abstract journals, so I appeal to the interested reader for examples of ingenious evaluations for future editions. My research in numerical methods for evaluating integrals, which led to some methods described in Chapters 4 and 5, was supported by an AFOSR grant in 1963-1965. However, the main support for this book came from West Virginia University and my department through lightened teaching loads, and from the computer center, which provided computer time. I am grateful to Mrs. Donna Moore and Mrs. Georgette Healy for typing the manuscript and to Dr. Richard Bellman for his editorial encouragement. Finally, I acknowlegde the inspiration of Dr. Norman Davids, who 25 years ago in a course on boundary-value problems introduced me to the concept

of Gaussian quadrature. Morgantown, West Virginia January 1970

WILLIAM SQUIRE

Contents CHAPTER 1

General Background 1 1.1. Introduction ................................................................................ 3 1.2. Riemann, Stieltjes and Lesbesgue Integrals .................................. 8 1.3. Multiple and Iterated Integrals ...................................................... 14 1.4. Improper and Infinite Integrals .................................................... 1.5. Mean Value Theorems .................................................................. 20 1.6. Inequalities .................................................................................. 21 1.7. Indefinite Integrals versus Definite Integrals ................................ 25 1.8. Fractional Integration and Differentiation .................................... 28 1.9. Line Integrals .............................................................................. 29 1.10. Surface and Volume integrals ...................................................... 31 1.11. Symmetry Arguments .................................................................. 31 Bibliographic Notes and Comments .............................................. 34

CHAPTER 2

Analytic Evaluation of Indefinite Integrals 2.1. Introduction ................................................................................ 2.2. Liouville's Classification of the Elementary Functions .................. 2.3. Basic Theorems for Integration in Finite Terms ............................ 2.4. Practical Integration of Rational Functions .................................. 2.5. Practical Integration of Algebraic Functions ................................ 2.6. Elliptic Integrals .......................................................................... 2.7. Integration of Elementary Transcendental Functions .................... 2.8. Symbolic Automatic Integration .................................................. 2.9. Derivation of Integrals from Differential Equations ...................... 2.10. Approximate Methods .................................................................. 2.11. A Practical Example .................................................................... Bibliographic Notes and Comments ..............................................

39 41 45 49 52

57 58 63 68 70 72 75

CHAPTER 3

Analytic Evaluation of Definite Integrals 3.1. Introduction ................................................................................ 77 3.2. The Gamma Function .................................................................. 79 3.3. Classical Calculus Methods .......................................................... 84 3.4. Series Methods ............................................................................ 90 3.5. Complex Variable Methods .......................................................... 93 3.6. Some General Forms for Definite Integrals .................................... 94 3.7. Use of Integral Transforms .......................................................... 96 3.8. Frullanian Integrals ...................................................................... 99 3.9. The Willis Expansion .................................................................... 105 3.10. Laplace's Method ........................................................................ 108 3.11. Integration By Parts Methods ...................................................... 111 3.12. Concluding Remarks and Examples ............................................ 115 Bibliographic Notes and Comments .............................................. 120

CHAPTER 4

Numerical Evaluation of Integrals 4.1. Introduction ................................................................................ 125 4.2. Simple Quadrature Formulas with Specified Nodes ...................... 127 4.3. Chebyshev's Equal Weight Quadrature Formulas ........................ 132 4.4. Gaussian Quadrature .................................................................. 135 4.5. Convergence of Quadrature Formulas .......................................... 138 4.6. Error Analysis .............................................................................. 139 4.7. Compounding and Adaptive Integration ...................................... 144 4.8. Extrapolation Methods ................................................................ 148 4.9. The Bernstein Quadrature Formula .............................................. 152 4.10. Monte Carlo Methods .................................................................. 153 4.11. `Best" Quadrature Formulas ........................................................ 154 4.12. Riemann and Riemann-Stieltjes Sums .......................................... 157 4.13. Integration of Periodic Functions ................................................ 158 4.14. Improper Integrals ...................................................................... 160 4.15. Product Integration ...................................................................... 164 4.16. Trigonometric Weight Functions .................................................. 170 4.17. Integrals Over An Infinite Range .................................................. 174 4.18. Indefinite Integrals ...................................................................... 178 4.19. Multiple Integrals ........................................................................ 181 4.20. Linear Integrodifferential Operators ............................................ 186 Bibliographic Notes and Comments .............................................. 189

CHAPTER 5

Quadrature by Differentiation 5.1. Introduction ................................................................................ 197 5.2. Compound Rules with Correction Terms ...................................... 198 5.3. Simple Quadrature Rules Using Derivatives .................................. 201 5.4. Summation Formulas .................................................................. 205 5.5. The Generalized Midpoint Rule with a Weight Function ................ 206 5.6. Linear Eigenvalue Problems ........................................................ 208 5.7. Boundary-Value Problems .......................................................... 214 Bibliographic Notes and Comments .............................................. 219 CHAPTER 6

Integral Equations 6.1. Introduction ................................................................................ 221 6.2. Classification of Integral Equations .............................................. 222 6.3. Conversion of Differential to Integral Equations .......................... 223 6.4. Direct Derivation of Integral Equations ........................................ 227 6.5. Exact Solution of Integral Equations ............................................ 231 6.6. Liouville-Neumann Theory .......................................................... 238 6.7. Fredholm Theory ........................................................................ 241 6.8. Hilbert-Schmidt Theory ................................................................ 243 6.9. Numerical Solution of Volterra Equations .................................... 245 6.10. Numerical Solution of Fredholm Equations .................................. 250 6.11. Practical Example ........................................................................ 254 Bibliographic Notes and Comments .............................................. 260 APPENDIX 1 List of Doctoral Dissertations on Integration and Integral Equations .... 267

APPENDIX 2 Integration Functions and Subroutines ................

..................... 273

APPENDIX 3 Subroutines for Solving Integral Equations ............................................ 287

AUTHOR INDEX ...................................................................... 293 SUBJECT INDEX ...................................................................... 299

Chapter 1

GENERAL BACKGROUND

1.1

INTRODUCTION

Mathematical expressions involving integrals frequently arise in science and engineering. These expressions can be classified as explicit integrals and integral equations. An explicitt' integral such as

y(x) _ `I

X2

exp(xt) costdt

(1.1)

0

has a known integrand, while an integral equation such as

y(x) = 1 + J (x-t)y(t)dt

(1.2)

0

has an unknown function in the integrand. Equations such as

y + I X (x-t)y(t)dt,

(1.3)

0

where both derivatives and integrals of the unknown function appear, are called integrodiferential equations. To a physicist or engineer, integration generally means either the inverse of differentiation or the area under a curve. Present-day mathematicians have developed more sophisticated conceptions of the integral. While these abstract logical structures are not immediately applicable, a brief account is necessary background. It is important to be able to read the mathematical literature and discuss problems with mathematicians. Experience has shown that pure mathematics when suitably interpreted has useful applications. Although we now use the same symbolism for the integral as an antiderivative and as an area under a curve, these concepts have separate histories. Obviously the concept of an antiderivative could not arise until the end of the seventeenth century, when Newton and Leibniz created the differential calculus. The concept of area under a curve (quadrature) goes back to the Greeks. Archimedes' method of exhaustion came very close to the idea of integration.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

2

An interesting determination of the area between curves is the "lunes of Hippocrates." Hippocrates of Chios, a fifth century B.c. mathematician, was a contemporary of the famous physician. His books have not survived but some of his theorems are included in Euclid's Elements. He proved that the area of a circle is proportional to the square of the radius, though his method did not determine the numerical value of the proportionality factor. From this result and the Pythagorean theorem for right triangles it follows that the area of a semicircle with the hypothenuse as a diameter is equal to the sums of the areas of the two semicircles with the sides as diameters. Therefore the construction in Fig. 1.1 involving an isosceles (45°) right triangle shows that the area of two moon-shaped shaded regions (the lunes) is equal to the area of the right triangle. If the sides of the triangle

Figure 1.1

are taken as 1, the area of each Tune is #. This remarkable result was the source of the persistent interest in squaring the circle. It was believed that if this area could be evaluated, so could that of a simple figure such as a circle. The evaluation of the area of a tune by analytical geometry and calculus involves calculating

A= J o

- ( x - )Z]}dx - I

([+-(x-})2]#--

}dx.

(1.4)

For this problem the classical geometrical approach is superior to the modern

analytical approach. The geometrical approach, however, depends on an ingenious device applicable only to the particular problem, whereas the analytical method is a general technique for determining the area between curves. This book solves many problems involving integrals. Some are handled by special devices, others by general procedures. No one is clever enough or has enough time to handle all his problems by special methods, so a knowledge of general procedures is essential. While the ability to invent

ingenious devices is to some degree an inherent talent, practice and the

(1.1)

3

GENERAL BACKGROUND

study of examples can enhance this ability considerably. We benefit by the work of our predecessors and the inspired proof of one generation becomes a routine homework problem in the next. The scope for ingenuity, however, always becomes wider. The general methods simply open up new fields. Exercises 1

Derive (1.4) by analytical geometry and evaluate the integrals.

2

(a) Show that the area between three equal cotangent circles (Fig. 1.2a)

is

r2(31/2-n;'2).

By using the concept of orthogonal projection, prove that the area between

(b)

three congruent parallel cotangent ellipses is (Fig. 1.2b) ab(3'"2-n/2).

Figure 1.2a

Figure 1.2b

1.2 RIEMANN, STIELTJES, AND LEBESGUE INTEGRALS

Toward the end of the nineteenth century, Lebesgue formulated properties that the integral of any bounded function should have. b+h

6

f(x) dx = I

(a)

Ja+h

a

(b)

f(x-h)dx;

Jbfdx+ `cfdx+Jafdx=0; a

b

c

f.b

f.b

f,.b

(c)

( fi+.f2)dx=

(d)

If f > 0 and b > a, b

J'fdx>0;

fidx+

f2dx.

six

INTEGRATION FOR ENGINEERS AND SCIENTISTS

4 1

1 dx = 1.

(e) 0

(f)

Iffn

(x) < fn+ 1 (x)

lim fn(x) =f(x)

and

for all x

n-ao

then lim

fbf(x)dx a

Although mathematicians have developed more than sixty kinds of integrals,

the general problem of satisfying all six conditions simultaneously is still

unsolved. Most of the theory is of little interest to the scientist, as the difficulties involve "pathological" functions that are not encountered in applications.

To provide background, a few important definitions of integration are outlined. These can be ordered in a systematic scheme (1), the Riemann definition being least general. Each more general definition reduces to the less general definition when it is meaningful, but gives a result for some additional cases. Lebesgue-Stieltjes Lebesgue

Stieltjes

Riemann (1) 1.2.1

Riemann Integration

The Riemann definition is closest to the one used in elementary calculus

courses and is in fact used in some modern texts. Consider a bounded function f, which needs not be continuous, defined over the range a 5 x < b. The range of integration is divided into arbitrary intervals by taking a set of division points

a=x0<x1 <x2<x3<

<x,,=b.

A lower bound for the integral is found by using the minimum value off in each interval, which is denoted by mink in the expression N

SN = Y, min(xk-xk-1). k=1

k

(1.5)

(1.2.2)

5

GENERAL BACKGROUND

This is called the lower Darboux sum. Increasing the number of intervals will either increase the sum or leave it unchanged. Similarly, an upper Darboux sum is defined as N

SN = I max(xk-xk_1) k=1

(1.6)

k

where maxk is the maximum value off between xk _ 1 and xk. When these two

limits are equal, the function is Riemann integrable and the common value is the Riemann integral. Functions that are Riemann integrable include (a) all continuous functions; (b) all bounded monotone functions; (c) bounded functions with discontinuities limited to a finite number of regions, the total length of which can be made arbitrarily small; (d) the sum, product, or quotient of two integrable functions, provided that the denominator of the quotient does not vanish in the range of integration. Moreover, (e) if f(x) is Riemann integrable, then I f (x) I is also.

1.2.2

Stieltjes Integration

Stieltjes' generalization of Riemann integration is based on the introduction of a weight function a(x). The definition of the lower Darboux sum is changed to N

SN = Y, min[a(xk)-a(xk_ 1)], k=1

(1.7)

k

and the definition of the upper Darboux sum is changed correspondingly. Again, if the limits are equal, the common value defines Saf(x)da(x). For a(x) = x, this reduces to the Riemann integral, When a(x) is differentiable, the Stieltjes integral is equivalent to the Riemann integral of f; f(x)d(x)dx. The purpose of the Stieltjes generalization is to handle cases where x(x) is not continuous. Thus defining a(x) = I when x is an integer and a(x) = 0 when it is not an integer makes it possible to consider sums as Stieltjes integrals. A typical example is a unified formulation of moments of inertia for both continuous distributions and particles. The usual definition for the moment of inertia of a collection of N particles is N

rk k=1

where mk is the mass and rk the distance from the axis of the k" particle. For

a continuous distribution,

INTEGRATION FOR ENGINEERS AND SCIENTISTS

6

b

I =

p(x)x2 dx.

J

(1.9)

a

By introducing a mass function M(x) defined by

dM = p(x) dx,

(1.10)

we get for the expression of the combined moment of inertia, b

I=

x2dM. a

In the case of particles, the function M changes abruptly at its location.

The Stieltjes integral exists when (a) f(x) and a(x) have no common discontinuities; (b) f(x) is piecewise continuous; (c) a(x) is of bounded variation. A function of bounded variation has a finite arc length in a finite interval; thus sin(1/x) is not of bounded variation in the neighborhood

ofx=0.

1.2.3 Lebesgue integration

Neither the Riemann or Stieltjes definition will work forf(x) defined over

the range 0 to I by

J(x) = 1 J(x) = 2

when x is irrational; when x is rational. No matter how small the divisions, the lower limit is I and the upper limit is 2. It can be evaluated by using Lebesgue's sixth property. Since the rational numbers are a denumerably infinite set, a sequence of functions is defined by

Jo = 1;

Jl = 1 except at the first rational number, where it is 2; J2 = 1 except at the first and second rational numbers, and so on. For a finite n there are only a finite number of discontinuities, so that (1.12) 1 J,,, the sixth property gives I for Jo Jdx. Usually, Lebesgue integration is treated in terms of sets and their measure.

An example of a set is the collection of points on a line segment, and the length of the line is the measure of such a set. The points corresponding to the rational numbers in the segment are a subset, but the subset in this case is a subset of zero measure. Even the smallest finite segment has a larger number of points than there are rational points in a much longer

(1.2.3)

7

GENERAL BACKGROUND

segment. The concept of the Lebesgue measure of a set can be used to unify

the conditions for the existence of a Riemann integral by the following theorem. THEOREM 1.1 A function f is Riemann integrable over an interval if and only if it is bounded and the set of discontinuities off that lie in the interval has zero Lebesgue measure.

Note: The short form iff is used occasionally throughout the book for "if and only if." Exercises 1

Prove that for any definition of an integral as the limit of a sum the following results hold. (a)

b

fdx.

b If I dx , Ja

fa

(d)

J0fdx=o. 2

Show that lo' x dx =

3

Agnew defines the Archimedean integral by

directly from the Riemann definition.

' lim Jfdx = N-. o

1

N

.f

k

N kt (N

(a) Generalize this definition to an arbitrary finite interval. (b) Evaluate lo' x dx.

(c) Show that for the example by which we introduced the Lebesgue integral this procedure gives 2 and discuss the implications.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

8

1.3 MULTIPLE AND ITERATED INTEGRALS

A double integral over a region of the xy plane can be interpreted geometrically as a volume, so the term cubature is sometimes used as an analog to the term quadrature. Although the discussion in this section is in terms of the double integral

1 = ff f(x, y)dxdy = iffdS,

(1.13)

R

R

the generalization to higher orders is straightforward. Equation (1.13) can be defined by a generalization of the Riemann definition. The region of integration is divided into parts and limits are obtained by using the maximum and minimum values off in each subdivision. When these limits are equal, the Riemann integral is defined. Altough this procedure can only be carried out explicitly for simple integrands and regions, it does make it possible to establish general properties of multiple integrals analogous to those of simple integrals presented in the previous sections. Since multiple integration is a linear operation, cfdS =

JJfdS

(1.14a)

JJR

and

jj(f+)dS=JJRfdS+ fJRfdS;

(1.14b)

also, if R = R1 + R2('('

JJRfdS=

I

fdS+ JJfdS.

(1.14c)

Other properties are

dS = area of R,

(1.15)

JJR

f.;,, area of R

JffdS

< fmax area of R,

(1.16)

R

JJRf dsl s fIR IfI dS.

(1.17)

(1.3.1)

9

GENERAL BACKGROUND

Iff, R2, J$R2 fdS < JJR` g dS;

(1.18)

(1.19)

JjR, If I dS HR2

T he last relation is not valid without the absolute value restriction. 1.3.1

Conversion to an Iterated Integral

The usual method for the analytical evaluation of double integrals depends

on the conversion of the double integral to an iterated integral that can be evaluated by two successive integrations. Such a conversion is only possible when the region of integration has the property: lines drawn parallel to at least one axis must not intersect the boundary more than twice. However, it is always possible to subdivide a region into parts having this property. For example, the region between two circles (Fig. 1.3) does not have this property; but drawing two lines parallel to the axis tangent to the inner circle divides the figure into four regions that have this property. In this example regions 1 and 3 have the property with respect to both axes but regions 2 and 4 only have it for lines parallel to they axis.

Figure 1.3

Regions having this property can be bounded by constant values of one of the variables (say x) x, and x and curves y = u (x)

and

y = 1(x).

(1.20)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

10

The double integral is then equivalent to an iterated integral xw

fdS =

a(x)

f(x, y) dy.

dx x,

R

(1.21)

1(x)

The right side is evaluated by first carrying out the integral with respect to

y, treating x as a constant. This gives. a function F (x, y) such that 8Flay = J(x, y). Then the y is eliminated by using the limits of integration; that is, Cu(x)

f(x, y)dy = F[x, u(x)] - F[x,1(x)] = G(x).

(1.22)

)(x)

This leaves the ordinary integral G(x)dx

JJR fdS =

(1.23)

Jx,

to be evaluated. 1.3.2 Volume of a Sphere

For example, the volume of a sphere of radius a is given by V = 2 JJ (hi -x2-y2)'rdS,

(1.24)

R

the region of integration being a circle of radius a. This region satisfies the condition for conversion to an iterated integral. Consequently, (a2-x=)'/2

+a

V=2

dx -a

[a2-x2-y2]}dy:

(1.25)

-(az-x2)''2

It is found that

G(x) = In(a2-x2);

(1.26a)

hence,

V = n (+a (a2-x2)dx = -na3 J

(1.26b)

a

1.3.3 Transformation of Variables and the Jacobian

The problem is simplified considerably if polar coordinates are introduced x = rcosO,

(1.27a)

y=rsin9,

(1.27b)

(1.3.3)

GENERAL BACKGROUND

11

so that 2x

('

d0fa

V = 2 f` rdS = 2

r2dr=Irra3.

J

o

(1.28)

o

In this example the new coordinates had an obvious physical significance

that made the transformation easy. In the general case, a transformation from x and y to a new set of coordinates u(x, y) and v(x y) is only permissible if the quantity J(u,v)=axay-ayax

au av

au av

(1.29)

does not vanish. If r = u and 0 = v, this corresponds to J (r, 0) = cos 0 - r cos 0 - sin 0 (- r sin 0)

(1.30)

= r. The quantity J, which is called the Jacobian of the transformation, can be written as a determinant au

ay au

ax

ay

av

av

ax

and is abbreviated as a (x, Y)

a(u, v)

The Jacobian determines the element of area in the new variables by the relation

dxdy = a(x,y)dudv.

(1.31)

a(u, v)

Since u and v can be considered as the original coordinates and x and y as new coordinates, it follows that a(x, y) a(u, v) a(u,u) a(x,y)

=1.

(1.32)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

12

1.3.4 The Morris Transformation

This has been used by Morris to transform a double integral to a form involving a constant integrand. We can write

JJ R xY

f

f(x, y)dxdy = J UUf[x(u, v), y(u, v)]Jdudv.

(1.33)

R

If we set a(u, v)

au ov

_ au av

f(x,y)=a(x,y)axay ay ax'

(1.34)

an infinite number of transformations that make the integrand 1 can be obtained. Either u or v can be chosen arbitrarily and the other obtained by treating (1.34) as a first-order linear partial differential equation. Thus we can transform (1.25) by taking

u=x

(1.35a)

and solving av

= (a2-x2-y2)}

(1.35b)

ay

to obtain

v = J(a2-x2) aresin

y

L a2-x2

+ y(a2-x2-y2)}.

(1.35c)

2

This looks complicated but since it is known that the transformed integrand is 1, it is only necessary to find the transformed boundary. The limits on u

are from -a to a and on v from -n/4(a2-u2) to it/4(a2-y2).Therefore fa `s/4(a2-u2) V=2

- J -a/4(a2-u2) dvdu

(1.36)

('a

= 2n J a (a2-u2)du = 1na3. -a

Actually, the principal use of the Morris transformation is for graphical rather than analytical evaluation of integrals. In engineering practice it is often necessary to evaluate quantities such as JJ(x2+y2)dS, if x2dS, or

(1.3.5)

13

GENERAL BACKGROUND

If xydS over irregular areas defined by drawings. A suitable Morris trans-

formation reduces this to evaluating the area of a figure obtained by a graphical transformation of the original area. 1.3.5

Interchanging the Order of Integration

In dealing with iterated integrals it is often convenient to change the order of integration. This is valid when the integrand is continuous, but requires care when there are singular points in the range of integration. Unfortunately, it is difficult to formulate explicitly the necessary conditions for the validity of the interchange. Calculus books generally give sufficient

conditions, such as continuity of the integrand. The problem can be simplified by using the theory of Lebesgue integrals. A theorem due to Fubini states that the interchange of order is permissible when If 11(x)) dS is finite. Exercises

1

2

Prove (1.15) through (1.19).

Morris gives the following transformations for some integrands that frequently arise in engineering for computing centers of gravity, moments of inertia, and products of inertia.

Transformation

Integrand x2 + y2 x2

u = (x2 -Y2)l2

u=(x'/3)+xy2 u=x2/2

v=y

v=xy

u=x2/2

v=y v=y

u = x2/2

v = y2

u=x2y/2

v=Y

u=x'/3

x xy

v = xy

(a) Verify that these transformation do reduce the integrand to unity. (b) Apply these transformations to (i) a quarter circle and (ii) an equilateral triangle, and plot the transformed region. (c) Examine the effect of the choice of the origin of the coordinate system.

3

Show that

x_y3dxdy = f 'of 00 (x+y) 1

4

1

X-Y dydx _

I f'o 1 f "0

(X+Y)3

1

2

Generalize the Jacobian to three dimensions and calculate it for the transformation from rectangular to spherical coordinates.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

14

1.4 IMPROPER AND INFINITE INTEGRALS

The definition of integration and most of the results presented in the previous sections are only applicable when the range of integration is finite and the integrand does not become infinite in the range of integration. An integral over an infinite range is called an infinite integral even though its value may be finite or even zero. When the integrand becomes infinite in the range of integration, the integral is referred to as improper. The two problems

are related and usually treated together. 1.4.1

Improper Integrals

To illustrate the difficulties, consider

fbdx

1

1

° x2

a

b

(1.37)

This breaks down when the range of integration includes x = 0. For example, according to (1.37),1(-&, --) = - 20. This is obviously ridiculous since the integrand is positive (and 100) over the entire range of integration. Similarly, consider b) (e°x-eb,)dx. I(a, (1.38) x

fo

If a > b, the integrand is positive over the entire range, whereas if a < b, the integrand is always negative. By formal manipulation, however, we obtain x °° 0,0

ebx

dx X

L= f-'o

a= ax

u

fo

= 1 °° ebx d bx = Jo bx

°°

u

e"

du.

(1.39b)

u

o

Therefore, subtraction apparently gives I(a, b) = 0.

In order to deal with an integral extending over a range including a singularity, it is necessary to exclude a small region surrounding the singularity and consider the limit as the excluded region approaches 0; that is,

fdx

f.b

f(x)dx = lien e-.0

a

++t s-c

fdx

+J

b

$+e

fdx . J

(1.40)

(1.4.3)

15

GENERAL BACKGROUND

If the singularity is at an end point, it is only necessary to consider 6

lira e-'0

a+e

fdx.

Since the end point can be transferred to x = 0 by a linear change of variable, it is only necessary to analyze the behavior of

1(n) = lim b E-0

dx

b1-n-E1-n

= lim

a X"

(1.41)

I- n

a-.o

It can be seen that if n < 1, the value of I(n) is finite. By using the mean value theorem (discussed in Section 1.5), the results can be generalized to integrals of the form !o f (x)/x° dx. For example, it can be shown that 5, cos xdx/x} is finite, while lo' cos xdx/x and J sin xdx/x2 diverge. 1.4.2 Cauchy Principal Value

In some cases the integration of a divergent integrand across a singularity gives a finite result because of cancellation. For example, applying (1.40) to dx/x3 gives 1

dx

dx

dx

-1/2 X3 + J -e

-112 X3

_

-

l \-

+2f+ 282

X3

1 dx

+,f

a

X3

(1.42) 282

282/ +

.

2 + 282

The terms e- 2 cancel out, leaving the value I The value obtained in this way

is called the Cauchy principal value of the integral. The existence of the principal value depends on the change of sign of the integrand across the singularity. It is easily shown that 5'_ 4 dx/x2 does not have a principal value and this is the reason for the paradoxical result obtained from Eq. (1.37). Similarly, the paradox in Eq. (1.38) is due to the fact that whereas (1.38) is a proper integral, the two parts in which it is split for Eq. (1.39a, b) are individually improper. 1.4.3 Finite Part of an Integral

A further generalization of the integral concept, which gives a finite value for some cases where the integral is formally divergent, is referred to as the finite part of an improper integral. This concept arose in connection with the solution of partial differential equations. For example,

u = K[(x-a)2 + (x-b)2 + (x-c)2]-1/2

(1.43a)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

18

is a solution of a zu ax2

azu

++02U

ay2

2

az

_0

(1.43b)

representing a source or sink of strength K. Since the equation is linear the superposition principle applies and any linear combination of (1.43a) for different values of a, b, or c is also a solution. Since differentiation and integration are both linear operations such functions as au/aa, J± i of (a)da, and alac $± 1 !o f(a, b) u da db are also solutions. A difficulty sometimes arises in trying to evaluate the integral of a derivative. For example, while (1.44)

(x-t)-I12dt=(x-a)-112

ax

a

is easily obtained by evaluating the integral and then differentiating, an attempt to evaluate ax(x-1)`112dt= --

f

Jo(x-t)-312dt

(1.45)

2

a'l

faces the difficulty that the integral is improper. The finite part of an integral is introduced by the definition

ax(x-t)-"dt =

f

a

f

(x-t)-lizdt.

a

ax

(1.46)

This can be generalized to give

fxa

A(t)

a ax (x-t)112

dt = a f s

A(t) dt ax a (x-t)112

(1.47a)

and

f= a"

A(t)

a ax" (X-t)1/2

=(-1)n1.3..... 2n-1 f" A(t)dt (1.47b) (x-t)a+}'

21

a

1.4.4 Infinite Integrals

The analysis of infinite integrals is based on considering the limit of a finite integral as the limit of integration approaches infinity. In analogy with Eq. (1.41) let us consider Rn+1-an+1 lim CR I(n) = xndx = lim (1.48) a R-. 00 R- OD n+1

(1.1.4)

17

GENERAL BACKGROUND

The situation is now reversed and the intergral diverges if n > -1; whereas if n < -1, a finite value is obtained. It is, however, necessary to consider the behavior of integrands, such as sin x and cos x, that do not behave as a power of x for large x. This can be done by invoking an analogy between infinite integrals and power series.

The integral can be considered a sum of integrals over subranges. For example,

f

1

dx

N-1

x2

1=1

i+1 dx x2

i

N-1

1

i=1 i(i+1)

(1.49a)

and

N-1

N

xdx =

i=1

N-1

i+1

xdx = i

2i+1.

(1.49b)

i=1

The first series and integral converge and the second diverges. Since a series of positive terms that converges is called absolutely convergent, an integral such as IT dx/x2 is also referred to as being absolutely convergent. It is rather surprising that it is not necessary for the integrand to approach

0 as x - oo for the integral to converge absolutely. Convergent integrals for which this condition is not met can be manufactured by defining the integrand as a function defined between n and n+ I in such a way that the maximum value remains the same but the integral decreases rapidly enough for the sum to converge. A classic example due to Dirichlet is

f(x) = 0,

n < x < n+1-(n+1)-2;

f(x)=(n+1)4(n+1-x)[x-n+l+(n+1)-2]

(1.50a) (1.50b)

for

n+1-(n+1)-2<x
It can be shown that Jr cos x dx/x} converges, although it would not if the absolute value of the integrand were taken. This shows that there is a basic difference between a Riemann integral and an infinite integral because if f (x) is Riemann integrable, j f (x)I is also.

Integrals that converge because of the oscillatory nature of the integrand are called conditionally convergent. It is an interesting example of the uneven

development of mathematics that the basic theorem for conditionally convergent integrals was proven by Chartier in 1853, before the concept of absolutely convergent integrals had been formulated. Chartier's theorem states that if f (x) approaches zero steadily as x - oo and if 11.1 g(x) dxj is

bounded, then J; f(x)g(x)dx is convergent.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

18

Integrals such as fR

sin x dx = 1- cos R

(1.51a)

cosxdx = sin R

(1.51b)

0

and J`R

J0

are neither absolutely nor conditionally convergent, but as the value remains

between ± 1 they are bounded. Therefore, when multiplied by a suitable steadily decreasing function, they give conditionally convergent integrals. 1.4.5

Summability Methods

The concept of an integral can be extended to give definite values for some nonconvergent integrals by a number of methods. Two of the most important are Abel's method, which defines

f(x)dx = lim J e-"f(x) dx,

J

(1.52)

S-0

.00

and Cesaro's method,, which uses

JR

J f(x) dx = lim R- ao

a

R-x f(x) dx.

a R-a

(1.53)

By methods to be discussed later it can be shown that

e-S:sinxdx = J0

1

(1.54a)

s2+1

and J(

e-SXcosxdx= 0

2 s+ s

(1.54b)

Taking the limit as s -+ 0 gives sin x dx = 1

(1.55a)

cosxdx = 0.

(1.55b)

0

and

0 J"o

However, it is not valid to put this result into (1.51a, b) and conclude that lim cos R = 0 (1.56a) R -,*

(1.4.5)

GENERAL BACKGROUND

19

and

lim sin R = 0.

(1.56b)

R-co Exercises 1

Determine whether the following integrals converge. (a)

cos x2 dx foo (b)

dx

1`° o

sin x xi/2/2

(c)

xexp(-x6 sin2x)dx (d)

dx f-,C x2 (sin x)213 (e)

°

x

sin(x+x2)dx"

o

(f)

e"""sin2x 2

dx x"

Show that (a)

f04

(a2

x 2 dx2x

it

-

2

)3/2

(b) b

dx

f:(a_x)3(x_b)h/2

fa

(x-b) (a-x)

(c)

fol foX

3"2 dx

b

dtdx _ (X-t)3/2t1/2

'

(d)

dxdt _ (x-t)3/2t1/2 j'o f i

- 27t.

_

0'

20

INTEGRATION FOR ENGINEERS AND SCIENTISTS

3

Evaluate f a sin x dx and f p cos x dx by Cesaro's method.

4

Construct an example analogous to the Dirichlet example that converges even though the integrand does not approach 0 as x-+ oo. Try to construct one where f(x) actually increases.

1.5 MEAN VALUE THEOREMS

Two important theorems covered in elementary calculus texts are the mean value theorems. THEOREM 1.2 (First Mean Value Theorem) b

fdx i (b-a)fmin.

(b-a)fmax %

(1.57)

a

If f is continuous, it can be written as b

f fdx = (b-a)f(x)

(1.58)

a

where b > x >-a. An important modification is obtained by splitting the integrand into factors and writing b

j'b

fmax J g dx

$fgdxfminj' a

o

b

g dx

(1.59a)

a

or b

b

Jf9dx = f(x) a

J

g dx.

(1.59b)

a

The second mean value theorem puts the undertermined value x in the limit of integration. THEOREM 1.3 (Second Mean Value Theorem) If f has a continuous derivative that does not change sign between a and b, then there is a value x

between a and b such that

5fdx=faJ sgdx+f( b)fXbgdx.

(1.60)

Exercises 1

2

Show that if f is continuous and > 0 in the range a to b and J6 .f dx = 0, then f = 0. (a) If g(x) is continuous and f(x) is a continuous differentiable function such that

f(a) = f(b) = 0, and f; fg dx = 0, show that g = 0. (b) Generalize the foregoing result to two dimensions.

(1.6)

GENERAL BACKGROUND

21

1.6 INEQUALITIES

A number of useful inequalities for integrals are analogs of corresponding inequalities for vectors. These are based on the concept of a function space where the integral of the product of two functions is the analog of vector multiplication. The Schwartz-Bunyakowsky inequality is

f g2dx. 6

6

[JJdx]2 a

(1.61)

Minkowski's inequality states that for p > I 1/p

b

1/p

1/p

b

_<[j' Iflpdt] [ f Iglpdt] a a

if+glpdt

,

(1.62)

while Holder's inequality states that

Jbfgdt fabIfgIdt

[Jb Iflpdt]l/p a

where l/p+ l/q = 1.

Fib x

Igl"dt1/Q

(1.63)

a

The individual inequalities are relatively easy to prove. For example, (1.61) is derived by noting that 1(A) = f". ( Af+g)2 dx

(1.64)

must be positive for all real values of A. Expanding the square gives b

I(A) = A2

fZdx + 2A

fgdx +

g2dx

a

=(A-AN)(A-A2) J0bf2dx

(1.65)

where Al and J2 are the roots of the quadratic I(A) = 0. It can be seen that if there are two distinct real roots, -there will be values of A for which !(A) is negative. The condition for the roots of I(A) to be equal or complex is that the discriminant 2

2

fgdx

-4 fZdx

('ab

{

,l

Transposing gives Eq. (1.61).

g2 dx ,0.

(1.66)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

22 1.6.1

The Chebyshev Inequality

A different type of inequality was first stated by Chebyshev. An increasing function is defined as one for which

J(x)-J(y) < 0

if y < x

(1.67)

when y < x.

(1.68)

and a decreasing function as one for which

f (y)-J(x) < 0 It can be seen that the double integral

ii

I

b a

(1.69)

[u(X)-u(y)][v(x)-v(y)]dxdy

is positive if u and v are the same type and negative if they are different. Multiplying out gives b ib

1(u, v) = Ja

[u(x)v(x) + u(y)v(y)] dx dy a

- TabI bs [u(x)v(y) + u(y)v(x)] dxdy.

(1.70)

Since the variables of integration are dummies, this can be rewritten to give ('b

b

o

Ja

41(u, v) = (b-a) J uvdt -

b

udt

vdt.

(1.71)

fd

b

(b-a)

uvdt a

is greater or less than b

I

b

udt

Ja

vdt, Ja

depending on whether u and v are of the same or different types. Obviously if u = v, they are of the same kind, so that

[$:udt]2
(1.72)

whereas u and 1 /u are of opposite kinds, so that b

S:udt fa

d >, (b-a)2. u

(1.73)

(1.6.2)

23

GENERAL BACKGROUND

A simple application of (1.73) is

JXxdx fsd->(x-1)2. 1

(1.74)

x

Since the integrals are elementary, this gives In

x>2x-1

In

11-x >2

x+1

,

1+x

x

x>1,

(1.75a)

x<1.

(1.75b)

The Chebyshev inequality can be written in the more general form b

b

pdx

fb

>f.b

pudx

pvdx (1.76) fa w here p >, 0 in the range of integration. This can be interpreted as an

f

uvpdx

a

extension of the simple inequality to Stieltjes integrals. 1.6.2 Bounds for Integrals

By using the Chebyshev inequality on remainder terms for numerical integration, which is discussed in Chapters 4 and 5, simple expressions can be obtained when a derivative of the integrand does not change sign in the range of integration. When f (3) is positive, (b

2

a) [f(a) +f(b)] +

f(a) - f (b)

(b ba)2

[ >J

b

fdx <

- a(a )]

(b 2 a) [f(a)

+f(b)).

(1.77)

The inequality is reversed if f (3) is negative throughout the entire range of integration. Higher-order approximations of this type are (b - a) 6

[f(a) + 4f(m) +f(b)] +

(b - a)2 144

[f(b) -f (a)]

- (b-a)3 [f(b) +f(a)] < J bfdx 288

< (b

a

- a) [f(a +f(b)] + (b 6a)2 Lf(a)2

f(b) -f(a)1

(b-a) J

(1.78)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

24

if f (4) is positive and

8f(m) +f(b)] + (b )31(m) +

(b-a) 10 U(a) +

+

(b-a)[ft3'(b) 28,800

-(b-a)5

_ft31(a)]

[ f(4)(b)-ft4'(a)]

57,600

< `b fdx < (612x) [f(a) + 10f(m) +f(b)] + a

+ (b-a)3f(m) + (b-a)5ft4'(m) 11,520

48

(1.79)

if f (5) is positive. In these equations m = (b+a)/2. Exercises 1

2 3

Prove the Holder and Minkowski inequalities. Prove the generalized Chebyshev inequality, Eq. (1.76), by conversion to a double integral. Prove the following. (a)

lnx < xt/z

for x > 1,

X-1

for

>

x < 1.

(b)

sin

_'x =

s

dx

fo (1-x2)t/2

< 4 [(1+x)112+(1-x)112-(1-x2)' 2-1]. x

(c) -'x)2<xIn1+x

(sin 2 4

1-x

By using (1.78) we find that

2 d < 0.6932.

0.6926 < J

I

X

Calculate the bound obtained by splitting the integral into two parts r' +k dx + dx t

X

Jt+kx

and tabulate the bounds for k = 0.1(0.1)0.9.

(1.7.1)

25

GENERAL BACKGROUND

1.7 INDEFINITE INTEGRALS.VERSUS DEFINITE INTEGRALS

The previous sections are based on the concept of the definite integral as contrasted to the indefinite integral or antiderivative

F(x) = JF(x)dx.

(1.80)

The basic distinction is that an indefinite integral defines a function of the variable of integration, whereas a definite integral is either a number or a function of parameters appearing in the integrand or limits of integration. The variable of integration is a dummy variable. This distinction remains even when an indefinite integral is written with a specific lower limit to determine the integration constant. Thus, although (`x

cos x dx = sinx

(1.81a)

x cos xt di = sin x

(1.81b)

0

and J1 0

have the same right side, there is a fundamental distinction. Differentiation of the right side gives the integrand of (1.81 a) but not of (1.81 b). Therefore, the value of an indefinite integral given in a table can be checked by differentia-

tion but there is no similar method of verifying a result from a table of definite integrals. 1.7.1

Derivative of a Definite Integral

In Section 1.4, it was necessary to consider the derivative of a definite integral with respect to a paramater. This was interpreted as the result obtained by carrying out the integration and then differentiating. As in many cases the integration cannot be carried out, an alternate definition is often useful. If 4(a)

fix, a) dx,

1(a) =

(1.82a)

((a)

then by the basic definition of a derivative r Cu(a+h)

%(a) = lim h-' h-.0

J L

!(a+h)

u(a)

Ax, a+h)dx -

f(x, a) dx 1(a)

(1.82b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

26

The first integral can be split into three parts f!(a)

o(ath)

u(a)

+I .JI ua+n)

+

JJ,(a)

It can be shown that fa(a)

h- 0

fu(a)

[f(x, a+ h) -f(x, a)] dx =

lim h-' ,(a)

a f(x, a) dx.

(1.83)

,(a) as

By using the mean value theorem, we get a(a+h)

f(x, a+h)dx = f[u(a+kh), a+ h} [u(a+h)-u(a)] (1.84)

u(a)

where 0 < k < 1. Therefore, in the limit as h -+ 0

h-'

u(a+h)

f "0(.)

?u (a)

f(x, a+h)dx = f[u(a), a] V

as

and similarly

al(o)

,(a)

f(x, a+h)dx = -f[l(a), a]

lim h-'

h-'0

(1.85a)

,(o+h)

(1.85b) as

Therefore

fu(a)

u(a)

f(x, a)dx as

a

f(x, a) dx

,(a) 8a

,(a)

+ au f [u (a), a]

- a! f[l(a), a]. 1.7.2

(1.86)

Discontinuity of Definite Integrals

It is rather surprising that a definite integral with a continuous integrand

may be a. discontinuous function of a parameter in the integrand. For example, later we will prove by several methods that 1(a)

C'0 o

sinaxd=- ifa>0, x

2

(1.7.2)

27

GENERAL BACKGROUND

ifa=0,

=0 n

ifa < 0.

(1.87)

2

Taking the derivative with respect to a gives

1(a) = I

(1.88)

cos ax dx. 0

It is easily seen that the derivative is infinite if a = 0 and it was shown in Section 1.4 that the integral in (1.88) vanishes when a 0 0.

Exercises 1

Evaluate the following both by integration followed by partial differentiation and by using (1.86) followed by integration. a

f3=

at

2

)(x2+12dx

a

(x+tt'2)dx

at 2

o

Show that fb11

a

alt

tat

f

(tx)

dx

X =0

where a and b are independent of t. 3

Investigate the following for continuity with respect to a. (b)

(a)

a0 cos ax

J0 1+ X,

°° [sin ax1J

dx

o

(c)

x

dx

sin x cos ax 00C

4

X

Show by differentiation that

y = I f(t)sink(x-t)dt 0

satisfies the differential equation

Y+k2y =f(x).

x

dx

INTEGRATION FOR ENGINEERS AND SCIENTISTS

28

1.8 FRACTIONAL INTEGRATION AND DIFFERENTIATION

The integral relation

J(x_t)f(r)dt

=

JJf(t)dtdt

0

(1.89)

fo

is obtained by applying integration by parts to the left side with

rof(t)dt,

u =x - t,

v= J

du = -dt,

dv = f(t)dt.

By induction it can be shown that for integral n's J

x (x !Qf(t)dt o

n!

-

f(t)(dt)R+1.

= 0

(1.90)

0f

This relation is used in Chapter 6 to convert differential equations to integral equations. It can be generalized to nonintegral values of n by replacing n! by r(n+ 1). The gamma function r is discussed in Chapter 3. We can then consider

f(t)dt

D--f= f

(1.91)

r(n)

J0

as a definition of an integration of fractional degree. It can be shown that D°+b= D°Db

(1.92)

whether a is integer or fractional, positive or negative. Therefore, we can obtain derivatives of fractional order by differentiating (1.91); for example D11Zf=7t-1/2dx

so that

J

of(t)(X-t)-112dt,

(1.93a)

D 1/2 1 = g -1/2 x -1/2

(1.93b)

= 2n -1/2 x 1/2

(1.93c)

and

D 1/2 x

It is also possible to extend Eq. (1.91) to imaginary and complex values of n since 1(n) is defined for complex arguments and

(x - t)'° = exp[ia ln(x - t)] = cos ln(x - t)+ isin ln(x - t)°.

(1.94)

29

GENERAL BACKGROUND

(1.9)

However, although fractional integration and differentiation has practical application (see Chapter 6), the author has not found any for complex n. Exercises 1

Evaluate

2

Show that D"e: = ex.

D'12x,12.

1.9

LINE INTEGRALS

The line integral is a generalization of the concept of the definite integral in which the value of the integral depends on a path of integration in addition

to the integrand. Consider an integrand h(x, y) that is a function of two variables. Geometrically, h can be interpreted as the height of a solid figure with the xy plane as the base. An ordinary definite integral Yb

fx,xb

h (x, y,) dx

or

l..

h(x,, y)dy

can be interpreted as the area intersected by a vertical plane parallel to either the x or y axis. A line integral involves the intersection of a vertical plane twisted in the xy plane with the solid figure. In some cases the interest is in the area this intersects, which can be designated as Jc h(x, y)dS where

dS is an element of the curve formed by the intersection of the twisted vertical plane and the xy plane. In other cases it is the projection of this area on vertical planes parallel to the x or y axes that is of interest. These are designated by

h(x, y)dx

h (x, y) d y.

or

Sc

Sc

In all three cases the value depends on the curve C, which determines the

path of integration. If this curve is known as a parametric expression

x =I(t),

y = g(t),

(1.95)

then

h(x,y)dS fc

=5 h[f(t),g(t)][J2(t)+92(t)]"2dt.

(1.96)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

30

The projected integrals are reduced to if h(x, y)dx = J h[f(t), g(t)] f(t) dt

(1.97a)

Sc and rf

f h(x, y)dy = { h[f(t), g(t)]g(t)dt. c

(1.97b)

r,

When the path of integration is given by an explicit expression y = y (x), it is necessary to split the curve into segments such that y is a monotone function of x on each segment. Then

h(x, y)ds =

1

f-'f h[x, y(x)](1+y2)112dx

(1.98)

c

reduces the line integral to definite integrals. In physical applications the grouping of projected line integrals

u(x, y, z)dx + v(x, y, z)dy + w(x, y, z)dz Sc

frequently arises. This integrand can be interpreted as the dot product of a vector such as force with components u, v, and w and distance along the

path of integration. There are important classes of functions where the value of such integrals becomes independent of the path and depends only

on the end points. In terms of forces this corresponds to conservative forces, which are derivable from a potential. In vector terminology this means that there exists a function P(x, y, z) such that

u _ ap , ax

v=

aP , ay

and w =

aP

(1.99)

az

Given a set of u, v, and w, the existence of such a function can be determined

by determining whether av az

aw,

- ay

aw

= au

and au

ax

az

ay

__ av .

(1.100)

ax

This can be recognized as the condition that the curl of the vector iu +jv + kw vanish.

It is obvious that if the value of a line integral depends only on the end points, when the path is a closed curve the value will be zero.

(1.11)

31

GENERAL BACKGROUND

1.10 SURFACE AND VOLUME INTEGRALS

The surface integral has the same relation to the simple double integral discussed in Section 1.3 that the line integral has to the ordinary definite integral. There is an important relation, Stokes's theorem, relating a surface integral to a line integral around the boundary of the surface

jJ oxVdS= #Vdl.

(1.101)

an d similar relations between surface and volume integrals are studied in vector analysis. Higher-dimensional generalizations, such as volume integrals and integrals over hypervolumes in phase spaces, are encountered in physics but will not be discussed here. We do want to point out, however, that the subject of higher-dimensional integrals is not completely exhausted. For example, Moon and Spencer recently generalized the familiar integral transformation theorems of vector analysis for retarded quantities. 1.11 SYMMETRY ARGUMENTS

In later chapters, extensive use is made of symmetry arguments. The basis of symmetry is the classification of functions as odd or even. An even function of the variable x satisfies the condition

f (x) = ft- x),

(1.102a)

whereas an odd function satisfies the condition

f(x) = -f(-x).

(I.102b)

This is sometimes referred to as the parity of the function (from the French pair and impair). Of course, most functions are neither odd nor even, but any

function can be split into an odd function and an even one by writing

f(x) = i[f(x) -f(-x)] + 11f(X) +f(-x)].

(1.103)

It is easily seen that f(x)+f(-x) is even and f(x) J(-x) is odd. Typical even functions are cos x and cosh x, while sin x and sinh x are odd. An example of the resolution of a nonsymmetrical function into odd and even parts is ex = sinh x + cosh x.

(1.104)

In the case of a function of several variables, parity can be considered with respect to each variable. Thus sin xy is odd with respect to both x and y but sin xy2 is odd with respect to x and even with respect to y.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

32

It is simple to see what happens to parity when functions are combined or operated on. The product of two functions of the same parity is an even function, while the product of an odd and even function is odd. Addition or subtraction of functions of the same type does not change parity, nor does multiplication by a constant. In fact, any linear combination of either even or odd functions retains its parity. Therefore, the power series expansion of an even function contains only even powers and the expansion of an odd function only odd powers. From this it can be seen that differentiation and

integration changes parity provided that the constant of integration is neglected.

Symmetry considerations can be a useful check on indefinite integrals, analogous to a dimensional check of a physical equation, in that they do not guarantee correctness but may detect an error. However, tables of integrals are frequently given in forms that conceal these symmetry considerations. For example, it is usually stated that

J=lnx. x

(1.105)

The integrand 1/x is an odd function, whereas In x is neither even nor odd. The difficulty is clearly seen if we consider J"' dx/x, which should be 0 by symmetry if it exists. Actually, the right side of (1.105) should be written either as In lxi or I In x2, so that it will be an even function. Another case is dx

= sin

_, x

(a2-xz)'/2

(1.106)

a

The integrand is an even function of both x and a, so the integral should be an even function of a and an odd function of x. However, sin-' x/a is odd with respect to both x and a. This can be corrected by either writing the right side as sin - ' x/Iai or using the alternate form dx (ag-x2)1/2 = arct an

x

(a2-x2)1/2'

( 1 . 107 )

which has the correct parity properties. Symmetry can be u sed to show that 2n

sin x dx = 0

f

(1.108)

0

by changing the point of symmetry from x = 0 to x = n. Showing that zR

f

0

cos x dx = 0

(1.109)

(1.11)

33

GENERAL BACKGROUND

involves splitting the integral in two parts, 0 to it and it to 21r, and noting that in each part the integrand is antisymmetrical about the midpoint. It is a general result that if the integrand is antisymmetrical about the midpoint in, then m+a

fdx=0.

(1.110)

Sm-a On the other hand, if the integrand is symmetrical around in, then m+a

m+a

=2

m-a

fdx = -2

fdx.

(1.111)

n. Jm-a' fm A typical symmetry argument occurs in the proof of the following theorem.

THEo M 1.4 If the integrals exist, then dx (a x) d x °f a +x) lnx-=1na JI f +x x x ( °°

J o

a

(1.112)

a

o

Introducing a new variable x/a = e" transforms the integral on the left side into +0 f(eY+e-'')(y+ln a)dy = J -ao

f+ 00

f(e+e-y)dy.

f(e+e-y)ydy + Ina

(1.113)

-ao

The first integral on the right vanishes because the integrand is odd. Then transforming back gives (1.112). Exercises 1

Apply (1.103) to In x.

2

Analyze a"(1 +e2")'' for symmetry.

3

Analyze the expression

fco In (1 +a2x2) 0

for symmetry.

A 1 + b2x2

=

it

b

In

a+b b

INTEGRATION FOR ENGINEERS AND SCIENTISTS

34

BIBLIOGRAPHIC NOTES AND COMMENTS

Most of the material in this chapter is covered in standard. works in advanced calculus, mathematical analysis, and applied mathematics. In preparing the chapter, the author used the following references because they were familiar and available, and contained material included in subsequent chapters. There are, however, many other equally useful references. [1] R. P. Agnew, Calculus (McGraw-Hill, New York, 1962) is a modern elementary calculus text that contains brief discussions of many advanced topics; the unusual problems

vividly reflect the author's personality. (2] I. N. Bronshtein and K. M. Semendyayev, A Guide Book to Mathematics for Technologists and Engineers (Macmillan, New York, 1964) is a handbook covering a remarkable range from elementary algebra through integral equations. [3] P. Franklin, Methods of Advanced Calculus (McGraw-Hill, New York, 1944) is an example of the older problem-oriented approach. [4) H. Jeffreys and B. Jeffreys, Methods of Mathematical Physics, 2nd ed. (Cambridge Univ. Press, London and New York, 1959) is an excellent advanced survey. [5] H. Lass, Elements of Pure and Applied Mathematics (McGraw-Hill, New York, 1957) is a compact survey for capable graduate students. [6] V. Smirnov, A Course in Higher Mathematics (Addison-Wesley, Reading, Massachusetts, 1964) is a five-volume translation of a work in the continental tradition (many worked examples but no exercises); it covers an entire sequence of undergraduate and graduate courses. (7) L. P. Smith, Mathematical Methods for Scientists and Engineers (Dover, New York, 1953) is an example of the survey designed for seniors and first-year graduate students.

It is somewhat more rigorous than most books of this type and its availability as a paperback makes it a "best buy."

[8] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, New York, 1927) is a classic reference in the old tradition now available in paperback. For many years it was the definitive reference on special functions. [9] D. W. Widder, Advanced Calculus (Prentice-Hall, Englewood Cliffs, New Jersey, 1961) is an example of the current abstract approach.

In later chapters we refer to these books by author's name and bracketed number. SECTION 1.1

The history of the calculus is traced in C. B. Boyer, The Concepts of the Calculus (Columbia Univ. Press, New York, 1939). A presentation stressing the historical development is 0. Toeplitz, Calculus, a Genetic Approach (Univ. of Chicago Press, Chicago, 1963).The material on the lunes of Hippocrates is from The World of Mathematics, James Newman,

ed. (Simon and Schuster, New York, 1956), Vol. I, pp. 90-91. Exercise 2 is from M. Klamkin and D. J. Newman, "The Philosophy and Application of Transform Theory", SIAM Rev. 3(1961), 10-36.

(1)

GENERAL BACKGROUND

35

SECTION 1.2

H. Lebesgue's classic is Lecons sur l'integration (Gauthier-Villars, Paris, 1928). A number of his more elementary publications are now available in translation in Measure and the Integral, K.O. May, ed (Holden-Day, San Francisco, 1966). The standard modern reference is E. J. McShane's Integration (Princeton Univ. Press, Princeton, New Jersey, 1947). A small but relatively readable book is R. Henstock's Theory of Integration (Butterworth, London and Washington, D. C., 1963). Two interesting survey papers are T. H. Hildebrandt, "Integration in Abstract Spaces," Bull. Amer. Math. Soc. 59(1953), 111--139, and E. J. McShane, "Integrals Devised for Special Purposes," Bull. Amer. Math. Soc. 69 (1963), 597-627. C. H. Page discusses the physical application of Lebesgue integration on p. 9 of Physical Mathematics (Van Nostrand, Princeton, New Jersey, 1955). SECTION 1.3

W. L. Morris, "A New Method for the Evaluation of f f f(x, y) dx dy," Amer. Math. Monthly 43 (1936), 358- 362. SECTION 1.4

The concept of the finite part of an integral was introduced by J. Hadamard in Lectures on Cauchy's Problem in Linear Partial Diryerential Equations (Dover, New York, 1952). However, we have used a slightly different definition, introduced by H. Lomax, M. A. Heaslet, and F. B. Fuller in "Integrals and Integral Equations in Linearized Wing Theory", NACA Rep. 1054 (1951), which simplifies the evaluation of multiple integrals.

The Dirichlet example and Chartier's theorem are in Whittaker and Watson [8]; Abel's and Cesaro's methods are covered in Widder [9). SECTION 1.5

A very general inequality, including those presented here as special cases, is given by S. lsayama, "Extension of the Known Integral Inequalities," Tohoku Math. J. 26(1926), 238-246.

The Chebyshev+ inequality was originally stated in a letter to Lcgendre, who presented a proof in lecture notes. The present proof and the first problems are from F. Franklin, "Proof of a Theorem of Tschebyshef's on Definite Integrals," Amer. J. Math. 7(1885), 377-379.

The generalized form and the first two bounds are from J. A. Shohat and A. V. Bushkovitch, "On Some Applications of the Tschebyshef Inequality for Definite Integrals," J. Math. Phys. 21(1942), 211-217. The third bound is due to A. K. Kharadze, "Mean Value Theorems Applied to Polynomials" (Tbilisi State Univ., Tbilisi, 1947), p. 137. SECTION 1.8

An extensive discussion of the history of fractional integration and differentiation, which goes back to Liouville, is given in H. T. Davis, Theory of Linear Operators (Principia

Press, Bloomington, Indiana, 1936). J. J. Smith and P. L. Alger, in "The Use of the Null-Unit Function in Generalized Integration," J. Franklin Inst. 253(1952), 235-250, claim to resolve a long-standing difficulty.

t There are several systems for transliterating from the Cyrillic to the Latin alphabet, so that alternate spellings are used for Russian names. Also, the names of references cited in translations of Russian material can be garbled.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

36 SECTION 1.10

A discussion of integrals in phase space may be found in any extensive treatment of statistical mechanics, for example, J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1964), Chapter 2. The vector trans-

formations for retarded quantities are in P. Moon and D. E. Spencer, Vectors (Van Nostrand, Princeton, New Jersey 1965), pp. 291-293. SUGGESTED RESEARCH EXERCISES 1

Explore the viewpoints expressed in the following quotations.

Does anyone believe that the differences between the Lebesgue and Riemann integrals

can have physical significance, that whether, say, an airplane would or would not fly could depend on this difference? If such were claimed I should not care to fly in that plane. - R.W. Hamming, "Numerical Analysis vs. Mathematics," Science 148(1965), 473-475.

There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they would like an integral to possess. This of course is quite annoying to us rigorous mathematicians; what is even more annoying is that by doing so they often come up with the right answer.

- E. J. McShane, "Integrals Devised for Special Purposes," Bull. Amer. Math. Soc. 69(1963), 597-627. 2

K. Menger, in Calculus, a Modern Approach (Ginn, Boston, 1955), criticized the conventional notation for integrals and proposed a system that he claims is easier to teach and use. Read his description and comment on it.

3

H. Macoll applied the calculus of propositions (a branch of symbolic logic) to

4

determining the new limits when the variables in a multiple integral are transformed. Study his method and explore the possibility of embodying it in a computer program. ("Calculus of Equivalent Statements," Proc. London Math. Soc. 9(1878), 177-186; "On the Limits of Multiple Integrals," Proc. London Math. Soc. 16(1884), 142-148). Show that

f'

dx

_

0 (1+x2n)t/n

I 0

dx (1-X2n)t/n

= Cos

it

Zti

for n = 2, 3, ... (Klamkin). Hint: It is not necessary to evaluate the integrals individually if you can find the correct geometrical interpretation in terms of a circle and inscribed polygon. 5

(a) Show that if v is a continuously differentiable real-valued function on the finite

closed interval a,< x,< b, then

f

b[v(x)-v(a)]2dx

(ba)j

°

a

and that for a< b the equality holds if v(x)=v(a)+csin('tx-a).

2b-a

(1)

37

GENERAL BACKGROUND

(b) Show that if v is a continuously differentiable real-valued function on the finite closed interval 0,< x,< 27C and

v(O) = v(2n),

(i)

(' 21t

vdx = 0,

J 0 then

ZR

2R

vdx

v2 dx

I

o

0

and that the equality holds if there are real numbers c and d such that

v(x) = d sin(x-c). [J. D. Diaz and F. T. Metcalf, "Variations on Wirtinger's Inequality," in Inequalities, O. Shisha, ed. (Academic Press, New York, 1967).] 6

Show that if u is a convex function continuous in the range 0< x< 1, then

v(x)=-l I s u(t)dt xJo

with v(0) = u(0)

is also convex. H. K. Wilson," On the Average of a Convex Function,"SIAM Rev. 10(1968), 115-116.]

Chapter 2

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

2.1

INTRODUCTION

In this chapter exact and approximate analytical methods for evaluating indefinite integrals are described. The indefinite integral can be defined as the inverse of differentiation: If

F(x) = f(x), then

F(x) = ff(x)dx. This is a rigorous descriptive definition since it is possible to determine whether a proposed F(x) is the integral off (x) by carrying out the differentiation. However, it is a nonconstructive definition, as it does not prescribe any method for finding F (x) whenf(x) is given. Since the derivative of a constant is zero, the function F(x) can have an

arbitrary constant added to it. This leads to some apparent paradoxes. Using integration by parts relations, such as

f

dx

xlnx

=1+J

dx

xlnx'

J cot x dx = I + J cot x dx,

(2.1a)

(2. l b)

are obtained. It does not follow from these relations that 0 = 1. Although we will usually omit the constant of integration, it is interesting to note that it can be used to show that the standard integration formula x dx

=

n+l

( 2 . 2)

does not really break down for n = - I but includes dx

T

= In x

(2.3)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

40

as a limiting case. This is done by writing xn+I_cn+1

x"dx =

n+1

which becomes indeterminate (0/0) as n -+ - 1. By using L'Hospital's rule,

lim J x"dx = lnx-lnc

(2.5)

n--1

is obtained. It is assumed that the reader has had the usual course in calculus covering methods for evaluating various integrals involving algebraic and elementary transcendental functions. This chapter presents a unified treatment of these

methods based on Liouville's work on integration in finite terms. In a remarkable series of papers published between 1833 and 1841 Joseph Liouville created a systematic theory and proved that certain apparently simple integrals, such as l ex2 dx, cannot be expressed in terms of elementary

functions. It is surprising that this work is not more widely known, but apparently no current text on either elementary or advanced calculus gives a

discussion of this work. To quote the mathematician J. F. Ritt, who has extended Liouville's work and published a treatment using more modern complex variable techniques, "The questions treated by Liouville are questions which occur to every strong undergraduate student of mathematics.

Nevertheless Liouville's work never received very wide attention. It has always been something everyone would like very much to know about but which very few undertake to study".

In addition to presenting an elementary exposition of this theory, we make use of certain devices for evaluating integrals, such as assigning complex values to parameters and partial differentiation, that are useful shortcuts but cannot be used in elementary courses. For example, starting with the standard integration formula

Jedx__k1e(2.6) and taking k as the imaginaryiw gives e'wx

dx = j'coswx + i sin wx dx = (cos wx + i sin tax) iw

J

(2.7)

Equating real and imaginary parts gives f cos wx dx = w-' sin wx

(2.8a)

41

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

(2.2)

J sinwxdx = -w-tcoswx.

(2.8b)

By taking advantage of the fact that integration and partial differentiation commute, that is, f tf k)dx

x,

- ak f f(x, k)dx

under reasonable restriction on f, such results as

f

xekx dx =

d ekx dx

8k

kx = a e = ekx x

ck k

k

1

(2.10)

kZ

are obtained more directly than by the usual integration by parts approach. Also, we cover methods using differential equations to derive integrals involving the higher transcendental functions, such as Bessel functions; discuss some recent work on the use of computers to evaluate integrals analytically; and describe some methods for obtaining approximate values for integrals that cannot be evaluated in closed form. Exercises 1

2 3

Develop a general method for deriving equations similar to (2.2a) and (2.2b). Evaluate f e°; sin bx dx and f e"" cos bx dx by setting k = a + ib in Eq. (2.6). Evaluate f x3 sin x dx and f x4 sin x dx.

2.2

LIOUVILLE'S CLASSIFICATION OF THE ELEMENTARY FUNCTIONS

In the preceding section we stated that Liouville proved that it was impossible to express f exp(x2)dx in terms of elementary functions. Obviously

this requires a definition of what is meant by an elementary function. The definition is purely conventional, consisting of these classes of functions:

(1) rational functions; (2) explicit and implicit algebraic functions; (3) exponential and logarithmic functions (elementary transcendentals); (4) functions defined by a finite combination of the preceding classes. A rational function is defined as the result of a finite number of additions

and multiplications on the variable x and can be expressed as the ratio of two polynomials in x. Only integral powers of x are admissible, but the coefficients can be irrational, transcendental, or complex numbers. It will also be necessary to consider rational functions of functions of x. For example, (1 +e")/(l-e2x) is a rational function of ex, whereas (l+ e-")/(1 +3e2") is not a rational function but it can be rewritten as either

INTEGRATION FOR ENGINEERS AND SCIENTISTS

42

(ex + I )/(ex+3 e3 x), which is a rational function of ex, or (e-

2x

+e -3 x)/(e- 2x+3),

which is a rational function of ex. Explicit algebraic functions are more general than rational functions because a finite number of root extractions are permitted in addition to the elementary operations. Examples of explicit algebraic functions are x3/2, x1/2 + (x+x'3)1'4 and (i+x)112 - (ir+ex)1/3 +`

are not algebraic functions because the exponents are algebraic, transcendental, and complex, respectively. The fundamental distinction can be seen by considering the roots of the equations However, x "i, x", and x 1

x3/2

= 1

(2.lla)

= 1.

(2.11b)

and xf2-

The first equation has three roots obtained by squaring the three cube roots

of unity. In the second case, as T is approximated by rational fractions 14/10, 141/100,1414/ 1000, and so on, more and more roots (all of modulus 1) appear. Liouville classified explicit algebraic functions into orders by defining rational functions as algebraic functions of order zero. Algebraic functions of order one involve roots of rational functions; those of order two involve roots of first-order functions; and so on. The general form for a k`"-order function is (2.12) Pk=Pk-1 +EQk-1\Rk-1)1/n.

where the subscript indicates the order.

If y(x) is an explicit algebraic function of x, it is possible to find an algebraic equation for y with coefficients that are polynomials in x. For example

y=

x1/2

+ (x+x1/2)1/2

(2.13a)

satisfies

y4-4xy2-4xy-x = 0.

(2.13b)

The converse, however, is not true; given an algebraic equation of degree five or higher for y, it is not generally possible to solve it explicitly for y in terms

of radicals. Of course, there are cases where a higher-order equation can be factored into equations of degree less than five, which can then be solved. For example y8 - x4 y4 = 0 (2.14)

can be factored and has the solutions

y=0, y=x, y=-x, y-=ix, and y=-ix,

(2.2.1)

43

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

which are zeroth-order algebraic functions. All functions defined by algebraic

equations with polynomial coefficients are classified as implicit algebraic functions and considered elementary. 2.2.1

Transcendental Function

The designation transcendental for such functions as ex and In x is based

on an analogy with number theory. Such numbers as e and it are called transcendental numbers because they are not the roots of an algebraic equation with rational coefficients. Similarly, neither ex nor In x is an implicit algebraic function, that is, the soluton of an algebraic equation with polynomial coefficients. A simple proof for ex has been given by Klamkin. Assume that ex is the solution of an n'-degree polynomial equation

ao(x)exp(nx) + a, (x)exp[(n -1)x] + - + a,,(x) = 0.

(2.15)

This can be rewritten to give

-ao (x)exp(x/ 2) = aI(x) + exp(x/2)

a2(x)

+ ... +

exp(3/2x)

a^(x)

exp[(n - })x]

(2 . 16)

which leads to a contradiction since when x- + oo, the left side becomes large (sign is unimportant) but by l'Hospital's rule each term on the right approaches 0 since differentiation of the polynomials ultimately gives a constant but a°x retains its exponential character. Because of the inverse relation between the exponential and log, if one is algebraic, the other is also; therefore it immediately follows that In x is also transcendental. In analogy with the classification of algebraic functions, a transcendental function of the first order is defined as a function where the arguments of

the logarithmic or exponential functions is an algebraic function as in e' , ex In(1 +x)}, or xex'. Also, a more complicated expression such as exp[y(l +x)] where y is a solution of

y5-y-x =0 is a first-order transcendental. So is the solution of the equation y5-y-elnx = 0

(2.17)

(2.18)

even though it cannot be solved explicitly. By using the definitions

sinhx = I(ex-e-x),

(2.19a)

sinx = I (ex-a-4x),

(2.19b)

44

INTEGRATION FOR ENGINEERS AND SCIENTISTS

arctan x =

1

2i

In

l + Ix

,

1-ix

1In1+x,

arctanhx=

2

1-x

(2.19c)

(2.19d)

it is possible to show that most of the nonalgebraic functions commonly encountered in engineering are first-order transcendentals. Transcendental functions of the second order are those in which the argument of a logarithm or exponential is a transcendental function of the first order. The simplest examples are ee" and In In x. However, because of the inverse relations between the exponential and logarithm, some functions, such as In ex, that appear to be second-order transcendentals may not even

be transcendental. The nonrational powers of x that were excluded from the category of algebraic functions are second-order transcendentals, as shown by the relations x" = exp(n In x), X` = cos In x + i sin In x.

(2.20a) (2.20b)

While it is possible to keep on generating transcendental functions of higher order, the theory is not as complete as for algebraic functions. For example, it is not possible to decide whether the solution of

y = xlny

(2.21)

is a transcendental function of finite order. In addition to simple transcendental functions, which are elementary, there are functions, called the higher transcendental functions, that are not considered elementary. These are defined by differential equations of the form (2.22) P(x, Y, l', Y, ..., y(")) = 0 where P is a polynomial. Typical examples are the Bessel functions, which are solutions of

x2y + xy + (x2-v2)y = 0.

(2.23)

There also exist functions, called hypertranscendental, that do not satisfy any differential equation such as (2.22). The best-known example of such a function is the gamma function, defined by the definite integral

r(x) =

f

tx- I e-r dt, 0

which are studied in the next chapter.

(2.24)

(2.3)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

45

Exercises Verify that (2.13 a) does satisfy (2.13b). Show that e"" is transcendental.

1

2

2.3

BASIC THEOREMS FOR INTEGRATION IN FINITE TERMS

It can be shown that the derivative of an elementary function is always an

elementary function, but the integral of an elementary function may not always be elementary. The simple idea underlying Liouville's theory was originally formulated by Laplace, who noted that some forms, such as radicals and exponentials, cannot be eliminated or introduced by differentiation. This makes it possible to make statements about the form of the integral. The basic theorems for integrating algebraic functions are the following. 1.

If the integral of an algebraic function is elementary, it must be of the form

jYdX = R0(x) + I Ck In Rk(x)

where the R's are all rational functions and the number of in terms is finite but undetermined. II.

If y is given by an Nth-degree equation and the integral is purely algebraic, it must be of the form ydx = Y_ Rk(x) yk. k=I

It is easy to see that it is not necessary to use higher powers of y than the (N- I )th, as the Nth-degree equation can be used to express these powers of y in terms of the lower powers. By using the second theorem, an alternate proof of the nonalgebraic nature of the logarithm is possible. Since

Inx = f

dx x

(2.25)

the integrand y = 1/x satisfies a first-degree algebraic equation xy = I. Therefore, by the second theorem, if In x is an elementary function, it must be a rational fraction N(x)/D(x). By differentiation

D& ND

1

x

=

DZ

(2.26)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

46

This, however, leads to a contradiction. It is necessary for x to be a factor of D so that (2.27a) D(x) = xk Do(x), N(x) = xI No (x),

(2.27b)

D(x) = xk-' D, (x),

(2.27c)

N(x) = x'-' N, (x),

(2.27d)

where Do, D No, and N, are polynomials with constant leading terms. We then have l = N, xI-k + No DI xI-k' (2.28) Do

Do

2

but this means that I = k and that 1nO =

No(0) (2.29)

Do (0)

which is finite. Since In 0 is not finite, the assumption that In x is algebraic must be false. It is relatively easy by means of the second theorem to show that integrals of many algebraic functions cannot be algebraic functions. When it can be shown that the integral also cannot be expressed in the form required by the first theorem, the integral is not elementary. However, this second part of the proof is difficult to carry out by the methods developed by Liouville, so that we will not present a complete proof. It should be noted that by introducing complex variable theory concepts such as singularities on Riemann surfaces, short proofs are possible. These methods are used in Ritt's book. To investigate the algebraic nature of the integrals of algebraic functions, it is only necessary to consider SR" dx. This can be written as JND-k dx where both N and D are polynomials, and it can be shown that if the integral is algebraic and the integration constant omitted,

N dx = flk

(2.30)

where T is also a polynomial. Differentiating both sides gives the differential equation

D't-kLT = ND.

(2.31)

It can be seen that the degree of T is I greater than the degree of D. Therefore, a set of equations is obtained for the coefficients of T and the existence

(2.3.1)

47

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

of a solution determines whether the integral is algebraic As an example, consider

f x dx (l +x2) which is algebraic except for k = 1. In this case

N = x,

(2.32a)

D = I +x2,

(2.23b)

D = 2x,

(2.32c)

T = a + bx+cx2,

(2.32d)

7' = b+2cx.

(2.32e)

The resulting set of equations is 0 = b,

(2.33a)

1 = 2 c - 2 ka,

(2.33b)

0 = b(1-2k),

(2.33c)

1 = 2c(I -k),

(2.33d)

which clearly has a solution except for k = I. 2.3.1

Basic Theorems for Transcendental Functions

For transcendental functions, a theorem analogous to Theorem I

is

Theorem III.

III. If the integral u = Jfe9 dx (where f and g are algebraic) is elementary, it has the form u = we9 where w is rational in x, J, and g. For the case where j and g are rational, the test of the elementary character of the integral is whether the solution of the differential equation

IL+gw=f

(2.34)

is a rational function of x. For Se - x' dx we have

u;-2xw = 1.

(2.35)

48

INTEGRATION FOR ENGINEERS AND SCIENTISTS

It tan be shown that there is no rational solution by the following argument. Assuming w= N/D where N and D are polynomials of degree n and d, we obtain DPI - ND- 2xND- D2 (2.36)

where degrees of the terms are n+d-1, n+d-1, n+d+l, and 2d. As the two terms of degree n + d- I cannot vanish unless both n and d are 0 (i.e., w

is a constant), it is not possible for 2d to equal nd+ I and n+d-1 simultaneously.

A similar argument can be applied to Je' dx/x where the differential equation is

to+w=1,

(2.37a)

x

which gives

D1 -ND+ND =

DZ

,

(2.37b)

x

so that the polynomial of degree n + d- I must cancel polynomials of degree n+d and 2d- 1, which is not possible.

On the other hand, if we consider Je-X2 x dx, which is integrable, the resulting differential equation w+2xw = x (2.38) has an obvious rational solution, w = #. Exercises 1

Show that

dx

f (x - p)(ax2+2bx+c)"2 is algebraic if x-p divides axe + 2bx +c (Hardy). 2

(a) Show that J R, (x) In x dx is elementary if R, (x) = (c/x) + R2 (x) where R, and

R2 are rational functions. (b) Show that f (In x)l(x-a) dx is only elementary when a = 0. (c) Show that

lnx

(x-a)(x-b)

dx

is only elementary when a = b (Hardy). 3

(a) Show that ! eR, (x) dx can only involve e'R2 (x) + E ax 5 e'/(x- f k) dx and is only elementary when all the xk's vanish.

(b) Obtain analogous results for 5 sin x R,(x)dr and J cos x R,(x)dx (Hardy).

(2.4)

4

49

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

Investigate whether the following are elementary.

exp(x2 )

J

X

I Q In x dx,

dx,

,I

x

x dx

lnx

(Risch).

2.4 PRACTICAL INTEGRATION OF RATIONAL FUNCTIONS

A rational function of x can always be integrated as the sum of a rational function and a number of logarithms or arctangents of rational functions. These two apparently different functions are related by ln(u - iv) = 2 i arctan u.

(2.39)

V

When the degree of the numerator is less than the degree of the denominator and the denominator has no repeated roots, the partial fraction expansion can be written explicitly as

N(x) _ F N(r`) i (x - ri) fj, j ri - rj D(x)

(2.40)

where the is are the roots of D(x) = 0. The first restriction is not important since if the degree of N is greater than or equal to that of D, N/D can be rewritten as a polynomial plus a remainder of the proper form by dividing. The second restriction is more serious, and two methods of handling multiple roots are presented in this section. Equation (2.40) is easily integrated to give

Ndx-Y_ N(ri)Inx-rj. D

i

fj#j rj-r;

(2.41)

This is formally correct even when the roots are complex, but it has been usual in the past to convert the logarithms of complex roots into the arc-

tangents of real arguments by combining conjugate terms. However, as it is possible to work directly with complex numbers in FORTRAN and other computer languages, this will not be as important in the future. To illustrate the method, dx

dx

X4-1

(x-1)(x+i)(x-i)(x+1) _ +{ln(x-1) + ln(x+1) + i-' [ln(x-i)+ln(x+i)]} X-1 - I arctan x. = J In x+1

(2.42)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

50 2.4.1

The Hermite-Ostragradsky Method for Repeated Roots

The classical method of dealing with repeated roots, which was developed independently by Ostragradsky and by Hermite, is presented in some of the better calculus texts. The denominator D can be written as

D(x) _ fl (x-ri)"

(2.43a)

J

where m; is the multiplicity of the jth root; the jth-degree polynomial obtained by ignoring the multiplicity, which is written as

Dr(x) _ fl (x-ri),

(2.43b)

i

and the polynomial

C(x) =

D Dr

are also used. The general(' form of the integral then is

N dx = J

fD

pDr dx + Q

(2.43c)

C

where P is a polynomial of degree one less than D, and Q is a polynomial of degree one less than C. Differentiating and bringing to a common denominator gives

NC = PC2+DQ-QD,C.

(2.44)

The coefficients of the P and Q are found by equating the coefficients of the various powers of x in Eq. (2.44). As an illustration, we evaluate f xdx/(x-a)2, for which N = x,

D = C2 = x2-2ax+a2,

Dr = C = x-a,

C =1, P = Po,

constants

Q =qo, so that Eq. (2.44) becomes

x2-ax = po(x2-2ax+a2) - qo(x-a) from which po = I and qo = -1, so that

(2.45)

(2.4.2)

51

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

xdx

dx

a

J x-a x-a

(x-a)2

= In (x- a) -

a

x-a

.

(2.46)

It can be seen that a rational term only appears if the degree of N is greater than or equal to that of D, so that division is necessary, or if D has multiple roots. In practical applications the calculation is simplified by a procedure that can be used to find D, (and C) without factoring D. From Eqs. (2.43 a) and (2.43 b) it can be seen that

D= D y

I

i x-rj

(2.47)

so D which is the greatest common factor of D and D, can be found by the Euclidean algorithm. The required differentiation, determination of the highest common factor, and division to obtain C(x) can be carried out on a computer.

This process makes it possible to determine the algebraic part of the integral without finding the roots of D. However, it is impossible to find the

logarithmic terms without factoring D,. Since it is impossible to solve polynomial equations of degree five or higher explicitly, it is also impossible to write an explicit integration formula when the denominator is a general polynomial of degree five or higher. In a specific case, however, the roots can be found numerically. It is much easier to do this for D, than for D since the conventional methods do not converge rapidly when there are multiple roots. 2.4.2

Hardy's Method for Repeated Roots

An alternate procedure using partial differentiation for handling multiple roots was developed by G. H. I tardy. In this method the integral IN/D, dr is evaluated and then the required integral is obtained by partial differentiation with respect to the ri's. For the example considered above

j

(x aa)2

8a

x Xa dz.

(2.48) 2.48)

Since the degree of the numerator on the right-hand side is the same as the denominator, we have

xdx =

f x-a j

I+

a

x-a

dx = x + aln(x-a).

(2.49)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

52

As the first term does not involve a, it makes no contribution to the partial derivative and we have

xdx

(x-a)2

=

d

oa

aln(x-a) =ln(x-a)-

a

x-a

(2.50)

This procedure avoids the algebra involved in finding P and Q, but the partial differentiation can become involved when there are several multiple roots. Exercises

I

Examine G. H. Hardy's method for the case where the degree of N(x) exceeds that of D, (x).

2

Evaluate

f

x -I 3 dx.

x+l

2.5 PRACTICAL INTEGRATION OF ALGEBRAIC FUNCTIONS

Two general techniques for evaluating the integrals of algebraic functions are rationalization by a suitable substitution and the use of general integra-

tion formulas. However, it should be remembered that not all algebraic integrands can be integrated in terms of elementary functions and considerable effort can be wasted on an insoluble problem. Unfortunately, there is no simple general method of determining when an algebraic integrand leads to a nonelementary function, but if one transformation leads to an integral known not to be elementary, no other transformation can circumvent it. 2.5.1

General Integration Formulas

A number of general integration formulas are listed in Tables 2.1 and 2.2. Those in Table 2.1 give an explicit expression for the integral, while those in 2.2. equate the integral to an explicit expression and another integral or to two other integrals. The expressions in Table 2.1 are generally derived

by introducing a new variable into

J f(x)dx = g(x)

(2.51a)

J f [U(x)] dx = g [U(x)].

(2.51b)

to obtain

(2.6)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

Table 2.1

53

General Integration Formulas

(aU+b)mUdx=(aU+b)m+1

a(m+1)

J

I

Udx _

aU+b

2a

Udx b2+a2 U2

__

Udx

ln(aU+b)2 1

arctanaU

ab

b

1

tab

3 b2-a2U2

In

aU-b aU+b

(UV+UV)dx = UV J

UV - UV

I

VZ

ov-u UV

dx =

U V

dx=In V

f UV - UV J U2+V2 dx = arctan U V

UV-UV J

U2-V2

dx

=ln U-V U+V

Most of the entries in Table 2.2 are derived by integrating d dx

F(U,V)=OFO+OFV

8U

aV

(2.52a)

to get an integral relation F(U, V)

=fG(U, V)Udx + I H(U, V)Vdx.

(2.52b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

54

General Integral Transformations

Table 2.2

J UVdx=UV - J VOdx _

dx

` ,1 (U-x)2 U

+

1

U

U-x

(U-x)2

dx = -x +

dx

2m

1-u2m

1-U2m

UciU (1 +

f

U2)1/2 dx = U(1+U2)112

VUdx= U2

- VU + _

dx

-f

Vdx U

_

1

dx

U(x2-a2)

2a

U(x-a)

U Udx

_

- U(1 +U U)"2

dx

02(1+02)1/2

0(1+02)112dx

dx

1

2a J U(x+a)

+ f(1+02)"dx

For example, taking F(U, V) as (U-x)-' gives _

dx J

+

I

U-x

(U-x)2

U dx (2.53)

(U-x)2

Other entries are obtained by such algebraic devices as using

_ U+V-V

U

(2.54a)

(U+V)m

(U+V)m to get

Udx (U+V)m

-J

dx

(U+V)m-I

Vdx

(2.54b) (U+V)m

(2.6.2)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

55

The relations in Tables 2.1 and 2.2 are not restricted to algebraic functions and

a large number of integrals can be derived for tabulation by using them. However, it takes intuition to be able to use such relations in practice to evaluate a particular integral. 2.5.2

Rationalization

The use of rationalization is illustrated by the simple example $dx/(1 +x}).

The substitution x = (z- 1)2 transforms the integral into

I

2(z-1)dz z

=

2(z -In z)

and the inverse transformation z = I +x} gives dx = 2(l+x"2-ln1 +x112). J 1+xtrz

-

(2.55a)

(2.55b)

This integral is a particular case of the class

j x' (a+bx")"dx studied by Chebyshev, who showed the integral to be elementary and solvable by rationalization if and only if either p, (m + 1)/n, or p + (m + ))/n is an integer.

An important class of integrands that can be treated by a systematic procedure are rational functions of x and y where

y = (axe+bx+c)1J2.

(2.56)

Since any even power of y is a polynomial, the rational function can be reduced to (P+Qy)/(R+Sy) where P, Q, R, and S are polynomials. It can then be further reduced to U+ W/y or U+ Vy where U, V, and Ware rational functions.

Euler developed substitutions that eliminate the radical and make the integrand a rational function of a new variable t. We note that either the roots of axe +bx+c are real or a and c cannot both be negative. Therefore, one of the following three transformations will involve only real constants. If the roots are real, designate them by r, and r2 and set

y = t(x-r,).

(2.57)

Squaring both sides and solving for x gives

x=

r2a-r,t2 a-t2

(2.58)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

56

It can then be shown that

y=

r2 - rlt

2

a-t2

(2.59)

Since x, y, and dx/dt are rational functions of t, the transformed integral can be handled by the methods developed in the preceding section. When a is positive, the transformation

y = tx+xa1j2

(2.60a)

gives

x

=

t2-c b - 2ta 1/z

(2 . 60b)

Similarly, when c is positive, using

y = tx+c'12

2.61a)

gives

x=

21c'12

-b

(2.61b)

a - t2

It is obvious in both cases that y and dy/dt are rational functions of t. These transformations are special cases of a general principle: The integral of a rational function of x, YI , Y2, .. ., y,, can be reduced to the integral of a rational function of t if there is a parametric representation

x = Ro(t) Y, = R, (t) y2 = R2(t)

y = R,, (t) where the R's are rational functions. In the case we have considered there is such a parametric representation

x=d- 2ad+b-2et

(2.62a)

a - t2

and

y=e-t 2at+b-2et a - t2

(2.62b)

(2.6)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

57

where d and e are constants connected by e2 = ad2+bd+c.

(2.63)

If d = 0, then e = c} and Eq. (2.62a) reduces to Eq. (2.61 b). For e = 0, the transformation corresponds to Eq. (2.58).

Exercises 1

Evaluate Chebyshev's integral f x(a+bx")" dx for the three cases for which it is elementary.

2

Investigate the possibility of using Euler's transformation with the wrong sign by by working with complex quantities.

3

Verify that Eqs. (2.62 a, b) and (2.63) do provide a parametric representation of (2.56).

4

Derive the "Christmas tree" integrals by starting with f e" du = e° and setting u = e' and continuing the process. 2.6

ELLIPTIC INTEGRALS

A class of nonelementary integrals, the elliptic integrals, arises from integrands that are rational functions of x and the square root of a cubic or quartic. The two cases can be considered together because it is generally possible to find a bilinear transformation

x= at+b

ct+d

(2.64)

that will transform a (quartic` in x to a ( cubic I in t. There are, of course, \ cubic //+

`quartic)

cases where an integral involving the square root of a cubic or quartic is elementary. For example, if there is a repeated root, it can be factored out of the square root, Also, cases such as

I (ax4 + bx2 + c)"2 can be reduced to a tractable integral by setting x2 = t. Such integrals are referred to as pseudo-elliptic. It can be shown that the general case can be reduced to evaluating three forms:

INTEGRATION FOR ENGINEERS AND SCIENTISTS

58

rl-MX

dx

I [(1-x2)(1-mx2)]112

L 1-x2

I

dx

(1-nx2)[(1-x2)(1-mx2)]1i2 that are nonelementary. Tables equivalent to these integrals are generally tabulated as

F(O/a) =

I

4

dO (2.65a)

o (I - sin2asin20)1/2 0

(1 - sine a sin 2 0)"2 d0,

E

(2.65b)

0

4

rz (n;/a)

=

d0

Jo (1 - nsin2O)(1 - sineasine0)U2

(2.65c)

Exercises 1

Use the methods of Section 2.3 to investigate whether the elliptic integrals are algebraic.

2

Show that if

f(x)+f(4.-)=0, then

f(x) dx

[x(1-x)(1-k2x)]'12 is pseudo-elliptic. (Goursat, O. Hedrich, ed., Cours d'Analyse, in 2nd Ed., Vol. 1, pp. 267-269. Ginn, Boston, 1904.)

2.7 INTEGRATION OF ELEMENTARY TRANSCENDENTAL FUNCTIONS

While algebraic integrands, including those leading to elliptic integrals, do arise in practical applications, really complicated algebraic integrands are rare in practice. It is, however, common for complicated combinations of elementary transcendentals to occur in practical applications. As there are relatively few general methods for dealing with transcendental integrands, there is considerable scope for ingenious devices. However, the possibility that the integral is not elementary must be taken into consideration.

(2.7.2)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

2.7.1

Integration of Inverse Functions

59

The first general result is that if 1'(y) = F(y)

is integrable, the corresponding inverse function arcJ(x) can be integrated in terms of elementary functions. A simple application of integration by parts gives the explicit expression

Jarcf(x)dx = xarcf(x) - F[arcf(x)].

(2.66)

To illustrate, we evaluate j In x dx by considering the natural logarithm as

the inverse of the exponential. In this case F(y) = J(y) = y and becomes

lnxdx=xInx-e"x=xlnx-x.

(2.67)

1

For a more conventional example, the arcsine, f(y) = sin y and F(y) _ - cos y, giving

J aresin x dx = x aresin x + cos(aresin x)

= xaresinx + (1 -x

(2.68)

2)1/2.

The simplification of the second term is based on

cos(aresin x) =I I - sin2(aresin x))'/2 = (1-x2)1 r2.

(2.69)

2.7.2 Polynomials of Exponentials

The second general result is that

j P[x, exp(k, x), exp(k2x), ..., exp(k"x)] dx (where P is a polynomial function of the arguments) is always elementary. It is easily seen that the integral can be expressed as the sum of a number of integrals of the form jx' ekx dx where J is 0 or a positive integer but k is unrestricted. The individual integrals are obtained as o /ak' ekx/k. Integrands that are products of powers of x, exponentials, sine, sinh, cosine, and cosh terms can obviously be brought into the proper form by using the definitions of these functions in terms of exponentials. Alternatively, we can in some cases (as is shouw later) simplify the procedure by considering complex values of k using Table 2.3, which lists the real and imaginary parts of the elementary functions of x+ iy.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

60

Table 2.3 Simple Functions of a Complex Argument

exp(x+iy) = cos(x) + isin(y) sin(x + iy) = sin(x) cosh(y) + i cos(x) sinh(y) cos(x + iy) = cos(x) cosh(y) + i sin(x) sinh(y)

tan(x+iy) =

sinh(2x) sin(2x) +i cos(2x) + cosh(2y) cos(2x) + cosh(2y)

sinh(x + iy) = sinh(x) cos(y) + i cosh(x) sin(y) cosh(x + iy) = cosh(x) cos (y) + i sinh(x) sin(y)

tanh(x+iy) =

sinh(2x) +i sin(2y) cosh(2x) + cos(2y) cosh(2x) + cos(2y)

Other forms, such as

f P(x, In x) dx, f P(x, aresin x) dx, and so on, can be brought into the proper form by obvious changes in variable, such as t = In x or t = sin x, but such transformations only work when there is one transcendental function involved. 2.7.3 Algebraic Functions of Commensurable Exponentials

The third and last general result is that the integral fF(exp(a1 x), exp(a2x), ...,

where F is an algebraic function and the are commensurable numbers can be reduced to an algebraic integrand by the substitution exp(al x) = t. If F is a rational function, the transformed integral is also rational and can be evaluated in terms of elementary function.

(2.7.4)

61

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

2.7.4 Examples

To illustrate the use of Table 2.3 and the second general relation, let us begin with sin k l x

J cos k, x dx =

(2.70)

k,

J

Setting k, = a+ib, from Table 2.3 it can be seen that cos(a + ib) x = cos ax cosh bx - i sin ax sinh bx

(2.71a)

sin(a + i b) x = sin ax cosh bx + i cos ax sinh bx.

(2.71b)

and Equating real and imaginary parts of Eq. (2.70) gives

f

cos ax cosh bx dx =

a sin ax cosh bx + b cos ax sinh bx a2 +

1

(2.72a)

62

sin ax sinh bx dx =

b sin ax cosh bx - a cos ax sinh bx

a2 + b 2

(2.72b)

The hyperbolic functions can be transformed back to trigonometric functions by letting a = k, and b = ik2 and noting that by Table 2.3 cosh ik2 x = cos k2 x

(2.73a)

sinhik2x = isink2x.

(2.73b)

and

The resulting expressions

Jcoskixcosk2xdx= k, sinklxcosk2 - kZ cosk,xsink2x

(2.74a)

k, - k2

2

J sin k, x sin k2 x dx =

k2 sin k, x Cos k2 x - k, cos k, x sin k2 x

(2.74b)

k2 -k22 2

can be evaluated by l'I-iospital's rule when k, = ±k2. Alternately, by setting a = ik, and b = k2, we obtain expressions for f cosh k, x cosh k2 x dx and f sinh k, x sinh k2 x dx. The process can be repeated by setting k, or k2 or both equal to complex numbers again to obtain expressions for the product of three or four terms, and so on. This is an efficient method for generating integrals but it takes ingenuity and experience to produce a desired result.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

62

The third form is the most generally applicable method for attacking transcendental integrands. The obvious substitution t = exp(a, x) is universally applicable but in many cases the work can be simplified by the substitution t = tantl'2 x,

(2.75a)

which transforms cosx =

1-t2

(2.75b)

,

1 + t2

sin x =

2t (2.75c)

1+t2'

and

dx =

2dt

(2.75d)

1+12

Exercises 1

Evaluate f arcsinh x dx, ! arctan x dx, and f arctanh x dx.

2

Evaluate dx

a + b cos x + csinx 3

dx

an d

(a + b cos x + csinx)

Start with

j sink(x+a)dx = cosk(x+a)

-k

and derive analogs of (2.72 a, b) and (2.74a, b). 4

Show that if f is a homogeneous function,

(' f(sinx,cosx)dx Fi j sin(x - kj)

=

i

A j In tan

j

where

A = f(sin kj, cos k j) r

f1i#j sin(kj- kr)

(Herniae; see Edwards, Vol. 1, p. 198).

x-k' 2

(2.8)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

63

2.8 SYMBOLIC AUTOMATIC INTEGRATION

The use of computers for numerical calculations is a commonplace, and in later chapters we discuss numerical procedures for evaluating integrals that are suitable for automatic computation. A subject on the present-day research frontier is the use of computers for symbolic manipulation, which may lead to even more important developments than purely numerical calculations. At present there are subroutines available that will carry out analytical differentiations. On being supplied with the input exp(-x2) in the appropriate symbolism, the computer will return - 2x exp(-x2). Such subroutines can differentiate sums, products, and functions of functions. This makes it possible to differentiate a composite of functions if the derivatives of the component functions are defined. Provision is usually made for defining derivatives needed for a particular case. Such a program can perform operations, such as evaluating d°

sin x2 + ex

dx4 cos in x + x"2 I' that would involve a large amount of manipulation. As we have noted previously, indefinite integration (antidifferentiation) is a much more difficult problem, and so far there has only been one major

investigation. Slagle for his doctoral dissertation at the Massachusetts Institute of Technology developed a program SAINT (Symbolic Automatic Integration) for evaluating elementary integrals. This appears to be the first

case where the development of a computer program was accepted as a dissertation, but there will undoubtedly be many more in the future. As the program utilizes a specialized computer language (LISP), it is not

possible to give a detailed discussion. However, the general principles can be understood without going into these technical details and they are of interest because they clarify the methods used by human mathematicians. Slagle defined elementary functions in the following recursive manner: 1. all constants;

2. the independent variable;

3. the sum or product of elementary functions; 4. an elementary function raised to an elementary function exponent; 5. a logarithmic or trigonometric function of an elementary function.

This is less general than Liouville's classification, as implicit algebraic functions are not included. The limitations were imposed to conserve computer memory and are not inherent.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

64

Table 2.4 Standard Forms for SAINT Program

a.

$cdv=cv

b.

Jeudv=e0

c°dv =cv-

c.

In c

d.

Jinvdv=vlnv-v

f.

j sinvdv= -cosv

g.

Jcosvdv = sin v

h.

J tanvdv = Insecv

i.

f cot v dv = In sin v

J k.

1.

M.

n.

sec v dv = In(sec v + tan v)

J

Jcosecvdv = In(cosec v - cot v) $arcsinvdv = v(aresin v) + (1- v2)1'2

Jarccosvdv = v(arccos v) - (1- v2)1i2 $arctan v dv = v(arctan v) - I in(1 +v2) (Continued)

(2.8)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

65

Table 2.4 Standard Forms for SAINT Program

o.

j

arccot v dv = v(arccot v) + I ln(l + v2 )

= v(aresec v) - ln[v + (v2 -1)"2]

p.

Jarccosecvdv = v(arccosec v) + In[v + (v2 -1)1/21

q.

sect v dv = tan v

r.

J s.

{ cosec2 v du = tan v

t.

I dv -=lnv v

U.

J ' vcd v=

vc + 1

c+1 v.

I sec v tan v dv = sec v

w.

fcosecvcotvdv = - cosec v

cos(m-n)v _ cos(m+n)v 2(m+n)

v= - 2(m-n)

X.

si nmucosnu d

Y.

sin m v sin nv dv =

Z.

(m # ± n in x, y and z.)

sin(m - n)v

sin(m+n)v

2(m+n) sin(m-n)v sin(m+n)v Jcosrnvcosnvdv= 2(m-n) + 2(m+n)

The integration procedure depends on (1) a list of 26 standard integrals reproduced as Table 2.4; (2) a set of algorithmic transformations; (3) a set of heuristic transformations. The algorithmic transformations are devices, such as taking constant factors outside the integral sign or breaking an integral into the sum of two

INTEGRATION FOR ENGINEERS AND SCIENTISTS

66

integrals, that when applicable are almost invariably appropriate. It must be

emphasized that the computer can only recognize that an integral is the standard form, and cannot, for example, evaluate J(bv+c)di- until it is transformed into bJvdv+Jcdv. The heuristic transformations are devices, such as rationalizing substitutions, integration by parts, and some of the other transformations in Table 2.2, that transform an integral into one or more integrals that (hopefully) are simpler than the original integral.

The program starts off with a single goal, the evaluation of a certain integral. If this integral is not on the standard list, the application of the algorithmic and heuristic transformations generates as subgoals other integrals, the evaluation of which will lead to evaluation of the initial integral.

The SAINT program has a method for arranging these subgoals in a hierarchical sequence called a tree, estimates their relative difficulty; and attacks them. They will either be evaluated as standard forms or generate new subgoals, which are added to the tree. A simple example of a tree is shown in Figure 2.1a. A plain branch A means that either of the subgoals is equivalent to the apex, whereas A means that both subgoals are needed to evaluate the apex. When a subgoal is

evaluated, the tree must be pruned to eliminate parallel branches that are no longer needed. When 9211 and 9222 are evaluated in Figure 2.1a, the tree simplifies to Figure 2.1b, in which 92 is known. The subgoal 922 was

9

,Q 92

91

911

n

922 923

912

I--1 I

I meons integral

1..._J is known.

(A)

Figure 2.1

(B)

Example of a tree before (A) and after (B) pruning

(2.8)

67

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

determined and discarded after being used to evaluate g2 ; subgoal g23 was by-passed; while g12 remains because it is a possible method of evaluating g, . As the program does not have any method of recognizing nonelementary integrals, it is instructed to stop after expending a certain amount of effort on a problem and may therefore give up prematurely on a soluble problem.

This is necessary to keep it from spending a large amount of time on an insoluble problem. though Dr. Slagle reports that when tested on$ exp(x2)dx,

it reported after a comparatively short time that it had tried all relevant transformations unsuccessfully.

Because of memory limitations, the scope of the program was limited to exclude rational functions and it was tested on 54 problems taken from MIT calculus examinations. It succeeded in 52 cases, including sect x dx + sect x - 3 tan x' but failed on two apparently simple problems f x(I +x)} dx and f cos xf dx. It actually was slower than a bright freshman, but this was due to the memory limitations; most of the time was spent erasing material from the memory so that locations could be re-used. Since that time, computers with larger memories have become available and an order-of-magnitude increase in the speed of operation is feasible.' Ultimately such programs will become widely available and eliminate considerable intellectual drudgery and frustration. It really is a frustrating experience for an engineer to struggle with an integral that is obvious in principle but requires some tedious algebra. Very few engineers work with such integrals frequently enough to maintain their skill, and a surprisingly large number of expensive man-hours can go very quickly (particularly if a few colleagues are consulted) on a problem that could be done by a sophomore.

The procedure used by SAINT is essentially similar to that used by a human mathematician. Probably the principal difference is that the program works in parallel, completely exhausting a level before going to the next level, whereas a human would normally work in series, pursuing a line of attack through several levels before giving up and proceeding to another approach. Exercises I

Evaluate sec

2

x dx

l+ sect x- 3 tan x'

y('

x(1+x) 12 ' dx, and

I cosx'12dx.

I

t This has been reported by J. Moses in a very interesting MIT dissertation "Symbolic integration."

INTEGRATION FOR ENGINEERS AND SCIENTISTS

68 2

3

Investigate the programs available at your computer center for such symbolic manipulations as differentiation. Write a program for integrating rational functions.

2.9 DERIVATION OF INTEGRALS FROM DIFFERENTIAL EQUATIONS

A differential equation and a set of initial conditions is a valid and useful definition of a function from which its properties can be determined and numerical values obtained for tabulation. For example, to define sin x as the solution of

y+y = 0

(2.76a)

subject to the initial conditions

y(0) = 0,

Y(0) = 1

(2.76b)

is as legitimate as the (opposite/hypotenuse) definition of elementary trigonometry or the power series definition of function theory. Many of the functions encountered in applied mathematics satisfy linear second-order differential equations of the form d -

Py

+ QY = 0,

(2.77)

which is called a Sturm-Liouville equation. Integral relations for such functions can be derived by considering two equations with the same P but different Q's: d

Q.Y. = 0

(2.78a)

d Ph + QbYb = 0.

(2.78b)

dxPYa +

and

Multiplying the first equation by yb and the second by ya and subtracting gives d Yb

PYo - Y.

dX

PYb + (Q.-Qa)YaYb = 0.

(2.79)

The first two terms are an exact differential, so that the integral relation J (QQa) YaYb dx = P(Y.Yb-YbYa)

is obtained.

(2.80)

(2.9.1)

69

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

To illustrate the use of this important result, Eq. (2.74a) will be derived.

Since y, = sin k, x is a solution of ya + k;ya = 0

(2.81a)

and yb = sin k2x is a solution of 1'b + k2 Yb

(2.81b)

= 0,

examination shows P = 1, Qa = k', and Qb = k2. Substituting into (2.80) gives

(k2-k;) J sin k1 x sink2 x dx = k, sink2xcosk1x - k2sink, xcosk2x. (2.82)

The derivation is easily adapted to obtain the other expressions involving a combination of two sin, cos, sink, or cosh terms. It is also obvious that the argument kt can be replaced by ki +a in this derivation. 2.9.1

Lommel Integrals

The method is particularly useful for obtaining integrals involving the higher transcendental functions. For example, the Bessel functions of order v denoted by J,(kx) satisfy the differential equation

x2y+xy+(k2x2-v2)y=0,

(2.83a)

which is equivalent to d

dx

xy + (k2x-v2/x) y = 0.

(2.83b)

Taking two different values a and b for k gives

f Jv(ax)J,(bx)dx = -xb2

aJv(bx)Jjax)].

a2

(2.84)

Using a recursion formula

J,(kx) = v J,(kx) - J,,+ 1(kx)

(2.85)

kx

makes it possible to eliminate the derivatives on the right side of Eq. (2.84). These expressions for integrals of products of Bessel functions that arise in connection with the vibrations of circular membranes are called Lommel integrals after the mathematician who first used them.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

70

Exercises

Show that for two functions satisfying differential equations

1

d

2 dx2(PY)+QY=0

the integral relation corresponding to (2.80) is

d

d

f (P. Q. - PbQb)Y0Ybdx = PbYb dx P.Y. - P.Y. dx PbYb 2

Evaluate f e-' cos x dx and f e-' cos x sin x dx by using either (2.80) or the preceding result.

2.10 APPROXIMATE METHODS

We have shown that many simple-looking integrals cannot be expressed in terms of elementary functions. Numerical methods for evaluating such

integrals are covered in Chapter 4; this section is a brief exposition of approximate analytical methods for dealing with indefinite integrals. 2.10.1

Series Expansions

The simplest such method is to expand the integrand or part of the integrand in a power series and integrate term by term. Thus 2 (-1)nx2n+1 x3 x5

exp(-x 10

3

n!(2n+l)

(2.86)

which converges for all x. It is a general principle that the radius of convergence of the integrated series is the same as that of the series approximation to the integrand, the proof is left as an exercise. However, that a series converges in the mathematical sense does not mean that it is useful for computational purposes. The convergence for large x may be very slow and considerable roundoff error may be introduced when there are large terms of opposite sign. It is often more convenient to use formally divergent series. By writing

J exp(-x2)dx =

i exp(-x2)xdx, J

(2.87)

X

integrating by parts, and repeating the process, we obtain the expansion

(2.10.2)

j'

71

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

= - exp(-x2)

exP(_x2)dx

2x

2 1

1

2x

+

1.35

13 22

(2x)

- (2xz)3 +

(2.88)

which is convenient for large values of x, 2.10.2

General Forms

Most of the common approximations can be obtained by using the three general formulas 2

3

Jf(x)dx = xf(0) + -J(0)+

(2.89a)

31f(0)+

x2 X. , i Udx=xU--U+-U+

(2.89b)

3!

2!

UVdx = UV, - 0V2 + UV3 + ,

(2.89c)

J

where Vj means that V has been integrated i times. Equation (2.89 a) is obtained by expanding f in a Maclaurin series and integrating term by term, (2.89b) is obtained by repeated integration by parts starting with dV = dx; while (2.89c) is equivalent to repeated applications of integration by parts

starting with dV = V dx. Applying (2.89 a) to ! exp(-x2)dx gives (2.86) Equation (2.88) can be obtained by making the substitution x2 = t to obtain J t-}e-` di and applying (2.89c) with U = t-1. Using (2.89b) gives

J exp(-x2)dx = exp(-x2)[x + 3x3 + sx5 ,

+ ],

(2.90)

which is a known result.

Interesting formulas are obtained from (2.89c) by taking either U or V to be e" (there being no restriction on the sign of a), so that differentiation is equivalent to multiplication by a and integration to division by a. When V = e", the differential series

J e"g(x)dx = es[9 a

9 a

+

9 a

+

]

J

(2.91a)

in descending powers of a is obtained, while if U = e°X, the integral series

Je9(x)dx =e"L J gdx - a f f gdxdx+a2JJJ

(2.91b)

72

INTEGRATION FOR ENGINEERS AND SCIENTISTS

in ascending powers of a is obtained. Generally it is the integral series that is convergent while the differential series is asympotic. However, if g(x) is a polynomial, the differential series will terminate and give an exact result. 2.10.3 Use of "Computer" Approximations

In the past decade, extensive work has been done on finding analytical

approximations of various functions for use in computers, which can evaluate expressions very rapidly, whereas looking up tables is relatively slow and requires considerably more memory. Simple examples of the type of expressions used are e-" ^_- (1 + 0.2507213 x + 0.0292732x2 + 0.00382783 x3)- 4,

which has a maximum error of 2.4 x 10-4 for 0 < x < 16, and /X_1 3 -1 In x - 0.86304 x

x+l

+ 0.36415 (

\x+1

,

(2.92a)

(2.92b)

which is accurate to 7 x 10-2 for I < x < 104'. In principle, by using approxi-

mations of this type for the integrand, very accurate approximations for nonelementary integrals could be obtained. In practice, it appears preferable

to obtain some numerical values by series expansions or the methods of Chapter 4 and then fit expressions such as (2.92 a, b) to these values. Exercises

I 2

Obtain several approximations to f e-11 sin bx dx. (a) Compare the approximations obtained for f In x dx obtained by substituting

(2.92b) into the integrand and integrating with that obtained by substituting it

into the exact result. (b) Do the same for (2.92 a) and f e-I dx. (c) Obtain an approximation for f e'/xdx.

2.11 A PRACTICAL EXAMPLE

In the previous sections the illustrative examples were purely mathematical. To conclude this chapter, a physical problem is discussed to show how approximations are based on physical considerations and to indicate

the physical interpretation of the mathematics. The theory of thermal explosions involves systems that generate heat at a rate that increases with

increasing temperature and that lose heat to the surroundings at a rate proportional to the temperature difference between the system and the

(2.11)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

73

surroundings. Two markedly different types of behavior are possible. A slow reaction may occur, maintaining a small temperature difference at which the heat loss balances the heat of reaction, or very rapid reaction may take place, whereupon the temperature rises almost to the adiabatic value.

The transition between the two regimes is generally very sharp, so that a small change in the parameters of the system changes the behaviour drastically. The critical mass of a nuclar reactor is an analogous phenomenon. One approximate method of analysis is the nonsteady theory of thermal explosions in which the partial differential equations of the exact formulation

are simplified to ordinary differential equations by assuming that the temperature of the system is uniform and approximating the heat loss to the surroundings by a constant heat transfer coefficient. In terms of dimensionless variables, the ordinary differential equations are d0

dt

= JAY - AO

(2.93a)

and

d,i_ _S dt

(2.93b)

A°ee

B

where 0 is the temperature difference, t the time, S the reaction rate, A. the concentration of reactant, A the heat transfer coefficient, n the order of the reaction, and B the adiabatic temperature rise.

The natural choice of reference quantities makes B large ("500), and normally the crucial phase occurs before an appreciable amount of fuel is used. It is therefore customary to neglect (2.93b) and take J. = I in (2.93 a). This permits integration, which gives

t = 6-1

ee - (Alb)0

+ const.,

(2.94)

the constant being determined by the condition t = 0 when 0 = 0. The nature of the solution depends on whether the denominator vanishes for a real positive 0. If there is such a root, it is a limiting temperature, which is approached by a solution of the first type. If there is no such root, the temperature increases indefinitely, since neglecting fuel consumption removes the mechanism for limiting the temperature. It can be seen that for small values of A/S there cannot be a real intersection of (A/S)O and ee, whereas for large Alb there will be two intersections. For a critical value

there is a single intersection, which is a point of tangency. It therefore satisfies the equations

INTEGRATION FOR ENGINEERS AND SCIENTISTS

74

(2.95a)

(2.95b)

The critical value of A/8 is easily shown to bee and the limiting value of 0 is 1.

The integral in Eq. (2.94) is nonelementary but Gray and Harper have obtained useful approximations by using

ee = I + (e-2)0 + 02.

(2.96)

This polynomial and its derivative (e-2) + 20 match ee at 0 = 1. The resulting integral d0 J

I + [e-2-(A/8)]0 + 02

is elementary. When A/5 = e, the denominator is (1-0)2 and Eq. (2.94) gives, after rearrangement, (2.97)

In the general case, the integral depends on the sign of the discriminant

D=(A -e)2+4 (A -el. 6

(2.98)

6

When A/8 > e, the roots are real; the integral is expressed in terms of logarithms, and the smaller root is a limiting value of 0. When Alb < e, the roots are imaginary, the integral involves an arctangent, and 0 is not limited. Exercise 1

Evaluate (2.94) by using the rational fraction approximation ee

+ (e-2)0 - l I-e-t0

(2)

ANALYTIC EVALUATION OF INDEFINITE INTEGRALS

75

BIBLIOGRAPHIC NOTES AND COMMENTS

A very detailed exposition of the standard methods is given in J. Edwards

"Treatise on the Integral Calculus," Vol. 1, Chelsea, New York, 1954. While the theoretical aspects of this book, which has a strong nineteenthcentury flavor, were old fashioned when it appeared in 1921, the extensive collection of examples and problems keeps it in print. A little book by H. F. MacNeish "Algebraic Techniques of Integration," Wm. C. Brown, Dubuque, Iowa, 1952, contains many ingenious devices. G. Petit Bois "Table of Indefinite Integrals," Dover, New York, 1961 and W. Grobner and M. Hofreiter "Integraltafel" Vol. 1, Springer-Vienna, 1961 are much more extensive than the usual handbook tables. M. Abramovitz and I. A. Stegun "Handbook of Mathematical Functions"

U. S. Government Printing Office, Washington, D. C., 1961 has many results for the higher transcendental functions. SECTION 2.1

J. Liouville, (Sur Ia determination des integrates dont la valeur est algebrique,)) J. Ecole Polytechn. 14(1833), 124-193.

J. Liouville, ((M6moire sur les transcendantes elliptiques de premiere et de seconde espece, consid6rdes comme fonctions de leur amplitude,)) J. Ecole Polytechn. 14(1833), 57-83. (Other papers by Liouville are referenced in Ritt or Hardy.) J. F. Ritt, Integration in Finite Terms (Columbia Univ. Press, New York, 1948). SECTION 2.2

Most of this section and a considerable portion of other sections are based on G. H. Hardy's The Integration of Functions of a Single Variable, 2nd ed. (Cambridge Univ. Press, London and New York, 1916). SECTION 2.3

In addition to the books by Hardy and Ritt, a recent Systems Development Corporation

Report (AD 651,587), "The Problems in Integration in Finite Terms," by R.H. Risch discusses the subject in terms of differential fields. This approach was introduced by A. Ostrowski, (<Sur l'integrabilit6 6l6mentaire de quelques classes d'expressions,u Comment. Math. Helc. 18(1946), 283-308. We have not consulted the work of D. Mordukhay-Boltovskoi, On the Integration of Transcendental Functions (Warsaw, 1913), which we are told is difficult even for those who read Russian well. SECTION 2.4

C. Hermite, n Integration des Fonctions Rationelles, n in Oeuvres, Vol. 3, pp. 35-54 (Gauthier-Villars, Paris, 1912). The paper was originally published in 1872, so there is no

doubt that M. Ostragradsky (1801-1861) has priority. However, the calculus books that attribute the method to him do not cite a specific reference.

76

INTEGRATION FOR ENGINEERS AND SCIENTISTS SECTION 2.5

P. Chebyshev, <Sur I'int6gration des diff6rentielles irrationellesn J. Math. (Set.

1)

18(1853), 87-111. SECTION 2.6

A very interesting treatment of elliptic integrals is given by L. V. King, On the Direct Numerical Calculation of Elliptic Functions and Integrals (Cambridge Univ. Press, London and New York, 1924), but the methods were considered too specialized for inclusion in this book. SECTION 2.7

A Wyle Laboratory Research Staff report (WR 66-24, available as NASA M 67-12156), "Some Integration Formulae which Simplify the Evaluation of Certain Integrals in Common Use by Engineers," by M. J. Crocker and R. W. White, has a number of interesting results but we believe that the methods of this section and Section 2.9 permit simpler derivations. SECTION 2.8

J. R. Slagle, "A Heuristic Program That Solves Symbolic Integration Problems in Freshman Calculus," Rept. 5 G 001 MIT Lincoln Lab. (May 10, 1961). An abbreviated version with the same title appeared in J. Assoc. Comput. Mach. 10(1963), 507-520. A subsequent MIT thesis by W. A. Martin, "Symbolic Mathematical Laboratory" (January 1967) (AD 657 283), is also of interest. SECTION 2.9

A fascinating set of three lectures by B. Van der Pol, "How to Obtain Properties of the Solutions of Diferential Equations without Actually Solving Them," is only available as Cornell Univ. School of Electrical Engineering Res. Rept. EE 403 (15 September 1958). Apparently the author's untimely death prevented publication. More advanced methods are described in two papers: L. C. Maximon and G. W. Morgan, "On the Evaluation of

Indefinite Integrals Involving the Special Functions: Development of the Method," Quart. App!. Math. 13(1955), 79-83. L. C. Maximon, "On the Evaluation of Indefinite Integrals Involving the Special Functions: Application of the Method," Quart. Appl. Math. 13(1955), 84-93. SECTION 2.10

The classic reference on polynomial and rational fraction approximations is C. Hastings, Jr., Approximations for Digital Computers (Princeton Univ. Press, Princeton, New Jersey,

1955). Abramowitz and Stegun have a number of useful results. A recent reference is J. F. Hart et al., Computer Approximations (Wiley, New York, 1968). SECTION 2.11

P. Gray and M. J. Harper, "The Thermal Theory of Induction Periods and Ignition Delays," in Seventh Symposium (International) on Combustion, pp. 425-430 (Butterworths, London and Washington, D. C., 1959). W. Squire, "Mathematical Theory of Self Ignition," Combustion and Flame 7(1963), 1-8.

Chapter 3

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

INTRODUCTION

3.1

In the preceding chapter it was shown that many apparently simple indefinite integrals such as ! exp(-x2) dx cannot be evaluated in terms of elementary functions. Although this means that a corresponding definite integral f exp(-x2) dx cannot be evaluated by carrying out the integration and substituting in the limits, there are, as we show in this chapter, many ways of evaluating such integrals. To illustrate, we consider an approximate method of evaluating

I(x)= J +xz exp(-t2)dt,

(3.1)

that becomes exact for infinite limits. Since the variable of integration is a dummy variable

r 12(x)

_

exp(-u2)du

x

[x fX

-x

x exp(-v2)dv J

x

x

exp[-(u2+v2)]dudv.

(3.2)

-x

This involves a double integral over a square. Thus while the grouping u2+v2 suggests introducing polar coordinates, the limits of integration cannot be expressed in a simple manner in polar coordinates. Williams in 1946 suggested obtaining an approximate value by replacing the square 2xn-' 12 having the same area. Because the value of by a circle of radius (u2+v2) in the area that is in the circle but not in the square is smaller than its value in the equal area contained in the square but not in the circle, this gives a value that is too large. However in the limit as x - oo the approximation becomes exact because both the circle and square cover the entire plane. Therefore 2n

12(x) fo

2xn' 1/2

Jo

exp(-r2)rdrd9,

(3.3)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

78

which is easily evaluated to give

ft [i

12(x)

-exp(-4x2/]

(3.4a) 2

and + cc

f

exp(-x2)dx = n'/2.

(3.4b)

-00

The methods presented in this chapter are only applicable to limited classes of integrals and there is no general theory analogous to Liouville's to indicate how a specific problem should be attacked. Unfortunately, the field is no longer of interest to mathematicians. The most extensive table of definite integrals was compiled by l3ierens de Haan in 1867 and is available in a recent reprinting. However, the book he published in 1862 explaining the methods used to evaluate the definite integrals has never been reprinted and

is available in only a few libraries. It should be emphasized that there is no easy way of checking a definite integral given in a table analogous to the check of an indefinite integral by differentiation. Caution schould be exercised in using such results, particulary if the integral is an unusual one. If the calculation is important, the value of

the integral should be checked by either inequalities or a numerical calculation. There can even be difficulties in evaluating a definite integral by

substituting limits in the corresponding indefinite integral, particularly when multiple-valued functions are involved. For example, many tables give 121 12X = (32)- 112 In 1 + 2'/2x + x2 +8- 112 tan (3.5) I +xs 1 - 2'/2x + x2 1-x2 and this can be verified by differentiation. However, to obtain correct

dx

numerical results for a definite integral, the tan - ' must be taken from 0 to it.

The usual computer library subroutine goes from -n/2 to n/2 and gives incorrect results. An alternate form 1/2X + x2 = (32)- 112 In 1 +2 + 1+x4 I - 2'/2x + x2

dx

+8- 1/2 (tan-' 21/1-x + tan-' 21/2+x) involves the conventional range -ir/2 to n/2 for tan-'.

(3.6)

/

Exercises 1

2

Derive a lower bound for Eq. (3.3) by using the inscribed circle. (a) Verify by differentiation the formal correctness of both Eq. (3.5) and (3.6). (b) Carry out some numerical calculation of !o dx/(1 +x4) taking a = 0.8(0.1)1.5.

(3.2)

79

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

3.2 THE GAMMA FUNCTION

The gamma and beta functions, which are defined in terms of definite integrals, are of sufficient importance to warrant discussion before proceed-

ing to methods for evaluating definite integrals, particularly since many definite integrals can be evaluated in terms of these functions. We begin by noting that

x le-`dt = 1, .o

(3.7a)

t2edi2,

(3.7b)

x t3e-'dt = 6,

(3.7c)

j0

and that for integer n's it can be shown by induction that

t"e`dt = n!

(3.7d)

0 I'C This motivates the convention that 0! = 1, which puzzles many students. The gamma function, introduced by Euler, is a generalization of the concept of the factorial to fractional and negative values. For historical reasons we use the notation

It is therefore obvious that for integral values of x,

T(x-1)=x!

(3.9)

T(x + l) = xF(x)

(3.10)

The basic recursion formula

is obtained by integrating (3.8) by parts. By using this, the value of T can be determined for any x if it is known for a unit interval that is usually taken as I to 2. However, T(0) and F(negative integer) will be infinite. It is easily seen from the definition and from (3.7a) that T(1) = F(2) = 1.

It is also easy to evaluate T(-) = tr"2 exactly since by introducing the variable(' 1 = .x 2, we obtain

t-112 exp(-t)dt = 2

J 0

fo

exp(-x2)dx

('

exp(-x2)dx (3.11)

J and the last integral was evaluated in the previous section.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

80

This implied that T (4) = 17r'12. For other x between I and 2 numerical value of F (x) can be obtained to any desired accuracy by expansion of a-` in a power series and term-by-term integration or by numerical techniques described in the next chapter. Some examples of definite integrals that can be evaluated as gamma functions are

exp(-zx")x°dx =

1

z_t"+' " r a+l

(n > 0, a > -1, z > 0),

n

(3.12a)

f

°0

exp\((

1

dx

- -i ) x°

/a -1

_ Tf

2

)

X

0

(a > 1).

(3.12b)

The proof is very direct and simply involves the obvious change of variable

t = zx" and t = '/x2, respectively. 3.2.1

The Beta Function

The beta function is a function of two variables defined by the integral fi(x, Y) =

tx-1(1-

t)y-' dt

1

J

(3.13a)

0

that is related to the gamma function by /3(x, Y) =

T(x)I (Y)

(3.13b)

F(x + y)

This relation can be proved by a method utilizing the transformation from rectangular to polar coordinates used in the preceding section. We note that b(y' introducing t = u2, we get T(x) = rl 0 tx-' exp(- t)dt = 0

2J u2"-' exp(-u2)du,

(3.14a)

0

and that since the variable of integration is a dummy variable

r(x) T(Y) = 2 fo 1/2x-'exp(-u2)du . 2 fo v2y-'exp(-v2)dv 4 fo fo

it2x-'v2y-'exp[-(u2+v2)]dudv.

(3.14b)

On introducing polar coordinates, we obtain /2

r

T(x)T(y) = 4

sin2x-' fo x

f0

0sin2y-'Or2x+2y-2exp(-r2)rdrdO,

(3.15a)

(3.2.2)

81

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

which can be separated into the product of two integrals fir,;2

I(x)1(y) _

2r2's+''I-' exp(-r2)dr.

2sin2z-' Ocos2,-' Ddb

(3.15b)

°

°

The first integral is the transformation of fl(x, y), obtained by setting t = sine 0 into (3.13a). By (3.14a) the second integral is T(x+v), proving the desired result. Some integrals that can be expressed as beta functions are

x°dx _

1

fo(I-x')" -fl(

p+1 n-1 '

in

(m>O,p>-1,n<1);

(3.16a)

n

sine°- ' x cos26-'xdx = + f(a, b)

(a, b > 0);

(3.16b)

0

x° o

(1 +x'")9

_l

p+1

In

In

q- p+1

(3.16c)

m

The first two are easily proven by the substitutions t = x' and t = sine x, respectively. The third can he transformed into the second by setting x' = tang 0. An interesting specialization of (3.16b) "li

"i2

tan" x dx = 0

cot" x dx = 0

>t

2cos(nir/2)

(3.17)

is obtained by setting 2a-I = I -2b and using a well-known identity

1'(x)1(1-x) =

>t

(3.18)

.

sin 7rx

3.2.2 Product Representations

A number of proofs of (3.18) are given in the literature. Every one that the author has seen uses some advanced concept. An instructive one uses infinite product representations. A general method for expanding a function that is analytic in the complex plane and has an infinite number of simple zeros is the expression

zl

f(z) =f(0) eXP Cf(0) Jk=I -[ exp f(O)

I

\ak/

(1

-?I

(3.19)

ak)

where the ak's are the zeros of f(z). Applying this to J'(x) = sin nx/x, which

INTEGRATION FOR ENGINEERS AND SCIENTISTS

82

has zeros at ak = ±kn/n, gives +

sinC x =n

k

x

-l

f (1

=n F k= 1

\1

exp(kn

knl eXp(kn) \1 + kn/ expL-

\kn/

n2 x2

0o

=nfl(1k=1 k2 n2

(3.20)

It can be shown that

l+zexp

r(z)-exp(-yz) k=1 H

z

k

_\\\

(3.21) k

where y is a number known as Euler's constant that will be encountered again in Section 3.8. It is easily seen that

r(z)r(-z) =

(3.22)

- Z2 kF1

1-(Z 2 /

r(-z) _ - r(1 -z)

k2)*

(3.23)

z

is obtained, and by setting n = I in (3.20) and inverting, the desired result

r(z)r(1-z) =

it

(3.24)

sin nz

is obtained. 3.2.3

Dirichlet's Multiple Integrals

An interesting reduction of a repeated integral to a single integral due to Dirichlet is

ffltj
'*'

f fly

r(aj) r(E aj)

o1

t,)IIt,`-'d:,

f(x) xy a' 1 dx.

(3.25)

We indicate the method of derivation by showing how to eliminate one

(3,2.3)

83

ANALY7IC EVALUATION OF DEFINITE INTEGRALS

integration from ,

C

2r

f(t1+t2+c)t' 1' dt1 dt2.

0

ii

Introducing av, defined by vtl = (I-vWt2, transforms this to

f

J

(C

,2/(,-r)

fo

)

2

+

2,-a2-1 dvdt2.

at+i

Then interchanging the order of integration and setting t2 = vx gives 1-c 0

f(C+X)(l-v)al llla2-1X01+a2-'dxdv,

0

in which v and x have been separated. Integrating with respect to v then gives

r(a1)r(a2) r(a1 +a2)

' 1 -r

f(c+x)xa,+a2- 1 dx.

o

This reduction is useful for evaluating the mass, center of gravity, and moments of inertia of figures bounded by curves of the form

\x/, + ly i a

(-) +

b

1.

c

Exercises

I

Show that ,

Jo x'In"1 dx = x

2

r(n+1) (m+1)"+'

Show that "12 0

rz

sine"x dx = 2

R/2

sine"-' xdx = 0

3

form and n> -1. 135

(2n-1)

2.4.....2n

2,4.....(2n-2) 1.3..... (2n-1)

For the area in the first quadrant bounded by the curve x'j2 +Y1/2 = 1 calculate the area, center of gravity, moment of inertia about the x axis, and volume generated by rotating about the y axis.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

84

3.3 CLASSICAL CALCULUS METHODS

In this section a number of classical calculus devices for evaluating definite integrals are presented. The methods covered can be classified as (1) differentiation or integration with respect to a paramater; (2) derivation and solution of an equation (algebraic, differential, or functional) for the value of the integral; (3) representation of part of the integrand by an integral and interchange of the order of integrations. The methods are introduced by simple examples; then a number of more difficult examples are worked. 3.3.1

Differentiation with Respect to a Parameter

Differentiation with respect to a parameter is analogous to the procedure used in Chapter 2 to evaluate indefinite integrals. Starting with the result

exp(-xz)dx = it 1/2

(3.26a)

it is easy to introduce a parameter, a, and obtain +« (!)1/2 J

-

exp(- axz ) dx =

(3.26b)

a

«

Differentiation with respect to a gives

f

«

(3.27a)

2a aJ

and by further differentiation J

x°exp(-axz)dx =

(a)1/z>

4a2

(3.27b)

and so on. It can be shown by the methods developed in Chapter 2 that the corresponding indefinite integrals are not elementary. 3.3.2

Integration with Respect to a Parameter

A simple example of integrating with respect to a parameter gives 1 xo- t - 1 0o e-t-e-at o

Inx

dxfo=dt=Ina I

(3.28)

(3.3.4)

85

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

by starting with

fxt_1dx_-r 1

(3.29)

t

o

and integrating both sides with respect to t from t = I to t = a. It is characteristic of methods involving integration that it must be possible to evaluate the integral for some value of the parameter that is taken as the lower limit. 3.3.3

Solution of a Differential Equation

Although differentiating or integrating with respect to a parameter is effective for building up a collection of definite integrals from a few known cases, it is of limited value for evaluating a specific problem. A more flexible technique involves differentiating with respect to a parameter one or more

times in order to obtain a differential equation with the parameter as the independent variable that the definite integral satisfies. For a simple example, consider

I(a) = so that

expl- (x2 +

(3.30a)

5)]dx,

o

!(a) = -2a

a')]dx.

expL- \x2 + x

'*

f o

(3.30b)

x

By using the substitution x = alt in the right side, it can be shown that

I(a) = - 21(a).

(3.31)

This is easily integrated to give 1(a) = ce-2a =0.57r'/2 e- 2a,

(3.32)

the constant c being determined by the value of the integral for a = 0. 3.3.4 Solution of Algebraic Equations

In some cases it is possible to obtain an algebraic relation for the integral. For example, if "/2

I=

sin 2 x dx,

(3.33a)

Cost x dx.

(3.33b)

0

then it can easily be shown that

I=

n/2 0

INTEGRATION FOR ENGINEERS AND SCIENTISTS

86

The equality of integrals involving sines and cosines over suitable multiples or submultiples of the period is frequently used. It can be justified by the

trigonometric relation sin x = cos[(n/2)-x] and the introduction of the variable t = (n/2)-x, but in most cases it is obvious from geometrical considerations. Adding (3.33a) and (3.33b) gives ail 21 =

(sine x + cost x) dx = IT.

(3.34)

2

0

In some cases a set of linear equations can be obtained for two related integrals. For example, if b

1 = J e°x cos qx dx

(3.35a)

a

and b

J=

e°s sin qx dx,

(3.35b)

a

then integration by parts gives

1- a°s sin qx b

p

a

q

q

J= e°F cos qx b q

a

J,

P

(3.36a)

(3.36b)

1.

q

Simple algebra then gives p cos qx + q sin qx

I = e°"

p2+q2

b

(3.37a) 0

and

psingx - gcosgx

J=,

b

p2+q2

(3.37b) Ia

3.3.5

Solution of a Functional Equation

Functional equations in the sense of an equation involving two different arguments are relatively unfamiliar to physicists and engineers except for difference equations, but they can sometimes give very elegant solutions. An interesting example is

I(a) = f o

- 2 a cos x + a2) dx.

(3.38)

(3.3.5)

87

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

It can be shown that 1(a) is an even function of a. This is not immediately obvious, as the integrand is not even with respect to a. The point is that changing the sign of a simply interchanges the contributions of the integrand from 0 to n/2 and from n/2 to it. Therefore, we have 21(a) =

JIn[l + a4 + 2a2(1 - 2cos2x)]dx

(3.39)

ln[1 + a4 + 2a2 sine x] dx = 1(a2). 0

The last equality depends on rearangements such as n

J X/2

3x/4

In(1+2acosx+a2)dx = 2

,(n! 2

ln(l + 2a sin 2x + a 2 ) dx.

(3.40)

The resulting functional equation 21(a) = I(a2)

(3.41)

1(a) = ina2.

(3.42)

has the solution Examination shows that the general solution of (3.41) is

1(a) = C(a)Ina2

(3.43a)

where C(a) is a solution of the functional equation C(a) = C(a2).

(3.43b)

which has the solution C(a) = 0

for a2 < I

(3.43c)

C(a) = n

for a2 > 1.

(3.43d)

and

The first part is obtained by noting that 1(0) = 0 and that repeated applications of (3.41) gives

1(a) = (1)rI (a2n)

(3.44)

Since both terms on the right side approach 0 with increasing n, 1(a) must vanish. For a2 > I the proof is based on the relation

ln(1 -2acosx+a2)=1na2+In 1 -2cosx+ 12). a

a///

(3.45)

Integration of the second term gives 1(1 /a), which vanishes for a2 > 1. The first term is easily integrated to give it In a2, completing the proof.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

88

3.3.6

Integral Representation

A very powerful technique involves replacing part of the integrand by an integral representation, such as I

a-"i dt,

x

(3.46)

o

and then changing the order of integration. For example

sinx

1= f

x

o

dx = fo sinx fo e-x'dtdx (3.47)

= f 0,0 0f e-

" sinxdxdt.

On evaluating the integral with respect to x, we obtain

1=

dt n _2 0 1+12 2 °'

(3.48)

A finesse is involved in evaluating the slightly more general integral

1(k) =

°° sin kx o

x

dx.

(3.49)

Obviously, if k = 0, the integral vanishes. Otherwise the integral can be rewritten as

1(k) _ Jo

sin kx kx

I k 00

k dx =

sin u du .

(3.50)

u

o

Therefore, when k > 0, I(k) = ir/2; but when k is negative, x

1(k)

sin u du u

o

=-

sin u du o

u

=-

(3.51)

-.

2

An interesting example that combines integration with respect to a parameter and interchange of the order of integration was used by Childs to evaluate the double integral 1(p, a, b) =

I

--

I

exp[-(ax2 + 2pxy + by2)]

dxdy,

(3.52)

xy

which arises in statistics. The method is based on differentiating with respect to p to eliminate the I/xy and then integrating after changing the

(3.3.e)

89

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

order of integrations. The expression to be integrated is v

1(p, a, b) =

-2J f exp(-byz) o

exp[-(axz + 2zxy)]dxdydz. "0

o

(3 .53)

The lower value 0 in the outside integral was taken because 1(0, a, b) = 0 since the integral is antisymmetrical about the origin when p = 0. This was also used to validate the interchange in the order of integration. The original integrand does not satisfy the requirements because of the singularity at the origin, but the integrand

exp[- (axz + 2pxy + byz)] - exp[- (axz + byz)] xy

gives the same result, is well behaved at the origin, and has the same derivative with respect to p.

The inner integral is evaluated by using a device known as completing the square, that is writing exptzz

exp - a

(xz

+ Z zy x) = a

yz

[_a(xZyl exp+

/

a

a

z

(3.54)

so that //n

1(p, a, b) = -2l

oJ -

`\a

-2n

dz

o (ab-z2)

exp

zz

- I b - ") yzldydz

\

a/

= -2naresinr

p L(ab)'1z].

(3.55)

Exercise

I

Show that the following equations are valid by derivations using methods similar to those shown in this section.

)21

w

ex 0

Io

xl/z

- x - a 1dx=-If 2 ( x

exp - (X2 + _ )]dx cos ax

o 1+xz

dx = I e-°. 2

-

.

7r2 z

exp(- 2a).

INTEGRATION FOR ENGINEERS AND SCIENTISTS

90 i/ 2

J

In sin xdx = --1n2. 2

0

-.

/2

n3

InsinxIntanx =

16

0

sin px cos qx X =

I

it

if p > q;

0

n

4

=0 sinm x dx

n

cos' x + sinm x

4

ax

It

(1 + x2)(1 + x`)

4

if p = q;

ifp
IQ +b arccos(tana) sinxcosxdx =

_ # sine (a + b) arccos

i'll'

/:ex

tan a

[tan(a + b)]

sin a arcco

cos(a + b) cos a

Xldxdy=+

0

3.4 SERIES METHODS

We consider three different types of methods involving series. First, the definition of a definite integral as the limit of a sum sometimes expresses the integral as a series that can be evaluated. Thus if we consider the integral j

1=

e-"dx = 1-e-'

(3.56)

0

and divide the range of integration in N equal parts, the lower Darboux sum

(3.4)

91

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

is

SN =

( k exp(-1/N) exp(-1/N)' exp( --_ N

N

N

k=o

1 - exp(- 1)

1- exp(-1/N)

,

(3.57)

while the upper Darboux sum is I

S

Y1

N

N k=o

exp - k l- I N)

1-exp(-1)

N 1- exp(-1/N)

(3.58)

The limit as 1/N - 0 is 1-exp(- 1) in both cases. There is an extensive body of work in older algebra books on methods for evaluating sums without using calculus, but this can be considered obsolescent. Today, given a relation between an infinite sum and an integral, we would consider the evaluation of the integral by the methods of the calculus as the logical method for evaluating the sum. It is unlikely that we could evaluate a difficult integral by an analytic noncalculus evaluation of the defining sum.

A more useful technique involves expanding part of the integrand into a series, integrating term by term, and then summing the resulting series. Thus

f

00 sin ax

o

ex-1

dx =

e-"xsinaxdx

n=1 o

(3.59)

a

n=1 a2+n2

--21cotair-11. ad

Third, it is sometimes possible to obtain a set of integrals by expanding both sides of an integral identity in a power series. Thus by starting with e -kx

Jo we obtain

w (-k)n n=O

n!

dx

ex+e-x

x o

lnx X1/2 = 0,

(3.60)

Xe-112

ex + e-x

lnxdx = 0,

(3.61)

which implies that

Jo ex+e-x

Inxdx=0.

(3.62)

This technique, however, must be used with caution. In some cases the

INTEGRATION FOR ENGINEERS AND SCIENTISTS

92

expansion gives a well-behaved integral as a sum of improper integrals. For example

singxcos2px'12 dx = 0

(3.63)

tcos2px112 dx = 0

(3.64)

0

does not mean that xZ

J

0

as the integrals in (3.64) are not convergent. Series Transposition

3.4.1

We refer to an elegant device involving a series expansion as series transposition. It involves splitting the range of integration into parts and then making the limits the same by a change in the variable of integration. Applying this to

-dx = k=0 sin x

0

X

(k+ I )x/2

k(x/2)

dx sinxY

(3.65) x/2 1

o

x

+

+

1

1

+

1

n-x n+x 2n-x 2n+x

+

)dx.

The summation in the parentheses is a known expansion for cosec x = 1/sin x. (This can be seen to be a plausible result because of the identity of the singularities.) Therefore, the integrand is constant and the result is n/2. The same result is obtained for !o tan x dx/x, but there is a complication because of the singularity in the integrand at (k+1)n/2. The integrals must be interpreted as Cauchy principal parts and I

tan x o

(x/2) dx = lim x C-0 fo

tan x I

1

L

n=o nn+x

(n+1)n - x

The sum is an expansion for cot x. Exercises

I

Prove that

' lnx

0 l-x

dx=-n-2

;

6

dx.

(3.66)

(3.5)

93

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

it

. x212 t/2 dx = 8 In e4

o(1-x) `

2

Obtain an expansion in t for

exp(t sin x) dx. J

0"

3.6 COMPLEX VARIABLE METHODS

Some of the most powerful methods for evaluating definite integrals employ complex variable methods. These methods are beyond the scope of this book but a number of references are given in the Bibliographic Notes

and Comments at the end of this chapter. We note here, however, that the device of assigning parameters complex or imaginary values, which was used for indefinite integrals in the preceding chapter, is also applicable to definite integrals. Thus, starting with

/ I/2

ao

f _ exp(- ix2)dx = I

(3.67)

exp(- ix2) = cos x2 - i sin x2

(3.68)

and noting that

enables us to obtain

i

cosx2dx =

Jo

sinx2dx = g J

l/2. (3.69)

Exercises 1

Interpret (` 1

x" dx = 0

n+1

for n = a + ib. 2

Show that it

Io

1/2 snsI--24

sinsc2x2sins2bxdx=

2c

2

INTEGRATION FOR ENGINEERS AND SCIENTISTS

94

3.6 SOME GENERAL FORMS FOR DEFINITE INTEGRALS

This section presents a number of relations for definite integrals involving an arbitrary function, such as

Jf(xn+x-Jnx=o X

(3.70a)

0

and

f(x"+x-")lnx fo

dx

1+x2

= 0.

(3.70b)

A simple proof is obtained by setting t = In x, which transforms the range of integration into - oo to 0o and gives an odd integrand. For (3.70b) the

symmetry of the term e'/(1 +e2') is not evident on inspection but it is equivalent to 11(e-+ e), which is obviously even; the integrand is then odd. An alternate proof involves splitting the range of integration into parts, from 0 to 1 and I to oo. Then, introducing the variable t = 1/x shows that the integral from I to 0o is the negative of the other integral. A pair of results that can be looked on as explicit mean value expressions are

xf(sin x, cost x) dx = 2 J0

Jo

f(sin x, cost x) dx

(3.71a)

and

f(x"+x-)arctanxdx = 2f' f(x"+x'")arctanxdx

J

x

o

x

o

(3.71 b) 2J0

f(x"+x-")

dx .

The proof of (3.71 a) is based on the invariance of the original integral when the variable is changed from x to rr - t. In proving (3.71 b) the relation arctan

-I = -it - arctan x x

is required.

2

(3.6)

95

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

A related result for an arbitrary odd function f(x) integrable over the

range -Ito+lis 2ka

x2f(sinx)dx = -2k7r2

` 1/0

kx

f(sinx)dx

J

0

(3.72) A

_ -2k27r2 J f(sinx)dx. 0

The proof begins with the relation (' 2ka

(x-k7r)2 f(sinx)dx = 0

J

(3.73)

0

based on the antisymmetry of the integrand. Therefore 2ka

2ka

x2f(sinx)dx = 2kn

xf(sinx)dx

(3.74)

0 J0 since ro ka f(sin x)dx vanishes because of antisymmetry. Using series transposition, we get

2k- 1

2ka

5a(x+jir)f[(-1)1sinx]dx.

xf(sin x) dx = I=o

o

0

The proof is completed by invoking (3.71 a) and the fact that 2k-1

(-I)' j = -k.

1=0 Exercises

I

Show that

fa +2 x2 1

dx

+ x2/ x =

f t

f (x +

dx x2)

(Wolstenholme). 2

Show that

+x

f(x

x

- 1}dx =I X

f(x) dx ao

and apply this to some standard results such as Co

(Slobin).

f

exp(-x2)dx = 7C'/2 -00

(3.75)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

96 3

Show that if f is an even function,

J:J(x2

)dx = Jf(x2+2)dx

+

s/2

n12

f(sin 2 x) see x dx = 2 0

f(cos2 x) see x dx 0

(Glaisher).

3.7 USE OF INTEGRAL TRANSFORMS

An integral transform is a relation of the form b

f(s) =

K(s, t) f(t)dt

(3.76)

fa

b etween an original f(t) and an image f (s). The notation f (s) f(t) will be used to show the relation. The best-known integral transform is the Laplace

transform

f(s) = f o e-Srf(t)dt

(3.77)

0

but there are about a dozen others that occur frequently enough for tables of transform pairs to be available. Obviously a tabulation of transform pairs is a table of definite integrals, but the important aspect is the use of general properties of the transform to obtain additional definite integrals. The examples that follow are based on Laplace transforms but others can

be used in a similar manner. Recently, Mellers developed a technique involving the simultaneous use of two transforms to obtain some very complicated integral identities, but it is beyond the scope of this presentation. Two basic relations for Laplace transforms are

f(t). sf(s)-f(0)

(3.78a)

and

tf(t)

- d f(s)

(3.78b)

These make it possible to find the Laplace transform of functions defined by

simple differential equations. A trivial example involves e", which is the

(3.7)

97

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

solution of

f +af = 0

(3.79)

subject to f(O) = 1. The Laplace transformation then gives the algebraic relation

sf(s) - I +a,j= 0

(3.80a)

for the transform so that

f(s) = f

e -sr e -ardt

I

=

s+a

o

(3 80b) .

This is easily verified by direct integration in this case. Putting s = 0 gives e-ardt =

1

(3.81)

o f'o

a

While this example was trivial, the idea that the integral was evaluated by the use of the general properties of Laplace transforms without an explicit integration is significant. A less trivial example is the evaluation of lu J0(t)dt by taking the transform of the differential equation

121+tf +t2f=0

(3.82)

subject to f(0) = I and f (O) = 0. In this case the transformed equation is a differential equation

2_

d

Ts s

J

d ds

d2

sj+

_

ds2

0.

(3.83)

The solution must approach 1/s as s - oo. This is easily found to be

f(s) = (52+1)Ii2 =

Jo

e-StJo(t)dt.

(3.84)

00

Putting s = 0 gives 0x J0(t)dt = 1,

(3.85a)

f

while setting s = 1 gives 10"D

11e-tJ0(t)dt = 2-2.

(3.85b)

98

INTEGRATION FOR ENGINEERS AND SCIENTISTS

Some caution should be exercised in obtaining such integrals by setting s = 0 in the Laplace trasform. For integrals such as Jo sin t dt and Jo cos t dt the value obtained in this way is the Abelian mean defined in Chapter 1. Another theorem involving Laplace transforms, that

f f(t)d . J 00 f(p) dp, t

000

(3.86)

o

corresponds to the interchange of limits device of Section 3.3. The conditions

for the existence of a Laplace transform of f(t) guarantees the validity of interchanging limits.

A very powerful tool is the theorem that if

k(t, u) # x(s)exp[-uf(s)],

(3.81a)

then

f(u)k(t, u)du±a(s)f[fl(s)]. J0*

(3.81b)

If the transform can be recognized, that is a(S)f[II(s)] = 9(s) # g(t),

(3.88)

the integral is evaluated. An important application of this theorem is in heat conduction, where integrals of the form

(nt)-li2 I

f(u)exp(-

Jo

`\

4t1 dt

arise. The transform of exp(- u2/4t) (ttt)-' I2 is s -'/2 exp(- us'12), so that a(s) = s-''2 and fl(s) = s'12. A typical result obtained in this way is cosauex o

- au2 4t

dt = (nt)'/2exp(-a2t).

(3.89)

A theorem relating the product of two functions or of two transforms to an integral is called a convolution theorem. For Laplace transforms the theorem that

f(s)9(s) T-± J0 f(x)g(t-x)dx = J0 g(x)f(t-x)dx

(3.90)

is very important. It will be used in Chapter 6 in connection with integral

(3.8)

99

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

equations. A less familiar result is fc+k,

f(t)' 9(t)

-ae

27ri

(3.91)

f(z)9(s-z)dz,

which expresses a Laplace transform as a complex integral. This theorem has been used by Weber in an important paper on nonlinear differential equations. Exercise 1

Show that the following equations are valid by derivation using Laplace transform methods. (a)

x sin Ix dt

it

2e

I+x2

o (b)

oJo(t) - cos t

dt = In 2

t

(c)

dx _ 2"

x Jk(x)xk+1

F(n/2)

2k+1 F[k+1-(n/2)]

if k + I > (n/2) > 0. (d)

exp+u so

2)dx = exp(-u2)Lir1/2

fo

exp(x2)dx - JEi(u2)I

where Ei (x) is the exponential integral X

e`

-A

f-00

I

3.8 FRULLANIAN INTEGRALS

The basic idea behind this class of integrals arises in the example °°

I= 0

sin xdx-, 3

(3.92a)

x2

which by using a trigonometric identity can be rewritten as

if 3sinx-sin3xdx 000

x2

(3.92b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

100

The individual terms of the integrand in (3.92b) become infinite in such a way that the integrals are improper. However, It is proper to write

-

1 =*lim f6'0 3sinxdx a-.0

1= I lim a-o

du

sin u a

35

_ I lim a-0

XZ

a

sin3xdx

x2

Ia

2-

sin u d

i

3a

U

U

du sin u du .

(3.93)

U2

a

As 6 becomes small, u-' sin u

1, so (`3a

1 = jlim

3-0J`a

du u

= iln3.

(3.94)

This technique can be used to resolve the paradox in Chapter 1 for

1(a, b) = j

°`

e-ax - e-bx dx

(3.95)

x

o

and show that the correct value is ln(b/a). The general result is

[f(ax) -f(bx)] xx = [f(co) -f(0)] In fo"

(3.96)

b

Most expositions only consider the case where f(oo) -+ 0, but it is not difficult to construct examples, such as

f'0 [tanh ax - tanh bx] o

dx a = In , x b

(3.97)

where f(O) vanishes. An interesting modification of (3.96) was given by Ostrowski, who showed

that f(oo) and f(O) can be replaced by

M[f] = lim t-I Jo

I-00

f(x)dx

(3.98a)

and

m[f] = Iim t r l f(x)dx r-.0

,

(3.98b)

101

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

(3.8)

to give

f(ax) -f(bx) Jo

X

A = [M(f) - m (f)] In !.

(3.98c)

A further generalization replaces ax and bx by functions u and v to obtain J0D

[4f(u) - vf(v)] dx = M [xf(x)] In v(00) - m [xf(x)] In a(0)

(3.99)

0

when u and v are positive absolutely continuous functions and the limits of u/v are positive, and that M[xf J and m [xf J exist.

An interesting specialization of (3.98c) is obtained when f is a periodic function with period p

M[f] =

c+p

fp(x) dx

(3.100a)

and m[f] vanishes so that

Therefore, since In Icos xl dx = a In 0.5,

(3.101a)

f 0"

it follows that

cosaxldx cos bx x

In0.5 In a.

(3.1Ofb)

b

AIU(axI)_f(bx4l different generalization of (3.96) is W

Jo

dx )]

x

= Moo)-I(0)]

1InPa

- In q

+ p-q f0'0 f(x)Inxdx. pq

Hardy developed an extension

(3.102)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

102

f`0 [f(ax) - f(bx)] lnN x dx 0

=

(-1)" N! [InN+' a n=o n+1 n!(N-n)!

- lnN+1 b]

f(x)Inxdx. (3.103) o

This is obtained by differentiating the integral with respect to a, which gives

1(a) _

f (ax) lnN x dx =

1 f. f(x) [in x - In a]N dx. a

fo

(3.104)

o

Expanding [In x-ln a)N by the binomial theorem and integrating term by term gives (3.103).

This can be generalized to give an analog of (3.102) [f(ax°) - f(bx9)] 1nN x dx 0 Joo

N+i

N+1 "

(N+1)!

In"a 1) n!(N+1-n)! [pN+I "

In"b

- qN+i] f

o

f(x)In N-a+I xdx. (3.105)

Hardy also obtained many results by devices such as starting with ao

1= o

e-ax-e-bx 2 In x dx.

x

(3.106)

and differentiating with respect to a to obtain

](a) = -2 J o {exp(-tax) - exp[-(a+b)x]}1nx dx. x

o

(3.107)

This can be evaluated by (3.103). The result involves ID- a-x in x dx. This frequently occurs and is denoted by - y where y is known as Euler's constant. The numerical value of y is 0.5772175

, and

I = (a+b)[ln2(a+b) - a In22a - bIn22b]

+ 2(y-1)[(a+b)ln(a+b) - aln2a - bin2b].

(3.108)

(3.8)

103

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

Even more useful is the evaluation of integrals involving two functions by the following device. Start with a Frullani integral, such as [cos ax - cos bx]

dx

x

J0

=1n

a

(3.109)

,

b

and multiply by a suitable function of b, such as I/(b2+P2), and integrate from 0 to co. This gives db

°°

o b2+2 Jo cosax

dx

- j' 0

cosbx

dbd x

JobZ+#2

Jo

In

a

db bb2+#2.

(3.110)

Using the integrals db

JO()cosbxdb 0

Io

n

b2+fl2

In

(3.111a)

2fl'

J o b2+Q2

e_0xe

2#

fl

b b2+p

2

00

in

(3.111b)

/,

(3.11 lc)

and setting fi = b gives °°

fo

[cosax - e -'x)

dx

x

a = -In b.

(3.112)

Other results of this type can be obtained by multiplying by a-6 instead of (b2+fl2)-1.

Elliot obtained a number of results for multiple tiple integrals. By adding o

Jo Jf0'0 J

[f(ax + by) -flax + b'Y)] d-d! = Y

In

J

(3.113a)

to

f

b [f(co) -l(ax)] d x

[f(ax + by) -f(a'x + b'Y)] dxdy =

o

f0a,

Y

Jo

In Q, [f(co) -f(b'y)] dY y

(3.113b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

104

the relation

Jo

[f(ax+by)-f(a'x+b'y)]dxdy=lnaln-[f(x)-f(0)]

Jo

(3.114a)

is obtained if a b = a' b'

(3.114b)

and f does not become infinite in the range 0 to oo. By a similar process, if

abc = a'b'c'

(3.115a)

and

In

a

a'

In a In b'

a

= In

c'

a b c', In In c' c' c

(3.115b)

then 0°°

o°°

[flax + by + cz) -f(a'x + b'y + c'z)]

dx dy dz xyz

fooo

a

a = In

In a [f(oo) - f(0)],

In

b'

a

(3.115c)

c'

and so on. The derivation is dependent on f being a symmetrical function of the variables and can thus be generalized as follows. The relation

if =

I

fo

Jo'o

dxdy xy

[s(ax, by) - s(a'x, b'y)]

[s(ax, by) - s(ax, b'y)]

dx dy

xy

fo, 0 f o

+ fo '0 J 0" [s (ax, b' y) - s(a' x, b' y)] o

= In

bf

[s(ax, co) - s(ax, 0)] o

+ In

dx d y

xy

dx

x

a

a' fo

[s(oo, b' y) - s(0, b' y)]

dy (3.116) y

(3.9)

105

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

gives (when ab = a'b', so that In(a/a')+In(b/b') = 0) if = In a' In

[s(oo, 0) - s(0, 0) - s(oo, oo) + s(0, ao)].

(3.117)

b

If s(ax, by) = f(ax+by), then s(co, 0) = s(0, co) = s(co, 00) =f(oo),

(3.118)

while s(0, 0) = f(0), so that (3.117) reduces to (3.114a). Using (3.117) gives such results as

f"*

fo [f(ax)f(by) -f(a'x)f(b'Y)]

dxyy

= -1n a, In b, [f(co) -f(0)]2. (3.119)

Other relations, such as I

f

o

-f(X Y6 )] r [f(x°Y6) lnxln yxy

Jo

dx d y = In

a In a [f(1) -f(0)], a'

(3 . 120)

be

are obtained by a suitable change of variable. Exercises

I

Show that

-Inp+4e-axdx=lnl 0

p+ qe-bx x

l +1 In P

a

2

Obtain a number of "cross" Frullanian integrals by using Eq. (3.111 a, b, c).

3

Make up some examples using Elliot's results for multiple integrals.

3.9 THE WILLIS EXPANSION

Willis developed a powerful method for obtaining series expansions for integrals of the form

1 = f" f(()g(t)dt

(3.121)

0

when f (t) and §(s), the Laplace transform of g(t), can be expanded in power

series. The derivation is manipulative and does not establish conditions for the convergence of the expansions, which are often asymptotic.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

106

The derivation starts by considering the more general integral

I(s) = I

e-'`f(t)g(t)dt.

(3.122)

0

When (3.121) is convergent, (3.122) reduces to (3.121) for s = 0; otherwise, it gives the Abelian mean. If J(t) is expanded in a Taylor series, (3.122) can be written as

I(s) = n=0

n!

Jox

estg(t)t"dt. l

(3.123)

Repeated applications of (3.18b) give

J

e-S`g(t)tndt = (-1)"

o

d" ds"

9(s)

(3.124)

Therefore, in terms of the coefficients of a power series expansion for #(s) cc

9ln>(0)

(S) =

nt

n=o

ao

s" _ Y Ans",

(3.125a)

n=o

the expression

I = Y (- 1)"A"f"(0) 00

n=o

(3.125b)

n!

is obtained.

Willis has determined the An's for some important cases:

f"(0).

e-ar f(t)dt = n=0

0 J'o

sinatf(t)dt = n=o

f0'0

Cosatf(t) fo"o

(3.126a)

an+1 '

(-1)" l`2a'(0 2n+1

(3.126b)

a

"Y- (-1)"fa2(n+1(0)(3.126c) =O

$sintf(t)=f(o_ oadt 2 )"E

(-1)nl(2n-1)(0)' 1

(3.126d)

(2n-1)a""

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

(3.9)

fo*

107

exp(-i2/a2)f(t)dt = 0.5n112 an- (2)RJ(2ni0)) + (3.126e) I

n=1 (2n-1)!

f

3() f(4)(0)-

dolaIJ(t)dt=a[J(0)-2f(2)(0)+

2!

000

(3.126f) 1

5 (2)3J(6)(0) + ...1;

3 I

f

Jo

J1

C)J(t)dt =

`

+ aLJ(1)(0) + "f(3,(0)+ 2! «'02

Exercise I

By using the forms given in (3.126), find expansions for (a)

F (b)

,fo

J1+tt)dt,

(c)

exp(- m2 t2)cos tdt, Jf0 (d)

I

sin x sin mx o

dx x

as series in m and examine the convergence.

JJ

...1

(3.126g)

108

INTEGRATION FOR ENGINEERS AND SCIENTISTS

3.10 LAPLACE'S METHOD

Obviously in many cases it will not be possible to evaluate a definite integral exactly either by the methods described in the preceding section or

by other methods, and it will be necessary to resort to either numerical methods (which are discussed in the next chapter) or approximations. Unfortunately, the more powerful approximate methods involve complex variable theory, so that this book is restricted to a brief treatment of methods involving only real variables. In many cases the integrand has sharp peaks that make the main contribu-

tion. By simplifying the integrand by means of an approximation that is accurate in these regions, an integrable form that gives a close approximation to the exact value is obtained. For example, consider 00

1(t) = jex(-tx2)ln(1+x+x2)dx

(3.127)

where the integrand has two peaks near x = 0. The peak is not at x = 0 because the in term vanishes there. For small x

ln(1 +x+x2);x++x2-+jx3,

(3.128)

so the integral is approximated by

I(t)

exp(-tx2)(x

+ x2

-

x3)dx.

(3.128a)

f'Q Because of symmetry, the odd powers make no contribution, so 1/2

I(t) - I

_co

exp(-tx2)x2dx = (i6

(3.129)

t3

By taking additional terms in the power series, it can be shown that the next term is of order t - 112 . The accuracy therefore increases as t becomes larger and the peaks sharper. The basic principle is also applicable to integrals over a finite range. To illustrate, we evaluate

1(n) = f x"sin x dx

(3.130)

0 o

for large n. The integrand has a sharp peak near x = it, so the approximation sin x - n-x can be used, which gives

1

1

J x°(n-x)dx = nn+2(n+1 o

_

1l n+2)

) (3.131

(3.10)

109

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

It is easily shown that

_

1

(n+1

_

1

1

n2+3n+2

n+2

so that for large n 7rn+2

I(n) -

(3.132)

n2

A somewhat more elaborate procedure, more in the spirit of Laplace's

method, is to note that the peak of the approximate integrand is at x = n/(n + 1)n and therefore set

1(n) - 2

f"I(n+l)+c

x"(7r-x)dx=2nn+2ft"(1-t)dt 1

(3.133)

s

where a = 1/(n+1) ^-' 1/n. Integration gives

1 -(1-s)"+1 _ 1 -(1-E)n+2

1( n )=2 nn+ 2

n+2

n+1

L

( 3 . 134)

Expanding by the binomial theorem through e2 checks (3.132).

The use of Laplace's method is facilitated by the use of three general theorems: 1.

If h(x) is a real continuous function having an absolute maximum at x = 0 and approaching - oc as x approaches both ± oo, then for large t eth(=)

dx =

eth(o)

[th(22n

)(0)]

II. If in addition to the conditions on h, g is an integrable function that can be expanded in a power series around x = 0, then for large t f,,9 (x) eth(x)

dx

g (0)eth(o)Ith

11/2.

2 0) (2)(

III. If h is a real continuous function having a maximum at x = a and h(x) < 0,then th(Q)

eth(x) ,.,

Ja

th(a)*

INTEGRATION FOR ENGINEERS AND SCIENTISTS

110

Perhaps the best-known application is Stirling's approximation to n! for large n. Using the integral definition, we have

n! =

t"e-`dt.

(3.135)

fo

I ntroducing

the variable t = n (I + x) transforms this into ac

n! =

n"(1+x)"exp[-n(1+x)]ndx

i

(3.136) OD

exp{-n[x - ln(1+x)]}dx.

= n"+' exp(-n) J i

The function

h(x)=In(1+x)-x=- 2x2 + x3 3

- x44 +

(3.137)

has a maximum at x = 0 and is negative over the range of integration. While the function In (I +x) is not defined for x < -1, from the power series expansion, it appears that for any number of terms the contribution from - oo to - l will be negligible. Therefore, the first theorem can be invoked to give n ! = (2 nn)'l2(!e)a.

(3.138)

It is obvious that the location of a peak can always be shifted to x = 0 by a change of variable and that integrands with several peaks can be treated by adding the contributions of the peaks. In some cases the integrand must be manipulated to bring it into a suitable form. For example

1(a) =

f

ex

Zdz = n1J2exp(-2a)

(3.139)

z

is not suitable as it stands because the parameter a does not multiply both terms. However, by letting z = a'l'2x, we get 1(a) = a'/2 J

exp[-a(x2 + X2 2 )]dx,

(3.140)

which brings the parameter into the right place. However, h has twin

(3.11)

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

111

maxima at ± 1. By making a linear change of variable x = I + u, we obtain

x2+ I =(1+u)2+(1+u)2 (3.141)

Therefore

1(a) = 2a112exp(-2a) J

exp[a(-4u + ---)]du

(3.142)

which, on applying Theorem I, gives the exact result. Exercises 1

Show that (a)

o

exp(-x2-tx-1)dx'=(3)1/2exp[-3(1)2/31

(Abramowitz). (b)

J/2 2"

ex(1+x2)"dx

Jo (De Bruijn). 2

Find an approximation for

J'exP(t sin x) dx valid for large t.

3.11

INTEGRATION BY PARTS METHODS

This section describes two methods that can be considered generalizations of the familiar technique of integrating by parts. The ordinary integration by parts technique splits the integrand into factors

1(a) = l

fg dx

(3.143)

a

in such a way that f (")g"+ 1 -0 as x

co (where gk is the k" integral of g).

INTEGRATION FOR ENGINEERS AND SCIENTISTS

112

Integration by parts gives

J0fdL

1(a) =fg1

(3.144)

Because of the condition only the lower limit contributes to the product and repeated applications give N

1(a)= "=0 (-1)"l`"'(a)ge+l + (- 1)N+1

(3.145)

f(N+1)gN+I dX.

1.

As an example, for J.- e-' dx/x, g (x) can be taken as a-', giving f (x) = l 1x and the expansion a0 a-s

e-a

ap

00

dx=-0 n!a-"+N!

N

a "=o

X

a-=

x

dx.

(3.146)

This is an asymptotic expansion that gives accurate results for large a but always diverges for sufficiently large N because of the N! multiplying the remainder. 3.11.1

Boley's Method

Boley has developed a modification that gives a convergent expansion. However, it is of limited practical value because it involves other integrals that are often as difficult to evaluate as the original integral. The method is based on splitting the range of integration into parts, beginning with

fgdx = +

E.

ao

fgdx + I fgdx,

(3.147a)

0,

and then applying integration by parts to the second integral on the right to obtain 00oC

00

1

Jfgdx =

fgdx +fg1 fo4

(

Of

-

fu)gIdx. f-C-0

(3.147b)

(3.11.2)

113

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

The process is then continued. This gives

ii(_1)"

f Joo

N

+n=0I (-1)"+1 f(n)gn+1 a"+, ao

+(- 1)N+1

(3.148)

f(N+1)gN+ldx.

aN+,

Under suitable restrictions on f and g the third integral will vanish as N - 00 and the second summation will only involve evaluation at a,,+,. This will be true if f ()(x) approaches zero steadily as x -+ oo for all n and g. (x) is bounded, or vice versa. The key feature of Boley's method is the choice of the ak's to make the two series converge. It can be seen that if ga+, is bounded, the second series will converge if r(")(an+

1

f(n- 1)(an)

< 1

(3.149)

for n > 1,

+fl

and it can be shown that this also guarantees convergence of the first series. f(2) _ Applying this to J' a-x/x dx, for which f = 1/x, f(') = -1/x2,

_ -2/x3, and so on, with ak = k +I gives 2 e-x

a-x

x

dx =

fI

x

- z23 xex+ 234 x3ex

(3.150)

+-+-+ . e-2

a-3

4

27

The integrals on the right cannot be evaluated in closed form, but by using tabulated values, Boley found that the three finite integrals and two correction terms gave a value accurate to one part in 103. 3.11.2 The Leplace-Winckler Expansion

The Laplace-Winckler expansion can be derived by introducing a set of auxiliary functions kv defined by

IV= -,

(3.151a)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

114

(3.151b)

k+ IV - lVkt1,

and starting with ac

fdx =

-,vf dx.

a f"o

(3.152)

o

Integration by parts gives

fdx = Iv(a)f(a) +

I

a

Ivfdx.

a

(3.153)

On the basis of Eq. (3.152) 1 of = - 24.

(3.154)

Integrating by parts and repeating the process gives

j

('x

N

ac

f(x) dx = f(a)

v(a) - J N+ 1 of dx.

n=1

a

(3.155)

a

An alternate derivation can be obtained by setting f(x)dx = u(a)f(a).

(3.156)

f."o

Differentiation gives

-f (a) = u (a)f (a) + u(a)f(a),

(3.147a)

which can be rearranged to give u(a) _

-f(a) [1+u(a)].

(3.157b)

f (a)

If this is treated as an iterative sequence

k+lu(a)=

f(a) [I+ku(a)]

(3.158a)

L(a) = v(a), a

(3.158b)

a

and the iteration is started with tu(a)

the result is equivalent to (3.155).

(3.12)

115

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

Applying this procedure to J a-"lx dx does not give a satisfactory result. In this case, tv(a) =I +a and kv(a) =I +a, so the series is obviously

divergent. However, on applying it to If exp(-x2)dx, the well-known asymptotic expansion

exp(-x2 )dx=

exp(-a2)

1

1

2

a

a3

a

+-+ 3

15

as

a'

(3.159)

is obtained. Exercises

I

(a) Show by Boley's method that (0

dx sinx= E (j-1)!

x

j=1

(x /2)

f("j -I)i/2) )

dx cos(jir-x)X

(b) Transform the integrals on the right into x/2

Jo

sin t dt

t+(j-1)( 7r/2)

and evaluate the first three terms of the series by using the approximation

sint=t-!--1)I-t13. 2

Find the Laplace-Winckler expansion of

dx f,.,o

and compare the result for n = -1 with Eq. (3.146). 3.12 CONCLUDING REMARKS AND EXAMPLES

When definite integrals that cannot be evaluated by substituting the limits into the indefinite integral arise in practical applications, a preliminary analysis should be made before attempting to evaluate the integrals. First it

should be determined whether the integrals exist or whether a Cauchy principal value or other generalization is involved. Particular care should be

exercised in cases where several integrals are involved because in such cases the existence of a solution to the problem does not guarantee that the individual integrals are well behaved. The Frullanian integrals are examples of well-behaved integrands that can be split into singular parts.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

116

After confirming the existence of the integrals, it is advisable to reduce the number of parameters involved to a minimum. For example, the integral

I (a, b) = fo'o exp[-(ax2 + bx)] dx

(3.160)

appears to depend on two parameters, a and b. However, by straightforward

manipulation based on completing the square, it can be transformed into

I (a, b)) = a-'"2ex

6

P(4a

ex p(- u2) du

(3.161)

ei2a""=

so that it can be expressed as an explicit function a-'/2 exp(b2/4a) multi-

plied by an integral depending on only one parameter, b/2a112. This appreciably reduces the amount of work required to tabulate the integral for

an extensive range of a and b. Dimensional analysis can sometimes be used to obtain such reductions. The parameters in the integral can be assigned dimensions in terms of the variables, and the Vaschy-Buckingham it theorem can be used to reduce the number of parameters. Thus in (3.160), since the arguments of functions such as exponentials must be dimensionless, a = [x- 2] and b = [x-'] while

I = [x]. The two parameters a and b form the dimensionless group ba'/ 2 so that

I = a-'/2f(ba-1/2)

(3.162a)

I = b-' g(ba-112).

(3.162b)

or

It can easily be seen that the two forms are equivalent since

b-lg(ba-1/2) =a- 1/2 9

bua 1/2) 2 =

a-I12 f(ba-112).

(3.163)

Applying this procedure to the simple Frullanian form 1(a, b) =

f '0 [f(ax) -f(bx)] dxx

(3.164)

o

shows that I (a, b) = f(a/b).

(3.165)

(3.12)

117

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

Furthermore since interchanging a and b changes the sign of the integral, must satisfy the functional equation

f

(3.166)

f(b/a) = -f(a/b).

This shows that (3.167)

I (a, b) = const. ln(b/a).

For a double integral such as

I(p, a, b) =

j-°-woo

-°°ao

exp[-(axz + 2pxy + by2)]

dx dy

xy (3.168)

_ which was evaluated in Section 3.3, the constants are assigned dimensions in terms of x and y. While I is dimensionless, a = [x- 2], b = [Y-2 1 and

p = [x-1y-']. Therefore

p

1(p, a, b) =f (ab)ii2],

(3.169)

which reduces the number of parameters to 1. The integral can be rewritten in terms of dimensionless variables u = a''2x and v = b'12y to give

I[ab '72] = I ( )

j

expL- I u2 + abP/2 uv + v2/J du

\

-00

(

)

(3.170)

uv v

After reducing the integral to the simplest form, the next step is to search

tables of definite integrals. Even if the integral is not listed, most of the exact methods begin with related integrals. It should be remembered, however, that tables of integrals are not infallible. The author encountered an incorrect entry while trying to evaluate

I(r) = - f J -oo

sin rx ln(l _ sin2xldx, rx xz J

(3.171)

\\\

which arises in the theory of a diode detector circuit driven by random noise.

This integral is proper since the singularity at the origin is integrable. It cannot be reduced by dimensional considerations since both r and x are

INTEGRATION FOR ENGINEERS AND SCIENTISTS

118

dimensionless. The natural approach is to expand the logarithm in a series to obtain for (3.172) sin rx sin2k x d + 1(r) = ?

i

rk=1

X

o

Since the integrand is an even function of x, the limits were changed to 0 to oo and the factor 2 added. An entry in Bierens de Haan (222 Table 159 No.17) reads

sin°xsin2gx o

dx

x'+

rz

=(-1)Q2o+i

2q
= 0,

2q > a.

(3.173)

Setting a = 2k and 2q = r gives

1(r)_2

(-1)2k

r k=i

Ir

2

2 +

-n

00

lk

E rk=i 4k

(3.174)

for r < 2. The sum is easily evaluated to give (3.175)

3r

However, this is incorrect. It can be seen that as r - 0 X2.

1 = -2

In1 - sinX1 )dx = const., fo

\

(3.176a)

and it has been shown that 1

2n

= 1.275659 - 0.125r - 0.039306r2 + ---

(3.176b)

for r < 2. Therefore (3.173) is erroneous, though the discontinuity at r =2 is real. When, as is often the case, the definite integral cannot be evaluated exactly, the choice is between analytical approximations as described in this chapter and numerical methods, which are covered in the next two chapters. It would appear that the development of the computer favors the purely numerical approach and that analytical approximations may become a lost art in the future. However, the development of symbolic manipulation programs may reverse this trend.

J3.12)

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

119

In practice, we should be alert for simple approximations adapted to the particular problem. For example, the integral

exp[- (st + at-°)] dt

1.(s, (X, p) = J

(3.177)

0

arises in statistical mechanical calculations. Approximations valid for either small or large s have been developed but a simple device gives a universal approximation. First the integral is converted to x

1=

Y S

exp[-y(u + u-p)] du

(3.178)

o

where y = (cc?)"( ' +p). For p = I the integral can be evaluated since

fooD exp[-y(u + u-')]du = 2K,(2y)

(3.179)

where K, is a Kelvin function.

The device that gives a general approximation is the rewriting of the integrand as

exp[-y(u + u-')] = exp[-y(u + u-)] {1 - (1 - exp[-y(u-p - u-')]}, so that

J 0'0

exp[-y(u + u-°)]du =

-

o

fo

exp[-y(u + u-')] du

f{1 -exp[-y(u-° - u-')])exp[-y(u + u-')]du.

(3.181)

It can be seen that the second integral is much smaller than the first integral.

Examination shows that exp-y(u+u- ')] vanishes at u = 0 and oo and has

a maximum at u = 1, while {l-exp[-y(u-p-u-')]) vanishes at u = 1 where it changes sign and has a maximum value of 1. There is therefore cancellation, which reduces the value of the integral. Therefore, neglecting the second integral gives

exp[- (st + at-p)] dt =, Jo

2

(asp)'ni+p) K, [2((xsp)11("+p)].

(3.182)

s

It is left as an exercise to find error bounds on this result by obtaining upper and lower bounds for the neglected second integral.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

120

Exercise 1

Show that (a)

xexp(xz)erf(x)dx = n1/z [exp(12) - 1] 2 Jo (t2-x2)'12 (SIAM Rev. Problem 64-8); (b)

a-"cos ix

°

°°

-O Jo

(t

z

2 uz +a)

it

dtdx = +t [bt + (t +aa z

2)1/2i n

(SIAM Rev. Problem 65-5); (c)

7-3

f x fx fx o

0

du dvdw

1-CosuCosvcosw=(4"3)-1l,4(1)

0

(Math. Magazine Problem 612).

BIBLIOGRAPHIC NOTES AND COMMENTS

The most extensive table of definite integrals is D. Bierens de Haan's Nouvelles Tables d'Integrales Definies (Hafner, New York, 1957). This volume, originally published at Amsterdan in 1867, contains 8339 entries. A supplement by C. F. Lindman, Examen des Nouvelles Tables d'Integrales

Definies de Bierens de Haan (G. E. Stechert, New York, 1944), contains corrections and additions. A paper by E. W. Sheldon, "Critical Revision of de Haan's Tables of Definite Integrals," Amer. J. Math. 34(1913), 89-114, gives a critical discussion, particularly of the treatment of principal values, in which de Haan's tables are often incorrect.

An earlier book by D. Bierens de Haan, Expose de la Theorie, des Proprietes, des Formules, de Transformation et des Methodes d'Evaluation des Integrales Definies, has only about 3000 entries but contains an exposition of the methods used to obtain the entries.

Good modern tables are those of I. M. Ryzhik and I. S. Gradshteyn, Tables of Series Products and Integrals (Academic Press, New York, 1966)

and of W. Grobner and W. Hofreiter, Integraltafel, Vol. II (Springer, Vienna, 1961). Y. L. Luke's Integrals of Bessel Functions (McGraw-Hill, New York, 1962), is an excellent treatment of a specialized but important field, and M. Abramowitz and I. Stegun, Handbook of Mathematical

(3)

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

121

Tables (Gov't. Printing Office, Washington, D. C., 1964) has many useful results for the higher transcendental functions.

M. Geller, "A Table of Integrals Involving Powers, Exponentials, Logarithms and the Exponential Integral," Jet Propulsion Lab. Tech. Rept. 32-469 (August 1, 1963), has many useful results. An extensive account of the classical methods with numerous examples is J. Edwards' Treatise on Integral Calculus Vol. II (Chelsea, New York, 1954). An excellent brief account on which we have drawn heavily is E.

Rufener's "Die Methoden zur Ermittlung der bestimmten Integrale", Inaugural dissertation, Basel, 1953. Many examples involving higher transcendental functions are worked out in 0. J. Farrel and B. Ross, Solved Problems (Macmillan, New York,

1963). The paper by M. Abramowitz, "On the Practical Evaluation of Integrals," SIAM J. 2(1954), 20-35, has some interesting examples and introduces approximate methods. SECTION 3.1

J.D. Williams, "An Approximation to the Probability Integral", Ann. Math. Statistics 17(1946), 363-369. SECTION 3.3

See Edwards and Rufener for more extensive coverage. Equation (3.52) is from D. R. Childs, "Reduction of the multivariate normal integral to characteristic form," Biometrika 54 (1967), 293-300. G. H. Hardy, in "The Integral f sinx(dx/x)," Math. Gaz. 5(1909), 88-103 and 8(1915), o for evaluating this integral and attempts to assess 301-313, collects a number of methods

their relative difficulty numerically. Hardy's collected works are in the process of publication; Volume 5 will contain some very interesting material. SEcnON 3.4

Probably the best-known old-fashioned algebra book is H. S. Hall's and S. R. Knight's Higher Algebra, 4th ed. (Macmillan, New York, 1950; first published in 1887). A number of series evaluations of integrals are given in L. Euler's Collected Works, Vols. 18-20. (The old-fashioned notation is probably a greater obstacle than the Latin text.)

A modern treatment is H. T. Davis, The Summation of Series (Principia Press, San Antonio, Texas, 1962). SECTION 3.5

The use of complex parameters goes back to Euler. Edwards has several sections on this topic (pp. 342-361).

For those interested in complex variable methods Smith [7] has a good introductory account. Chapter 6 of Whittaker and Watson [8] is a classic. Those wishing to go still further should consult E. L. Lindeldf, Le Calcul de Rfsidus (Gauthier-Villars, Paris, 1905), and M. L. Rasulov, Methods of Contour Integration (North Holland, Amsterdam, 1967).

122

INTEGRATION FOR ENGINEERS AND SCIENTISTS

F. Klein, On Riemann's Theory of Algebraic Functions and Their Integrals (Dover, New York, 1963) is a translation of a classic with an unusual approach based on an analogy with fluid flow. An interesting recent paper by R. P. Boas, Jr., and L. Schoenfeld, "Indefinite Integration by Residue" [SIAM Rev. 8(1966), 173-183], includes (3.5) and (3.6) among other examples. SECTION 3.6

Some of the expressions (3.70a, b) date back to Liouville. We do not know of a complete treatment of these relations but have picked them out from various sources, including Edwards. (The author is indebted to his colleage Prof. H.W. Gould for some interesting references and examples, particularly for this section.) SECTION 3.7

The most extensive set of tables are A. Erdelyi, Tables of Integral Transforms, 2 vols. (McGraw-Hill, New York, 1954). V. A. Ditkin and A. P. Prudnikov [Integral Transforms and Operational Calculus Macmillan (Pergamon), New York, 1965] have a smaller set of tables combined with an exposition of the theory and applications.

An excellent introduction to the theory is C. J. Tranter's Integral Transforms in Mathematical Physics (Methuen, London, 1956). A more advanced treatment is the article "Functional Analysis" by 1. N. Sneddon in Vol.

of Handbuch der Physik (Springer,

Berlin, 1955). An interesting development of the subject based on modern algebraic concepts is L. Berg, Operational Calculus (North Holland, Amsterdam, 1967). A very extensive presentation of the practical applications is G. Fodor, Laplace Transforms in Engineering (Akedemiai Kiado, Budapest, 1965).

H. A. Mellers, "On an Operational Method for Computing Definite Integrals," USSR Comp. Moth. Math. Phys. 1(1961), 683-704.

E. Weber, "Complex Convolutions Applied to Nonlinear Problems," Proceedings of Symposium on Nonlinear Circuit Analysis (Polytechnical Press, Brooklyn, 1957), pp. 409-428. SECTION 3.8

A. Ostrowski, "On Some Generalizations of the Cauchy-Frullani Integral," Proc. Natl. Acad. Sci. USA 35(1949), 612-616.

G. H. Hardy, "A Generalization of Frullani's Integral," Messenger Math. 34(1905), 11-18.

E. B. Elliot, "On Some (General) Classes of Multiple Definite Integrals," Proc. London Math. Soc. 8(1877), 35-47, 146-158. SECTION 3.9

H. F. Willis, "A Formula for Expanding an Integral as a Series," Phil. Mag. 39(1948), 455-459. SECTION 3.10

This section is based on Chapter 4 of N. G. De Bruijn, Asymptotic Methods in Analysis (North Holland, Amsterdam, 1958), which also covers methods using complex variable theory.

(3)

ANALYTIC EVALUATION OF DEFINITE INTEGRALS

123

SECTION 3.11

B. A. Boley, "A Method for the Numerical Evaluation of Certain Infinite Integrals," MTAC 11(1957), 261-264.

S. R. Boley, "Numerical Evaluation of Certain Infinite Integrals," Columbia Univ. Inst. Flight Structures Rep. (October 1959). The Laplace-Winckler expansion was revived in a thesis by W. Richardson,"Asympto-

tic Methods of Evaluating f'Of(x) dx," Univ. of North Carolina (Mic 60-6993). This contains some interesting results but is very difficult reading. SECTION 3.12

The basic theorem of dimensional analysis is usually attributed to E. Buckingham, "On Physically Similar Systems," Phys. Rev. 4(1914), 354-376. However, it was given by A. Vaschy, uSur Jes Lois de Similitude en Physique,>) Ann. Telegraph. 19(1892). 25-28. A discussion of Vaschy's work (including translations) appears in R. Esnault-Pelterie, Dimensional Analysis (Editions F. Rouge, Lausanne, 1948). This is a unique presentation

of the subject by a pioneer in aeronautics and astronautics. It is somewhat difficult to read because the author translated it himself but will repay the effort. Equation (3.176b) is derived by J. K. Mackenzie in"Evaluation of a Fourier Transform," SIAM Rev. 9(1967), 219-222. Many interesting integrals appear in the problem section of this journal. For example, (3.177) is Problem 67-10, SIAM Rev. 9(1967), 593-599. Research Exercises 1

Explore the use of the methods based on differential equations developed in Section 2.9 for evaluating definite integrals. (In particular, consider the case when the right side of (2.80) vanisher. This is known as a Sturm-Liouville problem.)

2

Consider the problem of the efficient arrangement of a table of definite integrals. (Consult Ryzhik and Gradshteyn.) Would it be feasible to put a really extensive table

in a computer and then have a program to check a proposed integral against this table? (The problem of tabulation was considered in the 1967 MIT dissertation by J. Moses ("Symbolic Integration") refered to in Section 2.8.]

Chapter 4

NUMERICAL EVALUATION OF INTEGRALS

4.1

INTRODUCTION

In this chapter numerical methods for evaluating integrals are presented. The basic form in most of the methods covered is fo6

N

Wjf(xj).

w(x)f(x)dx ^,

(4.1)

J=1

The points x j at which the integrand is evaluated are called the nodes or abscissas of the quadrature formulas and the W j's are the weights. The function w(x) in the integrand is called a weight function. Its use may result in a considerable increase in the efficiency of the numerical integration. Cases where one or both of the limits are infinite and where the integral has singularities are treated. A brief discussion of multiple integrals and some forms involving integrodifferential operators is also included. The history of numerical integration goes back to Newton and his contemporaries but the advent of the high-speed automatic computer has resulted in a marked revival of interest and change in emphasis. When calculations were carried out by hand or on a desk calculator with the help of tables of functions, quadrature formulas with equally spaced nodes were favored and simple weights were a considerable advantage. On a computer, functions are evaluated by analytic approximations of the type described in Chapter 2, so there is no advantage to simple nodes and the computer can store any eight-place weight as easily as a simple integer. One of the first results,

therefore, was a revival of interest in Gaussian formulas, which some mathematicians had considered museum pieces. The concept of "practical"

was revolutionized and procedures such as the Monte Carlo method, which involve a tremendous amount of computation, became feasible. One result of the new procedures is that extensive calculations involving

numerous integrations have to be performed on the computer without any opportunity to examine the numerical value of the integrands for features, such as rapid oscillations or singularities, that might result in loss of accuracy.

126

INTEGRATION FOR ENGINEERS AND SCIENTISTS

Considerable effort has been devoted to the development of automatic integration routines aimed at relieving the user of the necessity of thinking. The goal is a routine that, when supplied with a function defining the integrand, the limits of integration, and a tolerance, will return a value accurate

within the tolerance or the best possible value with a warning that the tolerance is not satisfied.

Our opinion is that this is not a proper utilization of the capabilities of the computer. It is an attempt to accomplish the impossible by brute force. It is true that such routines can be made to work for most examples encountered in practice, but in theory it is always possible to construct examples for which any predetermined program will fail because the a - S procedure of classical analysis is reversible. It is true that there always exists a step size b small enough so that the error involved in replacing an integral by a sum is less than any specified value. However, even neglecting the problems

resulting from working with a limited number of significant figures, in a numerical calculation the step size must be specified. Then it is always possible to construct functions for which the error will exceed any C. There-

fore, the generality that an automatic integration routine achieves at the expense of efficiency cannot be relied upon.

We believe that the proper approach is to utilize the capabilities of the computer for the computation of the weights and nodes of a quadrature formula adapted to the class of integrals to be evaluated. For this reason we cover a wide variety of quadrature formulas in this and the next chapter, with the goal of enabling the reader to choose or construct one adapted to

his requirement. To achieve this wide coverage it has been necessary to present many results without detailed proof and to minimize numerical examples. However, a number of illustrative programs are included in Appendix 2.

In this and the next chapter there appear statements that one integration procedure is "better" than another. These mean either that (1) the specified procedure gives a better result for some particular example that we hope is representative, or (2) it is exact for higher-degree polynomials. Or these

statements mean that (3) although both procedures being discussed are exact for polynomials of the same degree, the preferred one has a smaller coefficient for the remainder term. Such criteria are probably significant when comparing methods with same basis, such as the Chebyshev and Gaussian rules based on polynomial approximation, but the problem of comparing such methods with the Monte Carlo methods or the "best" methods is unresolved.

(4.2)

127

NUMERICAL EVALUATION OF INTEGRALS

4.2 SIMPLE QUADRATURE FORMULAS WITH SPECIFIED NODES

This section presents formulas for evaluating integrals over a finite range by expressions of the form N

6

f

f(x) dx

(4.2)

WW f(x;)

a

with specified values of x;. Expressions of this type are useful for dealing with experimental data. Apparently the ease with which sets of weights for specific cases can be calculated on a computer is not generally realized. The most commonly used formulas with specified nodes are those using equal spacing, which have been extensively tabulated for the forms IN

Jo

f(x)dx =;-,i Wf(j N -1 N

1

o

f(x)dx

W;f(2j-1

E ;=,

f1f(x)dx ^

j=1

o

(4.3a) l

(4.3b)

2N

;f

W

(4.3c)

N+1

Equations (4.3a) are the closed Newton-Cotes formulas, while Eqs. (4.3b) are the open Newton-Cotes formulas. Equations (4.3c) constitute a family of open formulas; developed by Steffensen, that are useful in the numerical solution of differential equations. Open formulas have all the nodes inside the range of integration, whereas closed formulas use the end points. There is no generally accepted designation for formulas using points outside the range of integration, some examples of which are considered in the next chapter. Recently there has been some interest in a class of formulas proposed by Clenshaw and Curtiss: +1

I j=1 N

f(x)dx ^

0-OR N-1

J,

(4.4a)

which for a standard range 0 to 1 becomes

J

I f(x) dx =

Wjr ! N

=1 2

f 10.51 l + cos (1-1)n)l. L

\\\

N-1 J

(4.4b)

NUMERICAL EVALUATION OF INTEGRALS

(4.2)

128

Although these are generally based on an expansion of the integrand in Chebyshev polynomials, the W j's can be determined conveniently by the methods of this section.

The weights are determined by making the quadrature formula exact for N powers of x beginning with x°. Since integration is a linear operation, this makes the formula exact for polynomials of degree N- 1. The justification for this procedure is the Weierstrass approximation theorem, which states

that over a finite range any continuous function can be approximated to any desired accuracy by a polynomial. This does not mean, however, that any sequence of polynomial approximations will converge, a point that is discussed further later. We present two general procedures for calculating the required weights. The first is an algebraic method using the set of N equations

b`-a'

_ YN

i = 1, 2, ..., N,

Wj(xj)I-1

j=1

(4.5)

to determine the W j's. The notation is unusual in that the xj's are known quantities and the W j's are the unknowns. The determinant of the coefficients is of a type, known as a Vandermonde determinant, that cannot vanish, so there is always a solution. Furthermore, if the xj's are rational numbers, so are the W j's.

The second method, based on the Lagrange interpolation polynomial, gives an explicit expression for the weights involving the integral of a

,

polynomial. Given a set of points xk and a variable v, the polynomial

Ij(U) -

is

i#j xj-xi

(4.6)

1 when v = x j and 0 when v is any of the other x's. Therefore, the

polynomial N

P(v) _ I f(xj)lj(U),

(4.7)

j=1

which is of degree N- 1, coincides withf(v) at the N points xj. The assumption underlying the quadrature formulas of this section is that b

b

J f(x)dx = J p(v)dv. a

a

(4.8)

(4.2.1)

129

NUMERICAL EVALUATION OF INTEGRALS

Comparison with the basic form shows that b

II(v_x&)dV

WJ = J l1(v)dv = a

a

X,-X;

( 4.9)

Since the integrand is a polynomial, the integral is easily evaluated once the

product is multiplied out. However, this involves considerable algebra, particularly for large N. The calculation could be done on a computer either by developing a suitable symbolic manipulation routine or by using the numerical integration routines described below, which are exact for polynomials. We leave this as an exercise for the interested reader and content ourself with giving a program using the first method in Appendix 2. As we show in Section 4.20, this program can also be used to determine weight

for interpolation, numerical differentiation, and approximation of other linear operators. It should be noted that if the nodes are symmetrically disposed about the midpoint, so that

x;-a =

(4.10a) b-xN-J+1.

then by a simple symmetry argument WN-J+I = WJ.

(4.IOb)

This reduces the number of integrals to be evaluated if (4.9) is used and reduces the size of the system of linear equations if (4.5) is used. As the work involved in solving a system of N linear equations increases as N 3, this represents a considerable saving of work. 4.2.1

Specific Examples

To illustrate the two methods, we calculate the weights for Eq. (4.3a) with N = 3 by both methods. We have ('1

f I(x)dx = WII(o) + W2 l(+) + W3I(1),

(4.11)

J{0

but since by symmetry W, = W3, the equations to be solved are

1=2WI+W2i

(4.12a)

I =WI+iW2.

(4.12b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

130

The equation for i = 2 is a multiple of (4.12a), a general phenomenon in symmetrical cases. The solution is W, = W3 = * and W2 = . Using Eq. (4.9) with x, = 0, x2 = }, and x3 = I gives W1

_

' (v-])(v-1)dv=a'9

fv(v-1)

W2 =

0 O(-)

(4.13a)

(-#)(-1)

0

dv =

-4f' (v2-v)dv = 3

.

(4.13b)

0

The resulting expression I

f0

f(x) d x = 6 E j'(0) +f(1)] + 2 f(4),

(4.14)

known as Simpson's two-thirds rule, is probably the most widely used integration formula. It is interesting to note that when applied to x3 it gives I

x3dx=

6(0"}'1)+32 (8)=q,

(4.15)

f

which is exact. It is a general principle that formulas of the type being considered in this section with an odd number of nodes are exact for a polynomial of degree N instead of N- 1, and in most cases are actually better than the same type of formula with an additional node. Since extensive tabulations of weights are available and a program for generating weights is given in Appendix 2, we list only a few low-order formulas here. Actually, for reasons that are discussed later, the high-order formulas are not often used in practice. The simplest closed Newton-Cotes formulas are the trapezoid rule o

f(x)dx

f(x)dx

I[f(0)+f(l)],

(4.16a)

$[f(0) +f(1)] + '' [f(;) +f(;)],

(4.16b)

J0,

which is called Simpson's three-eighths rule, and Boole's rule I

f

f(x) dx ^-

9

[f(0) +f(1)] + 3 [f(j) +f($)] +'- f(#)

(4.16c)

(4.2.1)

131

NUMERICAL EVALUATION OF INTEGRALS

The simplest open Newton-Cotes formulas are the midpoint rule J

1

f(x)dx ^' f(#),

(4.17a)

followed by

Jf(x) dx ^' # [f(j) +f($)],

(4.17b)

0

f t f(x)dx - e [f(6) +f(6)] + *f(2),

(4.17c)

0

f(x) dx = a e [.r(e) +f('-s)] + ae [f(-e) +f(S)]

(4.17d)

0

The Steffensen formulas also begin with the midpoint rule; then we have

J^' fix) d x = #[f(3) +f(3)],

(4.18a)

0

f(x) dx

3 [f(#) +f(j)] - f(#),

(4.18b)

3

J0'

f(X)dx = Z4 [f(S) +f(5)] + 24 [f(s) +f(-)]

(4.18c)

Jo0

In general, the open formulas are inferior to the closed formulas. The midpoint rule, which is slightly superior to the trapezoid rule, is the most important exception. Exercises I

Derive the general transformation to find the weights and nodes for b'

f Ja

N

Wj f(x;)

f(x) dx = j=l

when given those for b

j

f(x) dx = a

Wj f(x j). j=1

INTEGRATION FOR ENGINEERS AND SCIENTISTS

132 2

Calculate the first four Clenshaw-Curtiss weights.

3

We give the following results to be used as test examples of the various quadrature rules for finite intervals presented in this chapter. t

I

x1/2 dx =

0

J

1

x3J2 dx =

0

f'

t

i

0 1+?

J' s

o l+x

= 0 37988551

= 0.77750463

2 dx

Jo 2+sinlOttx = 0.86697299

dx

J

0 x4+x2+0.9

1+X4 4

xdx

o e-1

dx = 0.69314718

fodx

dx

= 1.1547005

= 0.79111645

Show that the value of the Nth-order Vandermonde determinant is

fl

(x, -x,).

I -i<

4.3 CHEBYSHEV'S EQUAL WEIGHT QUADRATURE FORMULAS

When evaluating an integral where the individual function values are subject to an uncertainty e, the maximum possible error is e Y I W11. When the errors are random, this is minimized when all the W1's are equal. This consideration led Chebyshev to develop a family of quadrature formulas of the form N

J

1

o

f(x) dx t. W Y f(xj) j=t

(4.19)

where the xj's are selected to make the rule exact for powers of x. Since the

N xj's and W are adjustable, the formula can be made exact for a polynomial of degree N. Chebyshev's quadrature formulas are used in certain technical fields, such as naval architecture, where it is necessary to evaluate many integrals using data from blueprints, but are not favored by numerical analysts. We present them here primarily because they are a useful bridge

between the formulas of the preceding section and the very important Gaussian formulas of the next section.

It is obvious that if (4.19) is to be exact when f(x) is a constant, then W = 1 IN. Then in order for it to be exact for x1 through x, the x j's must

(3.4)

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NUMERICAL EVALUATION OF INTEGRALS

satisfy the set of nonlinear equations N

N

= Y (xf)k,

k+1

k = 1, 2, ..., N.

(4.20)

;=1

It is possible by a systematic procedure to reduce this set of equations to a single polynomial equation of degree N the roots of which are the x1's. Therefore, the xj's are not rational numbers and may not be real. In fact, S. N. Bernstein proved that all the roots are real only for N = 1 to 7 and N = 9. Although it has been pointed out that when the integrand is analytic, there is no basic objection to using complex nodes, apparently no one has calculated the required nodes and investigated their use. We will carry out

the calculation for N = 2 and 3 and introduce a general technique for solving systems of nonlinear equations such as (4.20) that will be useful for other quadrature formulas. For N,= I it is obvious from symmetry considerations that x1 = There-

fore the midpoint rule is the simplest Chebyshev formula as well as the

simplest open Newton-Cotes and Steffensen formula. For N = 2 the equations to be solved are

x1+x2=1,

(4.21a)

x1+x2q.

(4.21b)

2

2

2

The first equation expresses the symmetry conditions and the second then becomes the quadratic

Xa +(I _X.)2 =3

(4.22)

where xQ can be either x1 or x2. The roots are 4f 12 I/2 For N = 3 the equations are

By symmetry x2 =

,

.=x1+x2+x3,

(4.23a)

1=X2+x2+x2

(4.23b)

3 4=x1+x2+x3.

(4.23c)

which reduces the system to a quadratic xa

- xa+ =0

where again xa is either x1 or X3. The roots are + f 8 - 1/2.

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134 4.3.1

Solution of Non linear Systems

A general method for solving a system of nonlinear equations modifies the system f1(x1 , x2, x3, ...,

0,

(4.24a)

(4.24b)

f3(X1,X2,X3,...,X.)=0.

(4.24c)

by introducing a parameter t and considers the system

j [X (01 = (1-t) f (x0)

(4.25)

where x0 is an initial guess for x. Note that when t = 1, Eq. (4.25) reduces to (4.24), while for t = 0 the equation is satisfied for an arbitrary choice of x0. Differentiation with respect to t gives the set of differential equations 1 afl

I

_-

(4.26)

If this set of differential equations is solved by using the initial condition x(O) = x0, then x(1) is the solution of the system of algebraic equations. Obviously it is not going to be possible to solve the differential equations analytically and the numerical solution will be laborious, as it is necessary to solve a system of linear equations to find the derivative at each step of the solution. The method can be programmed and appears to compare favorably with other methods for solving sets of nonlinear equations. However, the author's experience with this method indicates that it has limitations. Generally, the difficulty manifests itself in solving the set of linear equations that determines the derivatives at each step of the numerical solution of the differential equation. It can be seen that if the differential equation is solved by the Euler method with h = I, then (4.26) reduces to the conventional generalization of Newton's method for systems of equations. If the initial guess is close enough to give convergence, Newton's method will be faster since it will not be necessary to solve the set of linear equations as many times.

It was found possible to compute the Chebyshev modes by direct solution of the nonlinear system but it was necessary to use double precision arithmetic. The behavior for N = 8 where no real solution exists depended on

the details of the method used. Newton's method diverged violently, whereas a differential equation routine ran for a long time without completing the integration.

(4.4)

135

NUMERICAL EVALUATION OF INTEGRALS

4.3.2 Dufton's Rule

Examination of the nodes shows that for N = 4 they are very close to simple

decimals. Dufton therefore proposed the expression (' 1

f

f(x) dx = I [ f(0.1) + f(0.9) + f (O.4) + f(0.6)]

(4.27)

Jo

but despite its simplicity it is not well known. Exercises I

Derive the interpolatory weights corresponding to Dufton's rule.

2

Show that the interpolatory weights corresponding to the Chebyshev nodes are the Chebyshev weights.

4.4 GAUSSIAN QUADRATURE

Logically, the Gaussian quadrature formulas in which both the weights and nodes are adjusted to make the integration exact for polynomials of degree 2N-1 are the natural culmination of the two previous sections, but actually Gauss developed this method long before Chebyshev's work. Following the procedure of the previous sections, we find that the weights and nodes satisfy the set of 2N equations N

Yj=t

WW(xj)f = 1 ,

i+1

i = 0, 1, ..., 2N-1,

(4.28a)

but by invoking symmetry this can be reduced to a set of N equations 121

(NY

Wj [(Xj)' + (1- X j)i] +

1-

j=1

2

( + 11)N

W. =

1 i+1

(4.28b)

The [N/2] means the largest integer in N/2 that is N/2 for even N and (N- 1)/2 for odd N. Wm is the weight corresponding to (N+ 1)/2 when N is odd. Surprisingly, this system of nonlinear equations always has a real solution with all the x j's between 0 and I and all the W j's greater than 0.

The reason for this will become evident when we present an alternate derivation using orthogonal polynomials. This derivation is given in Section 4.15 in connection with the more general form N I a

w(x) f(x) dx =

Wj f(xj).

(4.29)

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136

It is easily seen that the Gaussian formula for N = 1 is the midpoint rule.

For N = 2 the Gaussian formula is the same as the Chebyshev formula of the preceding section. For N = 3 the algebra becomes complicated. Fortunately, extensive tables are available. The principle of determining the nodes to maximize the accuracy of the

evaluation of an integral can be used in designing experiments. Ower describes its use to determine the flow rate through a duct from velocity measurements, while Price gives an interesting application where it is used to simplify data reduction. 4.4.1

Constrained Gaussian Rules

A number of modifications of the Gaussian principle, in which some of the nodes or weights, or both, are specified in advance, have been developed.

The best-known formulas of this type are the Radeau formulas, in which one of the end points is utilized, and the Lobatto formulas, which use both end points. Recently, Kronrod developed a very important concept that is also included in this section. In Section 4.7 an ingenious variant developed

by Ralston is presented. The simplest Radeau formula is ('1

f(x)dx

W1 f(x1) + W2 f(0).

(4.30),

The equations for the W's and x's are 1 = W1 + W2

1

}= W, X1, = Wxi.

(4.319) (4.31b) (4.31c)

Dividing (4.31 c) by (4.31 b) gives x1 = + ; then (4.31 b) gives W1 = $ and (4.31 a) gives W2 = 4, so that 1

f

0

f(x)dx = if(,) + If(0).

(4.32)

While symmetry cannot be used to simplify the equations for Radeau quadrature because choosing one end point makes the nodes nonsymmetrical,

(4.4.2)

137

NUMERICAL EVALUATION OF INTEGRALS

symmetry considerations show that if we use f(l) rather than f(0) JI

f(x)dx ^' 1f(3) + kf(l)

(4.33)

o

and a similar transformation holds for any N. The simplest Lobatto formula o

f(x)dx

WIf(x1) + W2 f(0) + W3f(1)

(4.34)

is Simpson's two-thirds rule, since by symmetry x1 =I and w1 = w2. Tables of both Radeau and Lobatto weights and nodes for large N are available. 4.4.2 Kronrod Quadrature

Kronrod developed his quadrature formulas by asking how the two best independent estimates for the value of an integral could be obtained for a fixed number of evaluations of the integrand. If the Gaussian formulas of

orders N and N+ I are used, none of the nodes coincide, so the 2N+ I evaluation gives values exact for polynomials of degree 2N-1 and 2N+ 1. Kronrod showed that a better result could be obtained by using the N-point Gaussian formula in conjunction with a new class of formulas taking the N Gaussian points as prescribed and determining N+ 1 additional nodes and the 2N+ 1 weights to obtain maximum precision. The resulting formulas are exact for polynomials of degree 3 N + I when N is even and 3 N + 2 when N is odd.

For N = 1 the three-point Kronrod formula corresponds to the threepoint Gauss formula, so the pair is J I

f(x) dx ^ f(f)

(4.35a)

0

f(x) dx ^-
(3 )1/2]} (4.35b)

but for larger N the Kronrod formulas have nodes and weights different from those of the Gaussian formulas. A program is included in Appendix 2 utilizing a Gaussian quadrature with 20 nodes and the associated Kronrod formula using 41 nodes. This is

INTEGRATION FOR ENGINEERS AND SCIENTISTS

138

extremely powerful and the difference between the two values is usually a conservative indication of the error. An alternate method of obtaining an optimum pair of quadrature formulas is to use a (2N+ 1)-point Gauss formula combined with a formula obtained by using the method of Section 4.2 to obtain weights for a formula with the central node or two symmetrical nodes eliminated. Kronrod argues that the resulting formulas would be too similar to the Gaussian to afford a reliable check, but the matter warrants further investigation. Exercises 1

Show why the following scheme fails. The seven-point Kronrod rule is exact for a polynomial of degree 11, while the associated three-point Gaussian formula is exact for fifth-degree polynomials. Suppose instead we take the seven-point Gaussian formula, which is exact for a thirteenth-degree polynomial, and generate an auxiliary formula by the following procedure. Drop the first and seventh points, and derive interpolatory weights that will be exact for a fifth-degree polynomial; do the same with the second and sixth nodes. By taking a suitable linear combination of the two rules (see Section 4.6), a rule can be generated that will be exact for sixth-degree polynomials and that uses the same node as the original Gaussian rule. The combination is

therefore better than Kronrod's. 2

Let s

f(x) = fl (x-x1)2 J=1

where the xj's are the nodes of the. five-point Gauss rule. Evaluate the integral $f(x)dx by the Gauss rules for N = 2(1)6.

4.6 CONVERGENCE OF QUADRATURE FORMULAS

In the preceding three sections families of interpolatory quadrature formulas have been described that are exact for polynomials of increasing degree as the number of nodes increases. In this section the question of the convergence of quadrature formulas is treated briefy. The problem has two aspects: 1. Given a family of quadrature formulas, for what class of functions does it converge ?

2. Given a class of functions, what conditions must be imposed on a family of quadrature formulas to ensure convergence ? A typical result of each type is presented. A typical result of the first type is that the closed Newton-Cotes formulas described in Section 4.2 diverge for analytic functions that have a singularity inside an ellipse centered at the midpoint of the range of integration (b+a)/2,

(4.6)

NUMERICAL EVALUATION OF INTEGRALS

139

and a semimajor axis ()(b - a) and a semiminor axis (8)(b - a). This divergence is related to the Runge phenomenon in interpolation. As stated previously,

the Weierstrass approximation theorem guarantees the existence of a polynomial approximation to any continuous function over a finite range. It does not, however, guarantee that the sequence of polynomials generated by the Lagrange interpolation formula with equidistant points provides such an approximation. Runge showed that the sequence of such approximations to (1 +25x2)-' over the range -1 to + 1 oscillated with increasing

amplitude for x < - 0.726 and x > 0.726 because of the singularities at x = ±0.2i. This difficulty can often be circumvented by dividing the range of integration into parts, which replaces the region in which singular-

ities are forbidden with a number of smaller overlapping regions with a reduced total coverage. Sections 4.6 and 4.7 give a more detailed discussion of the advantage of subdividing the range of integration.

An important result of the second type is a theorem due to Polya, who showed that a family of quadrature formulas will converge for continuous functions if and only if (1) it is exact for polynomials; (2) the sum of the absolute value of the weights remains bounded.

As the sum of the weights is equal to the range, the Gaussian-typeformulas of Section 4.4 would converge, as would the Chebyshev formula, if the complex nodes needed for degree greater than nine were used. However,

the equally spaced Newton-Cotes and Steffensen weights do not satisfy this condition. S. N. Bernstein has developed a criterion involving the Legendre polynomials for determining whether it is possible to have a quadrature formula with positive coefficients for a given set of nodes. He also showed that it is possible to construct a formula with rational nodes and positive coefficients for any degree polynomial. However, the number of the nodes will be much higher than the degree of the polynomial. Exercise 1

Calculate the successive approximation for N = 3(1)12 to 1/(1+25x2) at a point inside and outside the region of convergence using (a) equal-spaced Lagrange interpolation; (b) Lagrange interpolation based on the Clenshaw-Curtiss nodes [eq. (4.4 a, b)].

4.6 ERROR ANALYSIS

In this section a brief analysis of the error involved in evaluating an integral by a quadrature formula is presented. There are two types of error involved: truncation error, which decreases as more values of the integrand are used; and roundoff error, which increases with the number

INTEGRATION FOR ENGINEERS AND SCIENTISTS

140

of function values used. Because of these opposing factors there is a limit to the accuracy, which may be much less than the number of significant figures carried in the calculations. A specific example is presented in Section 4.19.

Roundoff error is a specialized problem and depends in part on the characteristics of the computer. The basic result relevant to the present discussion is that when a large number of terms are added together, the maximum possible total error increases proportionately to the number of terms, whereas the probable error increases as the square root of the number of terms. In a normal numerical integration using a reasonable number of terms roundoff error is small. However, when an attempt is made to use the capability of a high-speed computer to achieve accuracy through a brute force approach

by using hundreds or thousands of nodes, roundoff becomes significant. It can be seen that roundoff is accentuated when the weights are large and of opposite sign. Thus a quadrature formula nominally exact for a very highdegree polynomial may in practice give an appreciable error for a constant integrand. The principal problem in such a case would probably be roundoff

in the calculation of the weights. It is often necessary to carry out such calculations in double precision. Roundoff may also be significant when evaluating an antisymmetrical or approximately antisymmetrical integrand.

The estimation of truncation error is being actively investigated at present, using methods based on the theory of analytic functions and functional analysis so that there may be important developments in the near future. At present, error estimates are of limited practical value because they are often extremely conservative and almost invariably require at least as much work as the calculations proper. 4.8.1

Remainder Estimates

Conventional error estimates are based on a high-order derivative and therefore implicitly assume a corresponding degree of continuity of the integrand, so that they can be approximated by a finite Taylor series plus a remainder term:

f(x)

` f li

L- 1

(t)

) (x-a)'

+ f (L)(m) (x-a)L L!

(4.36)

where m is an unspecified point in the range of integration. A quadrature formula that is exact for polynomials of degree L -I evaluates the finite sum exactly and gives a remainder term f(L)( m ) bL+ I - aL+ 1 R -

L!

L+1

N

-

;=1

Wi(x1)L

.

(4.37)

(4.6.1)

NUMERICAL EVALUATION OF INTEGRALS

141

The term in the bracket is the difference between the exact and approximate value of the integral for the lowest power of x for which the integration is not exact. While the point m at which the derivative should be evaluated cannot be determined, upper and lower bounds can be obtained from the

largest and smallest values of the derivative in the range of integration. Aside from the difficulty of evaluating high-order derivatives, the bounds are obviously very conservative and may be inferior to simple inequalities such as were presented in Chapter 1. To illustrate the application of (4.37) the remainder is computed for the midpoint and trapezoid rules, which are both exact for first-degree polynomials, so that L = 2. For the standard range 0 to I Rmidpolnt =

Rtrapezoid

=

f( m)

L(m)

[3 - (1)2] =

[I - j[O2+1217 =

(4.38a)

f24)'

f (m).

(4.38b)

12

Since the midpoint rule has a smaller coefficient for the error term, it is considered superior to the trapezoid rule; but since the value of m is not the same for the two rules, it is not difficult to construct examples for which the

trapezoid rule is more accurate. It can be argued that the foregoing calculation does not show the true superiority of the midpoint rule, which uses only one node, over the trapezoid rule, which uses two nodes, and that a fairer comparison is with

0

f(x)dx = 4[f(j) +f(j)]

(4.39)

This is obtained by splitting the range of integration into two equal parts and using the midpoint rule on each part. This procedure, which is called compounding, is discussed in the next section. A simple calculation gives the

remainder f(m)/96, which is one quarter that of the simple midpoint rule. It is a general result that when the range of integration is split into p equal parts and the same simple quadrature formula is applied to each part, the remainder is reduced by a factor p - L

Dufton's rule I

f(x) dx ^-, 1 [ f(O.1) + f(0.4) + f(0.6) + f(0.9)]

(4.40)

f0 is also only exact for linear functions, but the remainder -f(m)/1200 is very small. This is not entirely due to the number of nodes, as splitting

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142

the midpoint rule into five parts gives f I

f(x)dx ^'-'-[f(0.1)+f(0.3)+f(0.5)+f(0.7)+f(0.9)],

(4.41)

0

which has a remainder f (m)/600.

A method of deriving quadrature formulas is combining two formulas with the same degree of precision to cancel the remainder. Thus taking (2 Midpoint + Trapezoid)/3 gives Simpson's two-thirds rule, while (4 Eq. (4.39)

- Midpoint)/3 gives I

f(x) dx

f0

3

[f(3) +f(4)] - 3 f( j),

(4.42)

which is the three-point Steffensen rule. In these cases the resulting formulas are interpolatory, but this is not generally the case. Taking [2 Eq. (4.40)+ Eq. (4.41)]/3 gives

fo f(x)dx

3'--°

177[f(0.1)

+f(0.9)] + 5[f(0.4)+f(0.6)] +

+2[f(0.3)+f(0.5)+f(0.7)]

1

(4.43)

which is exact only for quadratics. However, a number of rules of this type were widely used for hand computation because they had simple positive weights. 4.6.2 Lanczos' Estimate

Lanczos has proposed an alternative method for estimating the error that seems very promising but the exact conditions for its applicability have not been determined. The basis of the method is a relation between the remainder R for J f(x)dx, which is wanted, and the remainder R' for !a d/dx (xf)dx, which can easily be determined. Since the kth term in the power series expansion of the second integrand is k times the corresponding term of the first integrand, Lanczos takes R

(4.44)

R

L+1

R' is determined by noting that as the integral is an exact differential its value is bf(b)-af(a). Also, carrying out the differentiation gives

f

d

a

dx

(xf)dx =

('

fdx.

xf dx + Ja

fa

(4.45)

(4.6.2)

143

NUMERICAL EVALUATION OF INTEGRALS

Applying the same quadrature formula to these integrals gives R

[bf(b) ^ af(a) -

i=i Wjf(xi) - Yi=i WJxif(xL)] L+1

(4.46)

The first sum is the approximate value of the integral being evaluated; the second is the approximate value of j", xf dx by the same formula. Lanczos claims that the sum of the approximate value and the remainder will generally be an improved approximation to the integral. There is a need for a standard terminology to eliminate some confusion in the literature. Such a terminology would distinguish between three situations: 1. The true value falls between the approximate value and the approximate value plus a correction term. 2. The true value is closer to the approximate value plus a correction term than it is to the approximate value. 3. The true value lies between the approximate value ± correction terms. If Lanczos' correction is added to the midpoint rule, the formula

$ f(x)dx = #[f() +f(1)] - kf(#)

(4.47)

is obtained. In the next chapter quadrature formulas of this type that use derivatives are discussed in detail. For the present we note that the remainder

term for this expression calculated by Eq. (4.37) is -f (m)148. Therefore, although it does represent an improvement over the midpoint rule, it is less efficient than Eq. (4.39), which uses two nodes and no derivative.

There are theoretical reasons for doubting the general applicability of the Lanczos error estimate. The first reason is that it is a linear estimate; that is, if R, is the remainder for f' f1(x)dx and R2 is the remainder for j; f2(x)dx, the remainder for $ (f1 +Cf2)dx is R1 +CR2. Therefore, it is usually possible to select C in such a way as to construct examples for which either (1) the correction term vanishes even though the error is not small, so that it is not always a type I estimate; or (2) the error vanishes but the correction term does not, so it is not a type 2 estimate. An alternate type of counterexample is based on a general method for circumventing quadrature formulas using predetermined nodes. Any quadrature formula of the type we are considering will give 0 for the integral

of the function C rj; (x - x;) 2, but as the integrand is never negative, the correct value can be made as large as desired by selecting C. If we consider

the slightly more general function C-(b-x)(x-a) f; (x-xj)2, both the quadrature formula and the Lanczos error estimate vanish.

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144

To investigate the practical use of the estimate, the integrals "

ex dx = e"- 1,

J

(4.48a)

0

(' Jl

xl/2 dx = 3 u3/2

(4.48b)

cos x dx = sin u

(4.48c)

J0"

and the Lanczos error estimate were evaluated by the Chebyshev rules for u ranging between 0.1 and 10. The principal conclusion from examining the results was that when the correction term was large, it was not a reliable estimate of either the class 1 or class 2 type. When the correction term was small, it was frequently but not invariably of type 1. In view of its potential usefulness, a more complete theoretical study of the Lanczos error estimate appears warranted. Exercise 1

Calculate the remainder for the interpolatory formula using the Dufton nodes and compare the accuracy of this formula and the Dufton formula for some simple cases.

4.7 COMPOUNDING AND ADAPTIVE INTEGRATION

On the basis of the results on the convergence and error of simple quadrature formulas presented in the two preceding sections, it appears that it is often better to split the range of integration into parts and use an integration formula of a moderate degree of precision for each part rather than to use a single high-order formula for the entire range. This procedure, which we call

compounding, is particularly useful when the integrand does not have continuous high-order derivatives or when the range of integration is so extensive that it is unreasonable to expect a single polynomial to give a good

fit over the entire range of integration. An example of the first type is fu x21j2 dx, while the 10100 sin x dx is an example of the second type. It is relatively simple to fit a polynomial to a trigonometric function over a half cycle or cycle but the degree increases very rapidly as the range of the approximation is extended.

Compounding may be based on equal or unequal subdivisions. Obviously the use of small subdivisions in regions where the integral varies rapidly is desirable, but it may involve considerable unnecessary work to use the same small divisions in regions where the integrand is slowly varying.

(4.7)

145

NUMERICAL EVALUATION OF INTEGRALS

In a hand computation it is easy to distinguish such regions, and in recent years computer programs have been developed to provide variable subdivisions automatically. This is called adaptive integration and is discussed after consideration of equal subdivisions.

It is straightforward to compound open formulas, the Newton-Cotes being particularly convenient because they retain their equal spacing characteristic (whereas the Steffensen formulas have a double step at the junctions). Closed formulas require careful formulation to avoid calculating

the integrand at the junctions twice. Typical compound formulas for an arbitrary range a to b are N

Jon f(x)dx = At Y i=1

f[a + (j-{)h]

(4.49a)

where h = (b - a)/N (compound midpoint rule); b

f

a

.f(x) dx

h

rf(a) +l(b) +

Y1

2

i=1

L

f(a +jh)]

(4.49b)

where h = (b-a)/N (compound trapezoid rule);

f.,

f(x)dx

2.Sh

f[a + (10j-9)h] + f[a + (10j-6)h] i=1 [+ f[a + (10j-1)h] + f[a + (10j-4)h]

j (4.49c)

where h = (b-a)/1ON (compound Dufton rule); 6

h

N-1

N

af(x)dx^=- 4,YIf[a+(2j-1)h]+2 Y f [a + 2jh] + f(a) + f(b)j i=1

(4.49d)

where h = (b - a)/2 N (compound Simpson two-thirds rule); fbf(x)dx -_ a

h" 2

f[a +(1- -(112)1/2)h]1 1

+f[a +(i-}+(IZ)112)h] j

(4.49e)

where h = (b - a)/N (compound two-point Gauss or Chebyshev rule).

In many cases it is desirable to calculate the integral for several values of h, so it is convenient to be able to reduce h while still using the nodes for which the integrand has already been calculated. It can be seen that for the trapezoid rule, Simpson's rule, or any closed Newton-Cotes formula if It is halved, all the nodes remain nodes. Similarly, for an open Newton-Cotes formula, all the nodes are retained if the step size is cut to }. For Dufton's

INTEGRATION FOR ENGINEERS AND SCIENTISTS

146

rule, reducing h to h/4 retains the nodes, but it is not possible to cut the step size in such a way as to retain all of a set of irrationally spaced nodes such as are used in Chebyshev or Gaussian quadrature. 4.7.1

The Ralston Method

Ralston devised ingenious modified Gaussian formulas that give a particularly high degree of accuracy for the number of points used when compounded. Weights and nodes can be found for

f f(x) dx = Y1

o

WV.f(x,) + WN+I [f(0) -f(1)J

N

j=1

(4.50)

that make it exact for polynomials of degree 2 N. When this is compounded, the junction points cancel, except for the end points, giving 6

K

N

l

f(x)dx -- h Wh+I[f(a)-J(b)] + Y Y W;j[a+(k-l+x;)h]} k=1 j=1

o

J

(4.51)

where h = (b - a)/K. For N = I the system of equations W, = 1,

(4.52a)

W1 x 1 - W2 = },

(4.52b)

- W2 = 3

(4.52c)

W, x 1

determines W1, x 1, and W2. Because of the asymmetrical treatment of the end points, x, 01. The system reduces to a quadratic

x;-x1+'-6=0

(4.53)

the roots of which are

The corresponding values of W, are ±(-1r)". Ralston used only the plus sign but it would appear that the

calculation with the minus sign would provide a useful check, as the remain-

der term would have the opposite sign. The average of the two values in this case corresponds to the use of Eq. (4.49e). For N = 2 Ralston gives

f f(x) dx 1

(72)-

0

[f(0) -f(l )J +

+ CI + ir(,)"ZJf[# + 11 (212 - 71/2)]

+[ -

1112(7)'r2]f[#+-'-U(2ll2+71/2)],

(4.54)

(4.7.2)

147

NUMERICAL EVALUATION OF INTEGRALS

which is exact for fourth-degree polynomials. The complementary rule can be written by inspection. In this case the average will be exact for fifth-degree

polynomials so it will be less precise than the compounded four-point Gaussian evaluation.

This suggests the possibility of generating Ralston-type formulas by starting with a Gaussian formula of even degree and selecting half the nodes in such a way that no two are complementary. One possibility is to determine the weights by the method of Section 4.2 so that the individual formulas would be exact for degree N. Another possibility is to take W, to WN as twice the corresponding Gaussian weights and determine WN.+, by N 11

WN+1 =1=1E Wixi-3

(4.55)

Then this formula and its complement would only be exact for first-degree polynomials but the average would be equivalent to the 2N-point Gaussian result. It might therefore provide a simple method of checking the accuracy of the integration. The possibility of course exists that some of the asymmetrical formulas may have a higher degree of precision than the minimum value guaranteed by the method of derivation. 4.7.2 Adaptive Methods

To illustrate the idea of adaptive integration, let us consider what can be done by a combination of the trapezoid and midpoint rules to evaluate an integral within a specified tolerance. We start by evaluating the integral by the compound trapezoid rule, Eq. (4.49b), using N subdivisions; then by the compound midpoint rule, also using N subdivisions. If the difference between the two values is less than the specified tolerance, the step size is small enough. By taking (2 Midpoint value +Trapezoid)/3, the compounded Simpson's two-thirds rule value is obtained and taken as the final value. If the difference is greater than the specified tolerance, the compound trapezoid

value for 2N steps is obtained by taking I (Midpoint+ Trapezoid). Then the corresponding midpoint value is calculated, which involves a completely

new set of nodes, and the difference is compared with the tolerance. The procedure can be repeated several times until the tolerance is met or some criterion for giving up is reached. Such a criterion may be based on a simple limit to the number of nodes, or we can decide to accept an increase in the discrepancy between the trapezoid and midpoint rules as an indication that roundoff error has become significant. A program for this process is included in Appendix 2.

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148

As pointed out previously, the simple process just described may be inefficient because the step size is decreased throughout the entire range of

integration. Greater efficiency can be achieved at the expense of more sophisticated programming by subdividing the range of integration in parts,

using an error criterion for each part and subdividing those parts where the estimated error is greater than its share of the total error. Several variants of such schemes have been published. However, no program of this type can be guaranteed to work in all cases, as it is not possible to obtain a reliable error estimate using only values of the integrand. It is always possible to construct examples for which it fails by the second method described in the discussion of Lanczos' error estimate. Another type of situation in which this procedure may fail is illustrated

by an attempt to evaluate s

dx

I

Y

= 0.40546511

(4.56)

by the Gauss-Kronrod program. The result with a tolerance of 10-6 was -0.93810800. As the two values agreed within the tolerance, the need for further subdivision is not indicated. However, adaptive integration procedures generally give satisfactory results if a moderate tolerance is specified. Exercises 1

Try out some of the possible alternatives to the Ralston composite rule. In particular test the exactness of (4.55) for higher powers.

2

Compare the values for the standard examples in Section 4.2 obtained by compounding with those obtained by twenty-point Gaussian quadrature.

4.8 EXTRAPOLATION METHODS

A common procedure among numerical analysts has been to evaluate an integral by using a compound rule with p panels and then with 2p panels and

take the difference as an estimate of the error in the second value. Essentially this is an extrapolation of the values for different values ofp to obtain an estimate for infinite p in order to eliminate truncation error. The assumption involved is that a plot against 1 /p behaves like curve A rather than of fig. 4.1. It is obvious that the procedure is not always valid, but since the quadrature

formulas presented have truncation errors that decrease faster than p-', it will be valid for sufficiently large p. In fact, Mikeladze proved that for odd-order Newton-Cotes formulas using 2 k + I nodes, ('2, - Ip)/(4k+' -1)

(4.8.1)

NUMERICAL EVALUATION OF INTEGRALS

149

(where IP is the value obtained with p panels and IZ P with 2p panels) is an error estimate of the second kind. Although he gave methods for estimating the required value of p, their value is limited because the (2k + 1)th derivative of the integrand is required. A subroutine using this procedure is shown in Appendix 2. Figure 4.1

I/P 4.8.1

Romberg Quadrature

A more systematic use of extrapolation is a procedure, suggested by Romberg in 1955, that has become the subject of considerable research. To clarify the presentation a specific example 2

dx = 0.693147180

(4.57)

x is used. The value of the integral evaluated by the trapezoid rule using

2"`-1 panels is denoted by Ik, so

li = #(1 +1) = $,

li =(')+3+3+) = is

(4.58a) (4.58b)

and so on. We then define

I2 = 41k+1 3

(4.59)

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150

It can be shown that this is equivalent to using the Simpson two-thirds rule with 2k -1 panels. The next step 2 Ik-1-_ 16Ik+1-Ik

2

3

(4.60)

15

is equivalent to Mikeladze's result and corresponds to a Newton-Cotes formula but further continuation by 4m Ik+11 -Ik-1 4m_1-I

Im k

(4.61)

does not correspond to Newton-Cotes formulas. It can be shown that the top row converges more rapidly than any of the columns. Thus in the present case the resulting array is as on the following page. 0.750000000 0.708333333 0.697023809 0.694121851 0.693391202

0.694444444

0.693174603

0.693253967

0.693174901 0.693147193

0.693154532 0.693147652

0.693147479 0.693147182

0.693147181

The array gives a remarkably good approximation. The differencing procedure

involves much less work than evaluation of the integral with more nodes. Since only the trapezoid rule is used, it is not necessary to store weights or nodes. It is possible to construct analogous schemes based on the midpoint rule using 3k-1 panels and 9m-Iim-1

Im k

-I

=

Im-1

k+1

9m-1

k

(4.62)

-I

instead of Eq. (4.61). Similarly, Dufton's rule can be used with 4k -1 panels and l6m-1Ik+11-Ik-1

m Ik-1°

16m-11

(4.63)

One drawback to these schemes is the rapidity with which the number of nodes increases, so the present tendency is to develop variants in which the number of nodes is increased less rapidly. The Romberg method can be considered an application of Richardson's deferred approach to the limit. This is a method of eliminating a leading

(4.8.2)

151

NUMERICAL EVALUATION OF INTEGRALS

error term of known order. If

x = x, (h) + Ah9 + 0(h9+(4.64a) x=x2\2)+A(2)e+O[(2)Q+11, (4.64b)

then

x=x,

+X1-x2+0(h9+I).

21-1

(4.65)

From this viewpoint it can be understood why the Romberg method does not work for certain classes of integrands for which the trapezoid rule has a much smaller error than indicated by the analysis of Section 4.6. 4.8.2 Aitken Extrapolation

An alternative extrapolation procedure that does not require knowing the order of the leading error term but uses three approximate values is the Aitken formula

(x2-xI)2

xl+x3-2x2

(4.66),

When applied to the first column of the previous problem it gives (excluding the first column, which is the same) the following array. 0.69381054 0.69312019 0.69314533

0.69314754

This array is inferior to the Romberg extrapolation. However, Overholt has suggested a modification in which more differences can be taken to obtain results comparable to the Romberg procedure. The question arises whether Eq. (4.66) can be used to extrapolate with respect to the order of the quadrature formula rather than with respect to

the number of panels. To investigate this, the integral in Eq. (4.57) was evaluated by Chebyshev's rule of order I to 7 and the Aitken extrapolation applied. The results show the greater accuracy of the Chebyshev procedure compared to the compound trapezoid rule, but the differencing procedure

does not improve the results when the extrapolation is with respect to the order of the integration formula.

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152

N 1

2

Integral

Extrapolation

6

0.69282060 0.69367187 0.69313736 0.69293840

7

0.69314677

0.69314677

4 5

Extrapolation

0.66666666 0.69230766

0.69281054 0.69312804 0.69313710 0.69314659

3

Extrapolation

0.69334352

0.69282044 0.69304017

0.69297517

Exact 0.69314118

Exercise

I

Write programs for extrapolation procedures based on the trapezoid rule (Romberg) the midpoint rule, and Dufton's rule. Compare their accuracy for some specific case.

4.9 THE BERNSTEIN QUADRATURE FORMULA

Although the quadrature formula described in this section is based on an

interpolation formula, it is not in the same category as those presented previously because no matter how many nodes are used, it is only exact for linear functions. While it is convergent for any continuous integrand, it is of little practical value and is presented because of its interesting theoretical basis.

S. N. Bernstein devised a family of interpolating polynomials that are of considerable interest to mathematicians. The Nth-degree Bernstein polynomial approximation to a function f is BN(x)

NC"fIa +

IN(b-a(x-(b-a)N)N

(4.67)

where NC,, is the binomial coefficient N!/[n!(N-n)!]. This differs fundamen-

tally from the Lagrange polynomial because for finite N except for x = a and x = b it is only by coincidence that BN is equal to fat the interpolating points. However, the factor

(x-a)" (b-x)N-" has a maximum at

x = a+(n/N) (b-a) that becomes sharper and sharper as N increases, so that in the limit BN does converge to f for continuous functions. Integration of (4.67) gives fb

f(x)dx ^- (N+1) a

n=0

f[a +(n/N)(b-a)].

(4.68)

This is an entire family of quadrature formulas with equal weights and simple nodes. It would be foolish to compound these formulas, as doing so

(4.10)

NUMERICAL EVALUATION OF INTEGRALS

153

would destroy their equal weighting. To compare its accuracy with the compound midpoint rule and the compound trapezoid rule, we apply all three to Jo x2 dx where it is possible to evaluate the summations analytically.

The results, (1/3)+(1/6N) for the Bernstein formula, (1/3)+(1/6N2) for the trapezoid rule, and (113)-(1112N 2) for the midpoint rule, show why it is not used in practice. Exercise 1

Use the Bernstein polynomials for the interpolation problem of Section 4.8.

4.10 MONTE CARLO METHODS

In this section the basic idea of the Monte Carlo method (also called the method of statistical trials) is presented. While the presentation is in connection with simple single integrals, the principal use of the method is for evaluating multiple integrals, where (as is explained in Section 4.19) the amount of work involved in the conventional methods increases very rapidly with the multiplicity. Figure 4.2

We describe two formulations that are essentially equivalent. In the first

formulation, consider the integral to be evaluated as the shaded area in

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154

Fig. 4.2 enclosed in a larger simple area. A large number of trials is then made. Each trial consists of (1) selecting two random numbers and converting them to the coordinate of a point in the simple area, and (2) determining whether the point is in the shaded area. The ratio of the shaded area to the total area is the limiting value that the ratio of the number of trials that fall in the shaded area to the total number of trials approaches. Unfortunately, the deviation only decreases as N_ 112 where N is the number of trials, so that it is necessary to use a very large number of trials to obtain a result comparable to that obtained by a conventional quadrature formula. The second approach, which is more closely related to conventional quadrature, involves taking a series of random numbers, converting each of them to a value of x, and using b

f(x) dx Ja

6- a

I f(xr). N r=1

(4.69)

This is closely similar to the Bernstein formula of the previous section except that a random set of points is used instead of equally spaced points. It can be seen that (4.69) will be exact if the integrand is a constant, whereas the Bernstein formula is also exact for linear functions. A number of devices have been developed for decreasing the number of trials needed in Monte Carlo procedures but apparently they decrease the coefficient of the error term without changing its N-'/2 dependence. Exercise 1

Show the equivalence of the two formulations of the Monte Carlo method by Morris's transformation (Chapter 1).

4.11

"BEST" QUADRATURE FORMULAS

There is considerable interest among theoreticians at present in quadrature

formulas with weights and nodes determined to minimize the remainder for a class of functions rather than by an exactness condition. As relatively few computational results have been reported as yet, our account is brief and confined to an analysis due to Wilf. A modification in which the nodes are prescribed is included because it simplifies the computations and may be useful in connection with Kronrod's work described in Section 4.4.

R(W, x) = E C (k+1)-I k=0

-I J=t

Wj(xj)k2

(4.70)

(4.11)

155

NUMERICAL EVALUATION OF INTEGRALS

is minimized. The weights and nodes obtained in this way will not be exact for any specific powers of x but should be the best overall choice. Differentiation with respect to W, gives aR

N

-2

(k+1)-1

k=0

aW;

- j=1 Wj(x;)k xk.

(4.71)

Setting this equal to 0 and interchanging the order of the two summations gives N

k

k=0 k+1

cc

- jY1

(4.72)

Wj kY- (x;xj)k.

The sums over k can be evaluated to give the set of N equations N

In

x;

Wj

=Y

1

(4.73)

j=1 I -x;xj

1 -x;

Similarly, differentiation with respect to x; gives another set of N equations N 1

1

x;(i-x;)

x;

In

1

1

-x;

= Y j=1

WjXiXj)2 xj

(4.74)

Wilf gives the solutions in Table 4.1. Examination of his results shows two interesting features: (1) the sum of the weights is not 1; (2) the nodes are not symmetrically disposed. The first feature means that these formulas will not evaluate the integral of a constant exactly, no matter how many panels are taken ! The second feature means that the solution is not unique and that an equally valid solution can be obtained by replacing xj by I - x j since x

z

R(W, x) _ > (k+l)-' k=0

W,(1-xj)k

(4.75)

j=1

is just as valid a starting point as Eq. (4.70). Table 4.1

N=3

N=2 x, = 0.48118 x2 = 0.95477

w1 = 0.83421

w2 = 0.15040

x1 = 0.99378 x2 = 0.89813 x3 = 0.37903

w , = 0.02002

w2 = w3 =

0.24302

0.72048

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156

4.11.1

Specified Nodes Variant

An interesting variant is to start with a specified set of nodes and find weights that satisfy Eq. (4.70) In this case (4.73) becomes a set of linear equations for the Wi's. By taking the Gaussian nodes and computing weights by Wilf's criterion, an alternative checking procedure to Kronrod's improved

formulas (Section 4.4) might be obtained, but we would expect that as N became large, the weights computed in this manner would approach the Gaussian weights. The results obtained by using single precision arithmetic N = 5 are given in Table 4.2. Surprisingly, even with symmetrically disposed nodes the Wilf weights are not symmetrical. If we evaluate jo x dx by using the Wilf weights and nodes, the value obtained is 0.4991417. The weights

can be symmetrized by taking the average value of W, and WN+, _j for both W; and WN+, _j. The values obtained in this way are listed in the fourth column. These values give 0.50016 for So' x dx, which is an improvement. Table 4.2 Nodes

Gaussian weight

0.04691007 0.23076534 0.50000000 0.76933465 0.95300992

0.11846344 0.23931433 0.28444444 0.23931433 0.11846344

Wilf weight

Symmetrized weight

0.25227057

0.19299515 0.05918453 0.49567287 0.05918453 0.19299515

-0.03894508 0.49567287 0.15731415 0.13371972

Sum = 1.00003223

Similarly, using the Gaussian nodes for N = 11 yields, after symmetrization, the values in Table 4.3. These give 0.4999998 for jo x dx. Apparently

the question of the convergence of the Wilf weights for large N to the Gaussian weights requires further investigation. Table 4.3

Nodes

Gaussian weight

0.01088567 0.05646870 0.13492399 0.24045198 0.36522842 0.50000000

0.02783428 0.06279018 0.09314510 0.11659688 0.13140227 0.13646254

Symmetrized weight 1.5316337

-5.3129814 9.5563703

-10.951548 10.153690

-8.9543297

(4.12)

157

NUMERICAL EVALUATION OF INTEGRALS

Exorcises 1

2

Develop a program for solving Eqs. (4.73) and (4.74) by using the method described in Section 4.3 and check the accuracy of Wilf's results. Compare the results given by the "best" rules using specified nodes with interpolatory rules for a number of cases.

4.12 RIEMANN AND RIEMANN-STIELTJES SUMS

Rall has pointed out that it is possible to compute lower and upper Darboux sums (defined in Chapter 1) on a computer. There are well-developed

techniques for using a computer to find maximum or minimum (or both) values of a function in a interval, as this is an important practical problem. It is therefore possible to program the computer to divide the range of integration in parts and to search for the greatest and least value of the integrand in each subinterval. The search is appreciably simplified if it can

be assumed that there is no more than one maximum and minimum in each interval and becomes trivial when the integrand is a monotone function. Although the process is slow compared to Gaussian quadrature, it has the advantage of giving rigorous bounds. Also, it is easy to make the spacing adaptive by setting a criterion on the difference between the upper and lower value for each subinterval. Decell and Lea have noted that the procedure can often be simplified by rewriting the integral as a Stieltjes integral. For example, Rall evaluated A12

1(x, A) = 0

In() - At cost) COs t + x2 sin2 I

A

(4.76a)

which can be rewritten as 1

I (x, A)

nl2

ln(1 - At cos t) d[arctan(x tan t)].

(4.76b)

Xo The transformed integral is easier to evaluate because In(1-At cos t) is monotone increasing in the range 0 to n/2.

Secrest has pointed out that using upper and lower Darboux sums provides a best result in the sense of the previous section for integrating a set of isolated experimental points when all that is known about the functional form is that the first derivative is bounded. Taking the average of the upper

and lower Darboux sums is equivalent to using the trapezoid rule for each subinterval. It would therefore be interesting to investigate the possibility of using the extrapolation techniques discussed in Section 4.8 on the Darboux sums to obtain more accurate bounds. Exercise 1

Verify the equivalence of (4.76 a) and (4.76b).

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158

4.13 INTEGRATION OF PERIODIC FUNCTIONS

There was a widespread belief among numerical analysts that the compound trapezoid rule was the best method for integrating a periodic function over a period. An investigation by P. J. Davis showed that (1) whenf(x) is

periodic and has a kth derivative that is of bounded variation, then the

error of the compound trapezoid rule is of order h"' rather than

h2.

This is an example of the type of integral mentioned in Section 4.8 for which the Romberg procedure fails. (2) For analytic functions that are entire or that are regular in a sufficiently large region enclosing the interval of integration, the trapezoid rule is better than the Gaussian formula using the same number

of points. (3) When there are isolated singularities close to the interval of integration, the Gaussian formula is superior to the compound trapezoid rule. A related result given by Krylov is that when f has a period 2n, f02n

f(x) dx

kk1f[a+(k-1) N]

N

is exact for trigonometric polynomials of degree N-1. It is easily seen that

when a = n/N, this becomes the compound midpoint rule and that the compound trapezoid rule is equivalent to averaging the expressions for

a = 0 and a=2n/N. A somewhat different approach is to look for an analog of the Gaussian quadrature rule based on Fourier series instead of polynomials, that is, a set of W;'s and x;'s for which +1

f(x) dx

-I

W; f(x;)

(4.77)

j=1

is exact for f(x) = sin inx and cos inx rather than for x`. The range -1 to + 1 was chosen because it made it easier to apply symmetry considerations. It can be seen that because of symmetry (4.77) will automatically hold for sines if W; = WN-j+1

(4.78a)

and

xj

xN-j+l

(4.78b)

When N is odd, x(N+1)12 = 0. The problem then reduces to solving (N-1)12

W(N+1)/2+2 Y W;=2 J=1

(4.79a)

(4.12)

159

NUMERICAL EVALUATION OF INTEGRALS

and N

W(N+1)/2 + 2Y_ W;cos inxi = 0

(4.79b)

J=1

where i = 1, 2, ..., N-1 for odd N's. For even N's the equations are N

I W;=1

(4.80a)

J=1

and N/2

wjcos iltxj= O

(4.80b)

J=1

for i = 1, 2, ..., N-1. The solution in both cases is (4.81a)

xj _

1+N-2j

(4.81b)

N

which corresponds to the compound midpoint rule with N divisions. Davis's paper involves an analogous set of equations, so it appears that the trapezoid rule corresponds to a Lobatto quadrature formula. The three general results given above for the trapezoid rule carry over for the midpoint

rule. The proof for the trapezoid rule is based on the Euler-Maclaurin summation formula, which is presented in the next chapter. By using a related summation formula also presented there the proof is easily extended

to the midpoint rule. Table 4.4 compares the error involved in evaluation

(1-r2)dx 2 o 1+r-2rcosx

-2n

by N-point Gaussian quadrature, the midpoint rule, and the trapezoid rule for r = 0.1, 0.2, 0.2577, and 0.8. The value r = 0.2577 represents the critical location of the singularity. ' As predicted by Davis for smaller values of r, the Gaussian formulas are inferior but at r = 0.8 they are markedly superior. A peculiar feature of the results is the superiority of the midpoint rule with an odd number of points, so that the three-point result is comparable to the six-point result.

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160

Table 4.4 Comparative errors in evaluating Zn

f

o

(1-r2)dx 1+r2-2rcosx

by N-point Gaussian quadrature, the midpoint rule, and the trapezoid rule N

Gauss

Midpoint Trapezoid

Gauss

4 5

6 7 8

9 10 16

32

-0.029

0.000013 0.0013 0.000000

0.0056 0.0076

-0.00016 -0.00002 -0.000005

-0.000012 0.000000 0.000000 0.000000

0.000000 0.000000

0.013 0.0013 0.0001 0.00001 0.000001 0.000000 0.000000 0.000000 0.000000 0.000000

0.000000 0.000000

-0.13 -0.0050 -0.0064 0.0085

-0.0025 0.000067 0.000007 0.000004 0.000000 0.000000

r=0.2577 3

-0.19

4

-0.031

5

6 7 8

9 10 16

32

0.0082 0.0034

0.0037 0.055 0.000017

-0.0037

-0.000026 -0.00023 -0.000034 0.000010 0.000000 0.000000

0.000000 0.00024 0.000000 0.000016 0.000000 0.000000

Trapezoid

r = 0.2

r = 0.1 3

Midpoint

0.000000 0.000032 0.000000

0.10 0.020 0.0040 0.00080 0.00016 0.000032 0.000006

-0.000001

0.000001

0.000000 0.000000

0.000000

0.00080 0.020 0.000001

-0.00080

0.000000

r = 0.8 0.22 0.056 0.014 0.0037 0.00095 0.00024 0.000060 0.000020 0.000000 0.000000

3.0 1.52 0.42

-0.15 -0.29 -0.24 -0.14 -0.060 0.0023 0.00001

2.61 3.65 1.21

13.0 8.7 6.1

-4.46

4.5

0.53 1.80

0.22

-1.51 0.34 0.0099

3.33 2.5 1.94 1.5

0.36 0.010

Exercises 1

2

Examine the use of the Bernstein rule for periodic functions. In particular, calculate values corresponding to Table 4.4.

Show that no N-point formula can be exact for a trigonometric polynomial of degree N (Krylov).

4.14 IMPROPER INTEGRALS

In this section four methods for handling an integrable singularity in the

range of integration are described. Integrals over an infinite range are treated in Section 4.17. The next section on product integration also describes a useful approach to both types of improper integrals. The methods

(4.14.1)

161

NUMERICAL EVALUATION OF INTEGRALS

covered in this section are (1) ignoring the singularity; (2) subtraction of a small region from the range of integration; (3) subtraction of the singularity from the integrand; (4) elimination of the singularity by a change in variable. 4.14.1

Ignoring the Singularity

Many computers return 0 when instructed to divide by 0. The singularity is

thus automatically ignored when a conventional integration rule is used. Davis and Rabinowitz made a study of the results of this procedure. They were able to prove that when a sequence of quadrature formulas with rational nodes were applied to an integrand that was monotone around an integrable singularity, the process converged. However, the convergence could be slow. For example, the compound Simpson two-thirds rule with 32 panels gave 1.8427 for j o x-' / Z dx. Increasing the number of panels to 1024 brought the value up to 1.9721. The accuracy was appreciably decreased

when the singularity was at an irrational point (an interesting phenomenon

related to the approximation of an irrational number by a sequence of rationals) and the procedure did not converge for oscillatory integrals such as Jo sin (1/x) dx/x. Ignoring the singularity is probably the simplest method for evaluating the finite part of an integral. Monacella has shown that when dealing with an integral

f

b=

f(x) dx

_

x

a2

rewriting it as z

f ±a=

f

+ a

f

or f bbl + ,1 -a

and evaluating the first integral by a formula with an even number of symmetrically disposed nodes gives satisfactory results.

However, Davis and Rabinowitz later warn in their book that "on the whole, ignoring the singularity is a tricky business and should be avoided whenever possible" (pp. 76-77). It would appear more logical to use methods which take the specific nature of the singularity into account. 4.14.2

Singularity at an End point

The basic definition of the integral of a function with a singularity at the limit of integration is b

b

f

a

f(x) dx = lim f e-O

a+i

f(x) dx.

(4.82)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

162

This can be applied by evaluating the integral for a sequence of a and extrapolating. Davis and Rabinowitz give the example

I

!=

f

1

(4.83)

x1/2dxx1/3 = 0.84111692.

o

For a sequence of lower limits e = 2-"

N

8

16

32

40

1

0.81280497

0.84029678

0.84111612

0.84111663

which gives an accurate result. However, it was necessary to use an adaptive procedure that went up to several hundred nodes to obtain the values in the table. 4.14.3 Subtraction of the Singularity

A very powerful technique originated by Kantorovich subtracts the singularity as a term that can be integrated analytically, leaving a remainder that is easier to evaluate numerically. Thus equation (4.83) can be rewritten as 1

dx

= f I (x-1/3 - x-116)dx +

Jo XI/2+Xf/3

O

I

dx

o I+XIJ6

(4.84)

The first integral is elementary and gives 0.3, while the second integral has no singularity. Evaluating the left side by seven-point Chebyshev quadrature

gives 0.801835, which is off by 5%. If the integral on the right side is evaluated by the simple midpoint rule, the value of the entire integral is 0.828849, which is only about a 1% error. Going to seven points gives 0.840171.

The Kantorovich method can also be used to eliminate singularities close to the range of integration or small positive fractional powers of x. Thus

'

=

e" dx

o (x+0.01)112

1

0

1/2

dx

I

0 (x+0.01)112

('t

X dx = I 1 +x

+

1

X 1/ 2 dx o

I

ex - I

0 (x+0.01)I12

x 3/2

o 1+x

dx.

dx; (4.85a)

(4.85b)

(4.14.4)

163

NUMERICAL EVALUATION OF INTEGRALS

4.14.4 Change of Variable

Finally, the singularity can usually be eliminated by a simple change of variable. Setting x = u6 gives dx

'

u3 du

6

0 X3/6+X2/6

(4.86)

1 +u

0

integration of the right-hand side by the seven-point Chebyshev rule gives 0.84111919.

This procedure can be made systematic by noting that if the singularity is at an endpoint, the integral can always be transformed so that the singular-

ity is at 0 and the range of integration is 0 to 1. If the singularity is at an internal point, it can be transformed into two such integrals and then to one integral for the sum of the two integrands. By the substitution X = jl/(1+k)

we get I

0

xk f(x)dx =

I ,f(11/(l+k))

fo

dr

l+k

(4.87)

hus Thus I

1

ax

j o x2/6 (1 + x

116

= 1.5

dtt

o (1+1 1/4)

)

(4.88)

Integrating the right side by the seven-point Chebyshev formula gives 0.839372, which is an improvement over the original form but slightly inferior to subtracting the singularity. Equation (4.86) is obtained by the further substitution u = t1/4 Exercises 1

Explain why the singularity is ignored by taking the value of the integrand as 0 rather than as a large number.

2

Evaluate

('I

x-'/2 In by several methods.

e

x

=6

INTEGRATION FOR ENGINEERS AND SCIENTISTS

164

4.15 PRODUCT INTEGRATION

In this section product integration rules of the form I

bw(x)f(x)dx

j=I

a

Wjf(xj)

(4.89)

are discussed. Splitting off part of the integrand as a weight function often gives a considerable increase in accuracy, especially for small N. It also makes it possible to handle singularities and certain types of integrals over an infinite range. Since only f(x) has to be evaluated there is an appreciable saving when a number of integrals with the same w(x) are evaluated. Two methods of obtaining weights and nodes are presented. The con-

ventional approach obtains interpolatory formulas exact when f(x) is a polynomial of a degree related to N by methods similar to those described in

Sections 4.2, 4.3, and 4.4. The other method is a generalization of the transformation procedure in the previous section. While the formulas obtained by this procedure are generally not as good as the interpolatory rules, they represent a significant improvement over the conventional formulas without weight functions and are easy to obtain. 4.16.1

Change of Variable Method

If w(x) does not change sign between a and b, introduction of the variable

w(x)dx

t(x) = ,

(4.90)

fa w(x) dx

transforms (4.89) into

Y_ Wjf[x(tj)]

j=1

(4.91)

Now if this is compared with a conventional formula

f(t)dt

Y Aj f(tj)

j=I

(4.92)

it is seen that a set of weights and nodes for (4.89) is obtained by taking

(4.16.2)

165

NUMERICAL EVALUATION OF INTEGRALS 6

Wi = Ai

w(x)dx

(4.93a)

a

and xi a solution of

xi = x(ti).

(4.93b)

When (4.90) can be inverted explicitly, the method is very simple; even when the nodes must be found by numerical solution of

t(x)=1;,

(4.94)

it is easier than the conventional method. Examination of (4.93 a, b) shows that a transformed Chebyshev formula retains its equal weight property, but

equal spacing rules lose this property. We refer to formulas obtained by this transformation procedure as quasi-Chebyshev, quasi-Gaussian, and so forth. 4.15.2

Interpolatory Methods

The generalization of the methods for obtaining interpolatory rules to product integration is straightforward. For specified nodes either the left side of (4.5) is replaced by Jo x` w(x) dx or Eq (4.9) is rewritten as

wi =

fb

w(v) F1 v-x`

Ja

I#i xi-x,

A.

(4.95)

The existence of interpolatory nodes of course depends on the existence of Ja xI w(x) dx. If these moment integrals exist, it does not matter whether w(x) has singularities or the range is infinite.

Similarly, for Chebyshev equal weight rules the right side of (4.20) becomes N-Ja xkw(x) dx. The problem of the existence of real nodes remains,

and relatively little work has been donc with these formulas. Yokota calculated nodes for 5I

xf(x) dx ti -I

41Z

jf

f

N i=I

(4.96a) 1

and

i'!, x2f(x)dx ~ 3 N i

=Xj f ( ) + f(-Lr 1

1

I

(4.96b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

166

that are useful for calculating moments and moments of inertia. Yokota's values are given in Table 4.5. As these formulas are intended for use with data taken from blueprints, the four-figure accuracy is adequate. Table 4.5 Yokota Nodes for Equal Weight Quadrature

Nodes for

'

xydx

1

I

.1

()_(_)

(21)2 Y_

N

i=

N

XI

Xs

4

0.2701

0.6034

6

0.2980

0.5683

0.7703

8

0.0204 0.0677

0.5821 0.1293

0.7494 0.7077

10

Nodes for J

Na

1

x2 y dx =

X3

23

2

0.7746

4

0.5815

6

0.5030

X5

0.8544 0.8330

0 .8970

f(-

N12

3N;=I

x1

X4

I

X2

9284 8133

X3

0.9410

° For N> 6, complex x,'s are encountered.

Recently, Ullman proved that there are real Chebyshev nodes for the weight function 1+W(X)

=(1 -x2)"2(1+4a2+4ax) '

-j < a

(4.97)

over the range - I to + 1, but did not give any numerical results. 4.16.3 Orthogonal Polynomial Derivation

Most of the work on product integration has been in connection with Gaussian formulas exact when f(x) is a polynomial of degree 2N- 1. Although the weights and nodes satisfy a system of nonlinear equations analogous to (4.28a, b) the usual method of calculation is based on the derivation in terms of orthogonal polynomials.

(4.16.3)

167

NUMERICAL EVALUATION OF INTEGRALS

A set of polynomials orthogonal with respect to a weight function w(x) over a range a to b satisfies the integral condition ('b

0,

w(x)

n :A in.

(4.98)

a

If we let the subscript indicate the degree of the polynomial, it can be shown that any integral power of x can be expressed as a sum of the polynomial up to and including pk. From this it follows that ('b

xkp (x)w(x)dx = 0,

k < n.

(4.99)

a

It can be shown that when w(x) does not change sign, has n distinct roots, all of which are between a and b. The situation when.w(x) changes sign, which has been treated in a dissertation by Struble, is beyond the scope of this book. The derivation of the Gaussian rule is based on the possibility of writing an arbitrary polynomial of degree 2 N-1 as 12N-1(x) = PN(x)QN-1(x) + RN-1(x)

(4.100)

where QN-1(x) and RN-1(x) are polynomials of degree N-1. Because of (4.99),

w(x)f2N_1(x)dx = ibw(x)RN_1(x)dx.

(4.101)

a

This means that there is always a polynomial of degree N-1 that has the same product integral as a polynomial of degree 2N- 1. This polynomial can be determined from its values at any N points. If these points are taken as the zeros of PN (x), by (4.100) the values of RN- 1(x) at these points are those of f2 N _ 1(x). By the Lagrange interpolation formula in a form slightly

different from, but equivalent to, that used in Section 4.2,

RN-1(x) =

N f(xj)PN(x) j = I PN(xj)(x-xj)

(4.102)

Therefore, the nodes of the Gaussian product integration rule are the zeros of the orthogonal polynomial and the weights are given by

W=

/1

PN(Xj)

`b w(x)PN(x) dx. Ja (X - xj)

(4.103)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

168

Table 4.6 lists the more important orthogonal that have been used for quadrature formulas. A number of the weight functions have singularities at the end points and the last three are for an infinite range of integration. Table 4.6 Table of Orthogonal Polynomialsa' Range Weight function 0 to I 0 to I

I (x-x2)- 1/2

0to 1

(x-x2)'/2

0 to 1

(1

-I to t v tow 0 to 00

-00 to 00

-x)°'9xa-' -x2)a- 1/2

(I

exp(-x) exp(-x) exp(-x2)

xa

Limitation

-P-g > - I g> 0

a > --

-

a > -1

Name Shifted Legendre Shifted Chebyshev, first kind Shifted Chebyshev, second kind Jacobi

Ultraspherical or Gegenbauer Laguerre Generalized Laguerre Hermite

° The standard form of the shifted polynomials are for a range -I to 1.

In practice methods based on the orthogonal polynomial approach are used, as it is easier to find the roots of a polynomial than to solve a set of nonlinear equations. A further difficulty with this direct approach is that in some cases the system of nonlinear equations is ill conditioned. For example,

using the method described in Section 4.3 on the equations for w(x) = 1 gave the following results for N = 5. xi

W{

0.04694096 0.23086893 0.50011991

0.11853394 0.23936843

0.76930371

0.23925299

0.95310542

0.11842510

0.28441955

The calculations were carried out in double precision and the results rounded to eight places. It can be seen that symmetry is violated in the fourth place, but if these numbers are used, the results will be correct to eight places ! Therefore, the proper role of the computer in this problem is for generating the system of orthogonal polynomials and solving them. A recent paper by A.B. Huang and D.P. Giddens (J. Math. Phys. 47 (1968) 213-218 calculates weights and nodes for the range 0- co for the weight function e'=2

(4.16.3)

NUMERICAL EVALUATION OF INTEGRALS

169

4.15.4 Specific Example

To illustrate the application of product integration, we use dx 0

_ 't = 1.50796

x112(1 +x)

(4.104)

2

as an example. Direct integration by a six-point Chebyshev rule gives 1.30. For product integration w(x) = x-' 12 . Three simple formulas with specified nodes, the one-point Gaussian and the one-point quasi-Gaussian, are

x-1/2f(x)dx

J

2f(j) 3

(4.105a)

f(0) + f(1)

(4.105b)

3

-L-2 f(0)

(4.1O5c)

+ I f(j) + 1?s f(1)

2f(-})

(4.105d)

= 2f(j),

(4.105e)

respectively. We note that the usual symmetry conditions do not apply to product integration unless w(x) is symmetrical with respect to midpoint. Applying these to (4.104) gives, for the respective parts of (4.105) a

b

1.3333

1.667

c

d

e

1.5333

1.50

1.60

all of which are better than the simple six-point Chebyshev. The superiority of the quasi-Gaussian to the Gaussian is apparently accidental. For higher N we have the values in Table 4.7, so the Gaussian is about as good as the quasi-Gaussian with an additional node. It should be noted that the conventional error estimate can be generalized to product integration, but the Lanczos method does not seem applicable. Table 4.7

N

Gaussian

2

1.568627 1.570730 1.570794

3

4

quasi-Gaussian 1.573781 1.570531 1.570804

Exercise 1

Compare weighted Gaussian quadrature with Eq. (4.93 a, b) for evaluating Ju x3 e" dx. (Weights may be found in Abramowitz and Stegun.)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

170

4.16 TRIGONOMETRIC WEIGHT FUNCTIONS

There is a class of integrals with a weight function that changes sign which is of such importance that it is essential to discuss methods for their numerical evaluation. The Fourier integrals of the form px dx

f(x) J'a

are difficult to evaluate by conventional methods when p(6-a)> I because the range of integration contains many cycles of the trigonometric function. A conventional compound interpolatory formula requires a number of nodes in each cycle and roundoff error can become serious when f(x) is slowly varying. Filon was the first to develop special formulas for evaluating such integrals

by generalizing the compound Simpson two-thirds rule. In principle, his method evaluates the Fourier integral for any p with an accuracy com-

parable to that given by the compound Simpson rule for fa f(x)dx. A number of improvements of Filon's method have been developed that represent analogous generalization of more precise quadrature rules. We go in the opposite direction, and generalize the compound midpoint rule so as to show the principle involved with a minimum of mathematical complication. By splitting the range of integration and changing the variable in each part, we get b

N

w(x)f(x)dx = I fs- f(mj+t)w(mj+t)dt j=

a

(4.106)

s

1

where S = (b-a)/2N and mj = a+(2j-I)S. Approximating f(mj+t) by f(m j) in each subinterval gives

w(x)f(x)dx

E Wj f(mj)

j=1

(4.107a)

where +s

Wj =

-s

u>(mj+t)dt.

(4.107b)

When w(x) = sin px, the weights are Wj = 2 p-' sin pSsin pmj,

(4.108a)

(4.16)

NUMERICAL EVALUATION OF INTEGRALS

171

and for w(x) = cos px (4.108b)

Wj = 2 Sp- ` sin pS cos pm;.

These expressions, which are much simpler than Filon's, show the basic

principle of weights dependent on a, b, and p, which can give accurate results using only one node in many cycles. The simplicity of the weights also clarifies a phenomenon that has caused some analysts to oppose the use of this method with less than one node per cycle. It can be seen that if p (b - a)/2 N = kn, the weights given by (4.108a) or (4.108b) vanish, but this is easily avoided.

To illustrate the power of the method, the integral ex cos px d x =

I

1 - e (cos p + p sin p)

(4.109)

1 + pz

0

was evaluated using 10, 20, and 40 points. The results for the percentage of error were as shown in Table 4.8; they show a loss of accuracy asp increases. The higher-order formulas (such as Filon's) do much better, and in the next chapter a modification of the midpoint rule using a correction term involving f is presented that also increases the accuracy markedly. Table 4.8

N=10

p

-0.0133

1

1.516

10 100

1000 10000

N=20

N=40

-0.0033

0.0008 0.0930

0.3735

-2.303

-4.147

7.584

7.691

-39.25

28.17

-0.5777 7.718 6.984

A. M. 0. Smith has developed generalized three- and five- point rules for evaluating integrals of the form n

f f(x)csinos [pg (x)] dx a

where f and g are slowly varying. The principle will be shown by developing

a compound one-point rule for z

Jiiz

x3 sin px4 dx =

cos(p/16) - cos 16 p

4p

(4.110)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

172

According to Eq. (4.107a, b) N

2 I

x3sinpx4dx

(4.111a)

Wi J=1

1/2

where s

sin p(mj + 1)4 dt.

Wj =

(4.111b)

s

The slow varia tion of g is used by setting +s

s

sinp(m; +4m3 t)dt.

sinp(m;+()4 dt

4.112)

-s

-s This gives

12 x3 sin px4 dx u 1/2

2

N

p 1=i

sin pmj3 S sin pm j

Pm;

Using this gave the results in Table 4.9, which show that the method gives reasonably good results. Table 4.9 p 1

10

100

1000

4.16.1

Exact

N = 40

N = 160

N = 640

0.488927

0.489790

0.488984

0.488940

0.0446648 0.00399453

0.04436636

0.0446417

0.0049024

0.0041104

0.0446569 0.0040258

0.00042816

0.00047649

0.000484205

-0.00007647

Euler Summation

An alternate procedure for oscillatory integrands, which requires more

work but is capable of better accuracy, involves splitting the range of integration into parts, using the zeros of the integrand as points of separation;

evaluating the individual integrals; and then adding a sequence of alternating terms of almost equal magnitude without losing significant figures by

roundoff. There are a number of transformations for accomplishing this, the oldest and best known being due to Euler. The derivation is presented in

Chapter 6. For the present we note that given a sequence of terms a,, a2, ..., aN, a second sequence b, , b2, ..., bN_, can be obtained by bk =

(4.16.1)

173

NUMERICAL EVALUATION OF INTEGRALS

= ak+ak+1 and a third sequence by ck = bk+bk+1, and so on. The sum of la1+Jb1+act+...+ M,

I

will be the same as the sum of the original sequence but the convergence will often be greatly improved. For example, the value of 1/(1 + 0.9) is 0.5263. The sum of the first two terms is 0.1; after applying the Euler transformation, the sum of the first two terms is 0. 525.

For evaluating the integrals over a half cycle, Lobatto formulas are convenient because the values at the end points are zero. Alternatively, Price has given simple rules of the form (k+ i)n

N

sinyf(y)dy = (-1)k Y HJf(ka+x1),

(4.113)

J=1

k,

which eliminates the calculation of the sine terms. Longman has shown the accuracy of this class of methods by evaluating (10,000ir2 - x2)112 sinxdx = 298.435716,

(4.114)

Using 16-point Gaussian quadrature and a modification of the Euler transformation gave 298. 43558. The generalized midpoint rule, on the other hand, can only give five significant figures before roundoff error sets in, (10,000ir2_x2)112 cannot be evaluated accurately by because the function the midpoint rule unless the step size is very small. Exercises 1

Evaluate °

f

1

2 3

x"

cos

sin

sin °° _ cos it x dx, fl x : sin 2 x dx and fo x In x dx cos

by integrating between successive zeros and applying the Euler transformation (SIAM Rev. Problem 68-8). Investigate the use of Aitken extrapolation as a device for improving the convergence of series. Write a program for determining Fourier coefficients using a "Filon-type" quadrature rule.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

174

4.17 INTEGRALS OVER AN INFINITE RANGE

In the preceding section formulas were included for evaluating integrals over an infinite range when the integrand had a proper weight function such as fo e-sf(x)dx. The use of these formulas is limited and it is not feasible to force integrands into the proper from. For example, rewriting

= f '0 e-x ex dx 1+x2 fo 1+x2 - Jo dx

°°

(4.115)

will not give satisfactory results. The product integration rules are based on

the assumption that f(x) can be approximated by a polynomial, and this does not hold for e"(1 +x2)-'. Replacing the upper limit by a large finite value u has drawbacks. In order to make the neglected part 5.- dx/(1 +x2) =

;1/u, which will be called the termination error, small, it is necessary to make u very large. This makes it difficult to use a simple interpolatory formula over the entire range, and compounding involves either a large number of nodes or variable spacing. 4.17.1

Transformation to a Finite Range

It has frequently been noted that a suitable change in variable can transform an integral over an infinite range into an integral over a finite range, but no systematic study has been made of the practical utility of this procedure. Our work will utilize the equivalent procedure of transforming the

quadrature formula rather than the integral to obtain rules of the form

oj=I

f

W

f(x) dx

^-N

Wjf(xj).

(4.116)

However, such rules are subject to a basic indeterminacy. Since

fo" f(x) dx = f ; f(Sx) dx, o

(4.117)

0

the weights and nodes can be multiplied by an arbitrary scaly factor S. When N is large enough, the value obtained will be insensitive t'

of S within reasonable limits. However, if S is too small,

,

'

-hoice Modes

will be small and the contribution for large x will not be properly accounts ' for; and if S is too large, the contribution from small values of x will not I evaluated correctly.

(4.17.1)

175

NUMERICAL EVALUATION OF INTEGRALS

It is easy to generate suitable transformations. Any monotone functions t (x) satisfying either

t(0) = I and t (co) = 0

(4.118a)

or

and

t(0) = 0

(4.118b)

t(oo) = I

will transform the integral, giving x

I

j f(x)dx =

f[x(t)]

0

dx (4.119)

dt.

dt

o

Some simple examples are given in Table 4.10.

Applying such a transformation to an open quadrature rule

f

I o

Y Ajf(ij)

f(t)dt

(4.120)

j=I

gives weights and nodes for (4.116)

wj=Aj

(4.121a)

and

xj = x(tj).

(4.121b)

Table 4.10

t(x)

dx

x(t)

Transformed integral

di

ex

I

-in t

f(-1n t) t

X

t

1+x

1-t

tank x

} 1n

o

(1-t) I I +t

I

1-t

1-r

dt

z

2

I f(1-tt)

o

dt 1-t

$'f(+ln-±.t) dt o

I-t 1-r

2

INTEGRATION FOR ENGINEERS AND SCIENTISTS

176

As pointed out earlier, these weights and nodes can be multiplied by a scaling factor S.

We have found the following transformation, for which the scaling factor has a simple interpretation, convenient to use. Splitting the range of integration into two parts 0 to S and S to oo and introducing t = x/S into the first part and t = Six into the second part gives 1(x) dx = S J o

Jo

[f(st) + t 2f

\S/Jdt.

(4.122)

This transforms an open N-point rule for the 0 to 1 range into a 2N-point rule for the infinite range with W; = SAi and

Wi =

SA

'-"

t 1-N

and

x; = Sty ,

xi =S-,

1 < j < N, N + 1 < j < 2N.

(4.123a)

(4.123b)

ti-N

This procedure is equivalent to approximating the integrand by a series in

ascending powers of x for x < S and by a series in inverse powers for x > S. None of the transformations appears to have any significant advantage for a fixed number of nodes, but (4.123 a, b) gives twice as many nodes in the infinite range for a specific conventional formula. The Appendix includes

a program in which the N = 20 Gaussian and Kronrod formulas are transformed to evaluate integrals over an infinite range. It is easy to change the program to use any of the other transformations. Experience with the program indicates the following. 1.

It will evaluate simple integrals such as j dxl(l +x2), Jo a-" dx, or J10 exp(-x2)/(1 +x2)dx to seven-figure accuracy for a considerable range of S. Surprisingly, excellent accuracy is also obtained for strongly damped oscillatory integrands such as Jo a-T sin dx or Jo exp(-x2) cos x dx.

2.

Integrals with weakly damped oscillations such as Jo (sin x/x)2 dx and integrals with singularities such as Jo X_ 1/2 e-' dx or Jo dx/[x'12(1 +x)] are evaluated to three or four significant figures. However, a suitable transformation can remedy this. For example, by introducing u2 = x, we obtain ox0'

I /2

exp(- x) dx = 2

exp(- u2) du = 1.772454. fof'0

(4.124)

(4.17.2)

177

NUMERICAL EVALUATION OF INTEGRALS

The program gave S

fox`'/2e_x dx 2 f o exp(-x2)dx

1.0

3.16

10

1.755811

1.742855 1.772454

1.719805

0.316

0.1

1.763095 1.772454

1.7671951 1.772437

1.772454

Underllow

3. The rules will not work for conditionally convergent integrals such as sin x

-- dx

°°

or

x

o

cos x2 dx.

o

The values obtained fluctuate wildly as S is varied. 4.17.2 Goodwin-Moran Method

Another approach to the evaluation of integrals over an infinite range begins by considering the application of the trapezoid rule to the range - 00 to + 00. The simple rule + N

+ co

Jf(x)dx

h

0o

j=-N

(4.125)

gives surprisingly accurate results in some cases involving integrands such as exp(-x2). The governing factor is the rate at which the derivatives vanish as

lxi -+ oo. Integrands such as (1 +x2)-' do not give this high degree of accuracy. Since +00

f

+00

f(x) dx = f-00 f(x+a)dx,

(4.126)

- 00

(4.125) can be rewritten as j=+N

+CO

f(x)dx = h E f(jh+6)

(4.127)

j=-N

00

to center the summation and avoid adding negligible terms for either large

positive or negative arguments. P...k. P. Moran has suggested taking advantage of the accuracy of this simple procedure by transforming integrals over other ranges to integrals over - oo to + co. Thus by taking x = e', we obtain +00

f(x)dx = fo

f(e`)e`dt. 00

(4.128)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

178

Applying (4.127) to the right side gives

f(x) dx

Sin r [f(S) + Y tJ f(Sr') + r-' f(Sr-')].

(4.129)

i=t

The nodes in this formula form a geometrical progression, which seems more logical than uniformly spaced nodes for an infinite integral. This idea was utilized by H. V. Norden in connection with the numerical inversion of Laplace transforms. A simple alternative way of obtaining weights for such a formula is to take nodes in a geometrical progression for the range 0 to I

and determine weights by either the interpolatory method described in Section 4.2 or by the method described in section 4.11 and then obtain the nodes and weights for x > I by Eq. (4.123 a, b). Exercises

I

Compare the methods based on transformation with the Goodwin-Moran procedure for some known integral over an infinite range.

2

Devise three additional transformations to convert an infinite range to a finite range.

3

Generalize the Goodwin-Moran expression to an integral with a finite lower limit. 4.18 INDEFINITE INTEGRALS

Since an integral with a variable upper limit can be transformed to the standard range 0 to I by

f

x 0

f(x)dx = xJ I f(xt)dt,

(4.130)

o

in principle the methods described previously can handle such cases. In practice, however it is usually desired to evaluate (4.130) for a number of values of x, so proper organization of the calculation can save considerable time.

We start by considering the problem of an integrand tabulated at equally spaced values of the independent variable and restrict ourselves to rules using only these values. The simplest procedure is repeated use of the trapezoid rule but obviously this will not be accurate unless the step size is small and the error involved will tend to accumulate. Furthermore, since the

values of the integrand are available, a higher-order formula involves relatively little work. The conditions set above require equally spaced integration rules with some nodes outside the range of integration to get the

NUMERICAL EVALUATION OF INTEGRALS

(4.18.1)

179

process started. The weights for such rules, which can be calculated by the methods of Section 4.2, have been tabulated by Mikeladze. The simplest procedure is to start by (n+ 1)h

f(x)dx ^-_ nh

h

{Sf(nh) + 8f [(n+l)h] -f[(n+2)h]}

(4.131a)

12

and obtain the last value by

J(n-1)h(x)dx . 12[Sf(nh)+8f[(n-1)h]-f[(n-2)h]. (4.131b) n This is more effective than starting by (4.131 a) and then using Simpson's rule to integrate two steps. The next higher-order procedure would begin with

f(x)dx = 24[9f(O) + 19f(h) - 5f(2h) +f(3h)],

(4.132a)

I

do the intermediate steps with (n+ 1)h

f(x)dx

J

124h{f(nh)

-

+f[(n+1)h]}

4{f[(n-1)h] +f[(n+2)h]},

(4.132b)

and obtain the final value with 1)hf(x)dx t

24{9f(nh)+19f[(n-1)h]-5f[(n-2)h]+f[(n-3)h]}.

,1(

(4.132c) 4.18.1

Stability of Indefinite Integration

A class of formulas can be derived by using available values of the integral. For example, combining Simpson's two-thirds rule and Eq. (4.131 a) gives (n + 3)h

f

f(x)dx - 5g[(n+1)h] - 4g[(n+2)h]

eh

+ 2hf[(n+1)h] + 4hf[(n+2)h]

(4.133a)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

180

where kk

f(x)dx.

g(kh) =

(4.133b)

0

This formula is very unstable, as is shown later by a numerical example. The theory of the instability has been studied extensively in connection with numerical methods for solving differential equations. If we consider only the g terms in (4.133 a), it is equivalent to a difference equation

g(n+2)+4g(n+1)-5g(n)=0,

(4.134a)

which has solutions of the form c.?" where A is a solution of the associated characteristic equation

,2+4k-5=0.

(4.134b)

The roots are a = + I and A = - 5. The second root causes the instability because its modulus is greater than 1, so that it increases rapidly with n. A root of modulus greater than 1 or a multiple root of modulus I corresponds to a strong instability. Krylov has shown how to construct stable expressions of high precision but they involve nodes at irrational points. As noted in the introduction, for computer calculations it is unnecessary to use only rational nodes. Although it is necessary to use a unit spacing to re-use computed values of the integrand, formulas of the form ('(n+1)k

f(x)dx - h

JIf

nk

i=+M N

i=-M J=I

W,;f[(n+i+t;)h]

(4.135)

can be developed. The simplest example of such a formula with N = 1 and

M = 1 is (n + 1)k

f(x)dx =

24{f[(n-})h] + 22f[(n+1)h] +f[(n+2)h]},

(4.136)

Ink

which is exact for cubics, as there are four parameters, the three weights, and t1 = 1. Krylov gives a number of such formulas. 4.18.1

Example

To illustrate the methods developed above, fX

es dx = ex - 1

(4.137)

0

was computed by a number of these methods from 0 to I in steps of 0.1 and the results are shown in Table 4.11.

(4.19.1)

NUMERICAL EVALUATION OF INTEGRALS

181

Table 4.11

x

es-1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.221403 0.349859 0.491825 0.648721 0.822119 1.01375 1.22554

Exact [Start Eq. (4.131a)] Trapezoid Simpson's rule Eq. (4.133a) Eq. (4.131 a, b) rule 0.221587 0.350150 0.492234 0.649262 0.822804 1.01460 1.22656

0.221393 0.349844 0.491803 0.648693 0.822083

0.224736 0.353538 0.499230 0.656900 0.834497

1.01371 1.22549

1.02743 1.23499

0.221385 0.349911 0.491502 0.650246 0.814383 1.052299 1.03269

0.9

1.45960

1.46082

1.45954

1.47999

2.42365

10

1.71828

1.71971

1.71832

1.74415

-3.10223

The instability of (4.133a) is clearly evident. The trapezoid rule appears to

be superior to Simpson's rule because Simpson's rule is used for a larger

interval, 2h as against h. Exercise

I

Evaluate Ix sec= irx dx directly and by rewriting it as

47r- z x +

1-2x

z

sect nx

Jo

-

41r

-2

(1-2x)2

dx

(Eisner-Squire).

4.19 MULTIPLE INTEGRALS

This section presents a brief account of methods for evaluating multiple integrals. The presentation is principally in terms of the two-dimensional

case to permit simple geometrical interpretations, but the method can obviously be extended to higher dimensions. In spite of considerable recent

activity in this area, the general theory is relatively undeveloped and no comprehensive account of the subject is available. The methods can be classified as (1) combinations of one-dimensional quadrature formulas; (2) interpolatory formulas for simple regions; and (3) Monte Carlo methods. 4.19.1

Combinations of One-Dimensional Rules

It was shown in Chapter 1 that for a region possessing certain geometrical properties multiple integrals can be transformed to iterated simple integrals,

and that arbitrary regions can be decomposed into such simple regions. A very general method of evaluating multiple integrals was first employed in 1829 by F. Minding, who used a combination of Gaussian rules.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

182

There are two drawbacks to this procedure. The first is the rapid increase

in the number of nodes with the increase in the dimensionality of the problem. For example, if four-point Gaussian quadrature is used (which is exact for a seventh-degree polynomial), integration over a two-dimensional region requires 16 evaluations of the integrand, integration over a threedimensional region requires 64 evaluations, and so on. The second difficulty, pointed out by Price in 1963, is that integration with respect to one of the variables can make the integrand poorly behaved for integration with respect to another variable. Price considered the integral of f(x, y) = I over the region between the x and y axes and the parabola y = 1-x2 (see Fig. 4.3). The integral can be written as either 1

I -X2

Ia =

f(x, y) d y dx 0

(4.138a)

0

or y) v2

I

ly =

JJ (1 0

f(x, y)dxdy.

(4.138b)

0

y=I-x2

0

x Figure 4.3

The value is 1, the integrals being easily evaluated analytically in either formulation. Let us consider the application of Simpson's two-thirds rule

(4.19.1)

183

NUMERICAL EVALUATION OF INTEGRALS

to these integrals. For (4.138 a) we get 1Q

r 6I

L

3/4

]

J(0, y)dy + 4J o

.f(G, y)dy+0.

o

(4.139)

A second application of Simpson's rule to the two nonvanishing integrals then gives 1a = 3-L6 [f(0,0)+4f(0,1)+f(0,

+

i

[f(#, 0) + 4f(},

s)

1)] (4.140)

+f(}, $)],

which for f(x, y) = I gives the correct value . For (4.138b) the first application of Simpson's rule gives 1b = 6L f l f(x, 0) dx + 4 J 0

5

f(x, })dx+0.I

(4.141)

0

The second application then gives

1b = 36[f(0,0)+4f(},0)+f(1,0)]

+ 9 [f(0, }) + 4f(}'3, }) +f(-, 5, }) ].

(4.142)

This gives a value j+} i, which is appreciably off. Price calculated the integral using the compounded Boole's rule and obtained the values in Table 4.12. These results show the effect of roundoff error on the nominally exact result given by (4.138 a) and the limitation of the accuracy that can be obtained by (4.138 b) as the result of the combinaton of roundoff and truncation error. Table 4.12

Number of points 5

9 17

33

lb 0.65652626 0.66307927 0.66539809

1, 0.66666668 0.66666667 0.66666676 0.66666672

65

0.66621795 0.66650784

129

0.66660944

0.66666813

257

0.66664446

0.66666923

0.66666688

INTEGRATION FOR ENGINEERS AND SCIENTISTS

184

In this simple example the reason for the difficulty is obvious, the introduction of an integrand (1-y)'12 with a derivative that becomes infinite. We have in Section 4.14 described methods for dealing with such cases; the difficulty is that in multiple integrals they are not apparent by inspection of the original integrand. Interpolatory Rules

4.19.2

The superior efficiency of interpolatory rules can be shown by deriving an expression

fl

P1-x2

f(x, y) dy dx

I o

o

fI

f(1-Y)if'

f(x, y) dx dy

I

Jo

0

W1.1(0, 0) + W2 f(0, 1) + W3 f(1, 0)

(4.143)

the weights being determined by making the rule exact for f(x, y) = 1, f(x, y) = x, and f(x, y) = y. The system of equations W, + W2 + W3 = 2

(4.144a)

W3-isa

(4.144b)

W2 = I

(4.144c)

determines W2 and W3 directly and W, = 3/20. Equation (4.140) gives the exact answer forf(x, y) = x but gives 13/18 for f(x, y) = y. Equation (4.142) also gives the exact answer for f(x, y) = x, but gives 5'/2/3 forf(x, y) = y. The following difficulties arise in generating interpolatory rules in more than one dimension. 1. It is not possible to evaluate the integral of even a simple power function over an arbitrary multidimensional region. 2. There is no general analog of the Lagrange interpolation formula for functions of a number of variables, so it is not possible to write a function that will assume specified values at an arbitrary set of points. Formulas are available, however, when the points form a lattice.

3. The theory of orthogonal polynomials in several variables is in its early stages. It is not in general possible to obtain multidimensional analogs of the Gaussian formulas because there are no equivalent theorems about the reality of the roots and their location inside the domain of the integration.

(4.19.3)

185

NUMERICAL EVALUATION OF INTEGRALS

4.19.3 The Center of Gravity Formulas

It is obviously impossible to tabulate weights and nodes for all the possible domains that may arise in practice. Therefore, simple domains that

can be combined to approximate the domain of interest are important. There are three and only three simple geometrical figures that can cover a plane region without gaps or overlap. These are the equilateral triangle, the rectangle, and the hexagon. The reason for this is that at a point where the figures join, the sum of the angles must be 360°. This can only be obtained by six equilateral (60°) triangles, four rectangles, or three hexagons.

We present a set of simple cubature formulas using the vertices and central point of these three figures:

Jjf(xy) dx dy A

3

A 12

[9fccnier + I Jvertices

(4.145a)

a

JJf(x,

y) dx d y

[2fccntcr 6

jJf(x, y) dx d y = 2

+ Y fvertices].

6 11 42fcenter + 5 [-, fvertices]

(4.145b)

(4.145c)

A collection of higher-order formulas for these regions and for other simple plane and solid figures is given by Abramowitz and Stegun. Exercises 1

Evaluate

t 4+x2+y2

t

f

(x2+y2)tf2dzdy.

o

o

(The exact value is } [25 In (1+ 2 1/2) +2 1/ 2].) 2

Evaluate 1/4 o

[1 1(4 o

exp[(x2+y2)2 (x2+y2)ti2

dxdy

(value - 0.5096).

This can be attacked as a double integral or converted to a/a

exp(Jsec©)dO - 2.

2 o

INTEGRATION FOR ENGINEERS AND SCIENTISTS

186

3

Find h

h

0

0

ln(x2+y2)1'2cos5xy2 dxdy as a function of h = 0.1(0.1)0.5. (The value for 0.1 is =0.0267.) (Price.)

4.20 LINEAR INTEGRODIFFERENTIAL OPERATORS

Functions arise in practice that are defined by a combination of integration and differentiation. Frequently the evaluation involves differentiation of an experimentally determined function. Typical examples are

fA(i)

F( x) =

o (x-t)

dt

(4 . 146 a )

and

I(x) = J

x

(4.146b)

0

which arise in the calculation of sonic boom; or j* b fb

T(x)T(t)lnIx-tj dxdt,

W= a

(4.147)

a

which occurs in the theory of wave drag in supersonic flow. T is either the first or second derivative of a tabulated function. While ad hoc methods have been developed for these problems, they are usually laborious. We will show how the methods developed in this chapter can be used to calculate derivatives and integrodifferential operators. 4.20.1

Numerical Differentiation

The problem of estimating derivatives from experimental values is an important problem in its own right and has been studied extensively. All numerical differentiation methods are essentially curve-fitting procedures that use the experimental points to determine the constants in an assumed functional form that is differentiated analytically. Unfortunately, two curves that agree very closely can have markedly different slopes. For example, the Weierstrass approximation theorem guarantiees that y =x# can be approximated over a finite range by a polynomial to any desired accuracy, but a polynomial will have a finite slope while y has an infinite slope at x = 0.

(4.20.1)

NUMERICAL EVALUATION OF INTEGRALS

187

Values of the derivative obtained by numerical differentiation of experi-

mental data cannot be considered experimental values. The numerical differentiation process involves a hypothesis, and markedly different results can be obtained by different procedures.

The method of undetermined coefficients described in Section 4.2 is easily modified to obtain constants for the expression N

f(x1)

Y_ Wjf(xi)

(4.148)

j=1

where the weights Wj depend on x; (except for N = 2). The weights are determined by the set of linear equations

kx;`-' _

j=1

Wjx;

(4.149)

for k = 0, 1, 2, ..., N- 1. For N = 2 this gives

AX) = f(x2) - f(xl)

(4.150)

which does not involve x; . For N = 3

f(xl)[X2 + 2Xi(X3-X2) - Xg]+ + J(x2)[x3 + 2x1(x1-x3) - x;] + + f(x3)[xi + 2x,(x2-x1) -x21

Axl) _

(4.151)

x1 x2(x2-x1) + x1 x3(x1 -x3) + x2x3(x2-x3)

and the complexity increases rapidly with N. A computer program for carrying out the process for arbitrary N is included in Appendix 2. Since the numerical differentiation process magnifies the effect of experimental scatter, Lanczos has suggested a procedure in which a quadratic is fitted to five equally spaced points by a least squares procedure. This gives

the simple expression

/(X) = 2[f(x+2h)-f(x-2h)]

[f(x + h) - f(x - h)] ; (4.152a)

Joh

for points at the end of the range Lanczos gives AX)

(x) =

- 21f(x) + 13f(x+h) + 17f(x+2h)- 19f(x+3h) 20h

(4.152b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

188

and

f(x+h) _ - llf(x) + 3f(x+h)20hf(x+2h) +f(x+3h), (4.152c) using only four points. Equation (4.152a) can be generalized to j=+N f(x) =

j=-N

/h

jf(x +jh)

j=+N

Y jf j=-N

(4.153)

when more points are available. 4.20.2

General Method of Undetermined Coefficients

The usual methods for evaluating integrodifferential operators such as occur in (4.146 a, b) or (4.147) is to use a numerical procedure for the differen-

tiation and a conventional approach to the integrations. Singularities such as arise in (4.146 a) are generally handled by a subtraction transformation such as

f4(t) o

(xt) 'iz

dt = 2 A(x)x'rz +

`o ,4(t) - A Zx)

(x-t)

J

dt

(4.154)

to obtain a well-behaved integrand. It is our opinion, as yet unsupported by calculations, that the method of undetermined coefficients is a better approach to the problem, and that in the spirit of Eq. (4.148) coefficients should be found for N

L[f ] ^

W; fl a j).

(4.155)

J=1

For Eq. (4.146a) this would be

A(t)dt

o (x-t)

,,, x3/z

W;A(xu;) j=1

(4.156)

The u j's could be specified and the W j's determined from the system of equations 1

k(k-1)

uk-2

N

liz du = Y_ W;xk,

o (I - u)j=1

k = 0, 1, 2,..., N-1,

(4.157)

taking N >, 3. It can be seen that the system of equations is basically the same as (4.5), so that the computer program in Appendix 2 can be used. It

(4.20.2)

189

NUMERICAL EVALUATION OF INTEGRALS

is interesting to note that interpolation can also be considered a linear operator, so that the program can be used to determine weights for expressions N

f(x)

E Wif(xf)

(4.158)

=t

for a specified set of xj's. Exercises 1

The heat flux is related to the reading of a thin film resistance gauge by

9(t)

k

' 0

E(r)dz (t-,r)312'

Devise a numerical scheme to evaluate this integral and test it on

E(t) = t, 5,

0
5, 40,

(5(t-35))''2, 40< t 100. (See W. J. Cook and E. J. Felderman, "Reduction of Data from Thin-Film Heat Transfer Gauges: A Concise Numerical Technique," AIAAJ 4(1966), 561-562.) 2

Carry out some numerical differentiations and interpolations for x'12, a", and sin x tabulated for 0.0(0.1)1. BIBLIOGRAPHIC NOTES AND COMMENTS

Three important recent monographs on numerical integration are S. M. Nikolskii, Quadrature Formulae (Moscow, 1958), which is available in English as AERE Trans. 1055; V. I. Krylov, Approximate Calculations of Integrals (Macmillan, New York, 1962); P. J. Davis and P. Rabinowitz, Numerical Integration [Random House (Blaisdell), New York, 1967). We have drawn very heavily on Chapter 13 of Sh. E. Mikeladze, Numerical Methods of Mathematical Analysis (Moscow, 1953) (available as AECtr-9285), an outstanding treatment not as widely known as it deserves to be.

J. F. Price's "Discussion of Quadrature Formulas for Use on Digital Computers," Boeing Sci. Res. Labs. Rept. D1-82-231 (May 1960), is a fine introduction intended for computer center personnel. SECTION 4.1

J. 0. Irwin, On Quadrature and Cubature (Cambridge Univ. Press, London and New York 1923), has an excellent bibliography of the early literature with a summary of each

paper. A. H. Stroud's "A Bibliography on Approximate Integration," Math. Comp. 15(1961), 52-80, is a very extensive listing but gives no information beyond the title.

190

INTEGRATION FOR ENGINEERS AND SCIENTISTS

R. W. Hamming, Numerical Methods for Scientists and Engineers (McGraw-Hill, New York, 1962), is a pioneer in showing the impact of computer techniques. The discussion by O. Morgenstern, "The Worth of the Space Program : Economics,"

Aeronaut. Astronaut. 6(1968), 45-50, uses the concept of partially ordered sets. This appears relevant in comparing basically different types of quadrature methods. SECTION 4.2

The most extensive treatment of equally spaced quadrature rules is J. C. P. Miller's "Quadrature in Terms of Equally-Spaced Function Values," Univ. of Wisconsin Math. Res. Center, Technical Summary Rept. 167 (July 1960).

C. W. Clenshaw and A. R. Curtiss, "A Method of Numerical Integration on an Automatic Computer," Num. Math. 2(1960), 197-205. Also see Algorithm 279 of the CACM', which refers to this as Chebyshev quadrature. Explicit inverses of Vandermonde determinants are treated by J. F. Traub in "Associated Polynomials and Uniform Methods for the Solution of Linear Problems," SIAM Rev. 8(1966), 177-301. An extensive literature is summarized. SECTION 4.3

P. L. Chebyshev, "Sur les Quadratures," J. Math. Pures App!. 19(1874), 19-34.

S. N. Bernstein, "Sur les formules de quadrature de Cotes et Tchebychef," Dokl. Akad. Nauk SSSR 14(1937), 323-326.

A. F. Dufton, "A New Method of Approximate Evaluation of Definite Integrals Between Finite Limits," Nature 105(1920), 354-359.

The relation to Chebyshev quadrature was pointed out by C. F. Merchant, Nature 105 (1920), 422, who noted its prior use by naval architects.

For recent work consult A. Meir and A. Sharma, "A Variation of the Tchebicheff Quadrature Problem,"Ill. J. Math. 11(1967), 535-546. The method of solving a system of nonlinear equations is from N. N. Yakolev, "The

Solution of Non-Linear Equations by a Method of Differentiation with Respect to a Parameter," USSR Comp. Math. Math. Phys. 4(1964), 198-213. A related method is described by S. G. Mikhlin and K. L. Smolitskiy, Approximate Methods for the Solution of Differential and Integral Equations (American Elsevier, New York, 1967), who refer to it as "reduction to a Cauchy problem." SECTION 4.4

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, New Jersey, 1966), is a comprehensive treatment of the subject.

A. S. Kronrod, Nodes and Weights of Quadrature Formulas (Consultants Bureau, New York, 1965).

E. Ower and R. C. Pankhurst, The Measurement of Airflow, 4th ed. (Macmillan (Pergamon), New York, 19661. P. C. Price, "Gauss's Formula for Numerical Integration and the Design of Experiments," Proc. Carob. Phi!. Soc. 50(1954), 491-494. SECTION 4.5

C. Runge, "Ober empirische Funktionen and die Interpolation zwischen iiquadistanten Ordinaten," Z. Math. Phys. 46(1901), 224-243.

* Collected Algorithms of Communications Association for Computing Machinery.

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NUMERICAL EVALUATION OF INTEGRALS

191

G. Polya, " Uber die Konvergenz von Quadraturverfahren," Math. Z. 37(1933) 264-286. S. N. Bernstein, "On Quadrature Formulas with Positive Coefficients," Izv. Akad. Nauk SSSR 4(1937), 479-503. SECTION 4.6

The three monographs have extensive discussions but do not consider the Lanczos estimate, introduced by C. Lanczos in Applied Analysis (Prentice-Hall, Englewood Cliffs, New Jersey, 1956). SECTION 4.7

A. Ralston, "A Family of Quadrature Formulas which Achieve High Accuracy in Composite Rules," J. Assoc. Comput. Mach. 6(1959), 384-394.

R. A. Pukk, "Investigation of Algorithms for Optimizing the Number of Nodes of Quadrature Formulae for a Given Accuracy of Quadrature," USSR Comp. Math. Math. Phys. 6(1965), 21-39.

J.N. Lyness, "Notes on the Adaptive Simpson Quadrature Rule," JACM 16(1969), 482-495 SECTION 4.8

W. Romberg, "Vereinfachte numerische Integration," Norske Vid. Selsk. Forh. Trondheim 28(1955), 30-36, is the original source. There is now an extensive literature on

the subject. The example is taken from F. L. Bauer, H. Rutishauser, and E. Stieffel, "New Aspects in Numerical Quadrature," Proc. Symp. Appl. Math. 15(1963), 199-218. Algorithm 60 of the CA CM, which embodies this method, is widely used. K. J. Overholt, "Extended Aitken Acceleration," B.I. T. 5(1965), 122-132. SECTION 4.9

Mikeladze derives the formula but does not indicate the original reference. G. Lorentz, Bernstein Polynomials (Univ. of Toronto Press, Toronto, 1953), an extensive treatment of the subject, does not give the quadrature rule. SECTION 4.10

H. Kahn, "Multiple Quadrature by Monte Carlo Methods," in Mathematical Methods for Digital Computers, A. Ralston and H. S. Wilf, eds. (Wiley, New York, 1960). H. M. Hammersly and D. C. Handscomb, Monte Carlo Methods (Methuen, London, 1964).

R. A. Paranjpe, Monte Carlo Approach to the Flow Problems of Turbomachinery (Juris, Zurich, 1963), may appeal to engineers more than the mathematical treatments. An interesting recent paper is L. Rosenberg's "Bernstein Polynomials and Monte Carlo Integrations,"J. SIAM Numer. Anal. B 4(1967), 566-574. SECTION 4.11

A. Sard, "Best Approximate Integration Formulas; Best Approximate Formulas," Amer. J. Math. 71(1949), 80-97, is the pioneer work. Nikolskii discusses the problem in considerable detail. The treatment in this section is based on H. S. Wilf, "Exactness Conditions in Numerical Quadratures," Num. Math. 6(1964), 315-319.

192

INTEGRATION FOR ENGINEERS AND SCIENTISTS

SECTION 4.12

L. B. Rall, "Numerical Evaluations of Integrals and Solutions of Integral Equations," SIAM Rev. 7(1965), 55-64.

H. P. Decell and R. N. Lea, Numerical Integrations and Riemann-Stieltjes Sums SIAM Rev. 8(1966), 196--200.

D. Secrest, "Numerical Integrations of Arbitrarily Spaced Data and Estimation of Errors," J. SIAM Numer. Anal. B2(1965), 52-68. SECnoN 4.14

P. J. Davis and P. Rabinowitz, "Ignoring the Singularity in Numerical Integration," J. SIAM Numer. Anal. B2(1965), 367-383.

P. Rabinowitz, "Gaussian Integration in the Presence of a Singularity," J. SIAM Numer. Anal. B4(1967), 191-201.

V. J. Monacella, "On Ignoring the Singularity in the Numerical Evaluation of Cauchy Principal Value Integrals," David Taylor Model Basin Rept. 2356 (AD 649842) (February 1967).

L. V. Kantorovich, "Approximate Calculation of Certain Types of Definite Integrals and Other Applications of the Method of Subtraction of Singularities," Mat. Sb. 41(1934), 235-245. SECTION 4.15

S. Yokota, "New Rules for the Calculation of the Linear and Quadratic Moments of a Plane Area about the Middle Ordinate," Shipbuilder 35(1928), 261.

J. L. Ullman, "A Class of Weight Functions that Admit Tchebycheff Quadrature," Michigan J. Math. 13(1966), 417-423.

R. E. Beard, "Some Notes on Approximate Product Integration," J. Inst. Actuar. 73(1947), 356-403.

A. Young, "Approximate Product Integration," Proc. Roy. Soc. London A224(1954), 552-561.

G. W. Struble, "Orthogonal Polynomials Variable Signed Weight Functions" Thesis, Univ. of Wisconsin, Madison, Wisconsin, 1961.

A good elementary introduction to orthogonal polynomials is D. Jackson, Fourier Series and Orthogonal Polynomials (Math. Assoc. Amer. Monograph, 1941).

An advanced work is G. Szego, Orthogonal Polynomials (Amer. Math. Soc., Coll. Publ. No. 23, Providence, Rhode Island, 1939). P. Sweigert, "Computer Generation of Quadrature Coefficients Utilizing the Symbolic Manipulation Language Formac," NASA-TN-D-3472 (1966), describes computer programs for calculating weights for specified nodes.

G. Gautschi, "Construction of Gauss-Christoffel Quadrature Formulas," Math. Comput. 22(1968), 251-270, gives a general procedure for constructing Gaussian rules which is embodied in Algorithm 331 of the CACM. Stroud and Secrest have an extensive treatment of the subject supplemented by extensive tables of weights and nodes. Z. Kopal, Numerical Analysis, 2nd ed. (Wiley, New York, 1961), and H. Mineur Techniques de Calcul Numerique (Beranger, Paris, 1952), are also good textbook presentations. SECTION 4.16

The original paper is L. N. G. Filon, "On a Quadrature Formula for Trigonometric Integrals," Proc. Roy. Soc. Edinburgh 49(1929), 38-47. Higher-order modifications and additional references are presented by A. 1. Van de Vooren and H. J. Van Linde, "Numeri-

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NUMERICAL EVALUATION OF INTEGRALS

193

cal Calculation of Integrals with Strongly Oscillating Integrands," Math. Comp. 20(1966), 232-245.

The restrictions on their use is discussed by R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York 1959). The "Filon" generalization of the trapezoid rule is presented by E. O. Tuck, "A Simple 'Filon-Trapezoidal' Rule," Math. Comp. 21(1967), 239-241, and in a 1967 West Virginia University doctoral dissertation, "General Impact Force Determination in the Dynamic Response of Elastic Members," by H. F. Yuan. 1. M. Longman, "A Method for the Numerical Evaluation of Finite Integrals of Oscillatory Functions," Math. Comp. 14(1960), 53-59, utilized a modified Euler transformation. J. Price should also be consulted.

A. M. O. Smith, "Numerical Integration of Oscillating Functions Having a NonLinear Argument," Douglas Aircraft Co. (20 February 1967) (AD 648817). SECTION 4.17

E. T. Goodwin, "The Evaluation of Integrals of the form J 1_0,,f(x)e-11 dx," Proc. Comb. Phil. Soc. 45(1949), 241-245. P. A. P. Moran, "Approximate Relation between Series and Integrals," MTAC12(1958), 34-37.

H. V. Norden, "Numerical Inversion of the Laplace Transform," Acra Acad. Aboensis Math. Phys. 22(1961), 1-31. A discussion of the Goodwin - Moran procedure and some numerical examples are given in W. Squire, "Numerical Evaluation of Integrals Using Moran Transformation," West Virginia Univ. Dept. of Aerospace Eng. TR-14 (July 1969). SECTION 4.18

Both Mikeladze and Krylov consider indefinite integration in some detail. The Clenshaw-Curtiss method of Section 4.2 is recommended for indefinite integrals. The effect of a singularity near the range is treated by E. Eisner, "Numerical Integration of a Function That Has a Pole," Communs. ACM 10(1967), 239-243. Some comments by W. Squire and E. Eisner appear on page 610 of the same journal. SECTION 4.19

Irwin reviews the older work and gives a detailed bibliography. A. H. Stroud, "Quadrature Methods for Functions of More Than One Variable," New York Acad. Sci. 86(1960), 776-791, lists all papers covered in Mathematical Reviews after 1940. Mikeladze has a good chapter on the subject and J. F. Price, "Examples and Notes on Multiple Integration" Boeing Sci. Res. Labs. Rept. DI-82-0231 (February 1963), is an excellent introduction. SECTION 4.20

Some material on the approximation of linear operators may be found in R. E. Bellman et al., Invariant Imbedding and Time Dependent Transport Processes (American Elsevier, New York, 1964).

R. E. Bellman, R. E. Kalaba, and J. Lockett, Numerical Inversion of the Laplace Transform (American Elsevier, New York, 1966). The paper by Traub cited in Section 4.2 should also be consulted.

An interesting survey of methods for numerical differentiation is S. Fillipi and H. Engels, "Altes and Neues zur Numerischen Differentiation," Elektron. Datenverarbeit.

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194

8(1966), 57-65. Another recent reference is H. C. Hershey, J. L. Zakin, and R. Simha, "Numerical Differentiation of Equally Spaced and Not Equally Spaced Data," Ind. Eng. Chem. Fund. 6(1967), 413-421.

The examples cited at the beginning of the section are from E. Eminton, "On the Numerical Evaluation of the Drag Integral," ARC R & M 3341 (October 1961) and W. B.

Igoe, "Richardson Extrapolation Applied to Numerical Evaluation of Sonic Boom Integrals," NASA-TN-D 3806 (1967).

Problems and Research Exercises 1

Show that if P3 (x) is a third-degree polynomial and a and b are two zeros, then

f aP3(x)dx = 3(b-a)

f(a+b\.

2

l1

[W. R. Talbot, "An Integral Property of Cubics and Quadratics," Math. Mag 37(1964), 325.] [Hint : Consider the product integration rule with (x-b) (x-a) as the weight function.] 2

Quadrature rules can be used to obtain approximations for functions defined by integrals (Davis and Rabinowitz/Hart). Find approximate expressions for

In t = f1 Xx, arctan t =

3

j'tdX o 1 +x2,

Ei(t) =

J - dx. 0,

X

Compare the weights and nodes obtained for

f

t-ln(x)f(x)dx = E W, f(x;)

o

i=1

by a Gaussian product formula [D. G. Anderson, "Gaussian Quadrature Formulae for I,, -ln(x)f(x) dx," Math. Comp. 19(1965), 477-481) with those obtained by transforming the ordinary Gaussian weights and nodes by (4.93 a, b) and (4.94). 4

Consider the problem of inverting a linear operation, that is, by using

Lf(x,) = E Wuf(xi) Solve Lf = g(x) as f = IW,j]-'g by finding the inverse of the matrix. In particular, (a) convert the Lanczos differentiation procedure (4. 152) into an indefinite integration

procedure; (b) analyze the numerical inversion of Laplace transforms from this viewpoint (Bellman). 5

R. E. von Hold and R. J. Howerton, "The Definite Integral of the Product of Linear Functions," Math. Comp. 17(1963), 419-424, derive formulas based on approximating

(4)

NUMERICAL EVALUATION OF INTEGRALS

195

the integrand by a product of linear factors. Show that

f.b

f(x)9(x)dx

' b

3

a

Cf(a)9(a)

+f(a)9(b) 2 f(b)9(a) +.f(b)9(b)]1

is equivalent to Simpson's rule. 6

L. D. Gate, Jr., "Numerical Solution of Differential Equations by Repeated Quadratures," SIAM Rev. 6(1964), 134-147, derives integration schemes for differential equations using combinations of quadrature rules. Investigate the possibility of using the Ralston formula (4.54) instead of the two-point Gauss rule in his method.

Chapter 5

QUADRATURE BY DIFFERENTIATION

5.1

INTRODUCTION

The methods presented in this chapter are based on the use of derivatives of the integrand in addition to functional values of the integrand. Following

Lanczos, this will be referred to as quadrature by differentiation though Schonberg has objected to this description, arguing that this designation should be applied to the method for evaluating definite integrals by differentiation with respect to a parameter (described in Chapter 3). In Section 5.2 a generalization of the procedure of undetermined coeffi-

cients developed in Chapter 4 is developed in which derivatives of the integrand are used. It is shown how the use of the-derivative of the integrand

at the end points of the range of integration can increase the precision of compounded quadrature rules, and methods of approximating these corrections in terms of differences are presented.

Section 5.3 describes generalizations of the trapezoid and midpoint rules in which increased precision is obtained by using higher derivatives instead of more nodes. It should be noted that because of the persistence of form noted in Chapter 2 it is easier to compute a derivative at a point where the function is also being calculated than to calculate the function at another point. Section 5.4 discusses the Euler-Maclaurin summation formula and a related formula. Their relation to the integration rules in the preceding chapter is pointed out. Section 5.5 presents a compound generalized mid-

point rule for integrands with a weight function using derivatives for increased precision.

In Sections 5.6 and 5.7 the application of quadrature by differentiation to the determination of eigenvalues and to boundary-value problems is presented. We believe that this is a very powerful approach that will find more extensive application in the future, though it does have limitations. In general as subroutines for carrying out differentiations analytically become widely available the methods described in this chapter will find increasing acceptance.

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198

5.2 COMPOUND RULES WITH CORRECTION TERMS

The method of undetermined coefficients introduced in Section 4.2 can be used to find weights for quadrature rules of the form M

N

I

f(x)dx 0

Wjf(Xj) + j=1

pp

VkJ(Xk) k=1

by solving the system of equations l = W1 + W2 +

+ WN,

(5.2a)

i=WIx1+W2X2+...+WNXN+VI+VZ+...+VM, W I X j +W2XZ +

+WNXN + 2 VIXI + 2 V2X2 +

+ 2VMXM,

(5.2b) (5.2c)

and so on, for the W's and V's. For this more general form, however, it is possible for the determinant to vanish so that an arbitrary choice of the nodes is not possible. A simple example is +I f(x)dx '=' W1 f(0) + W2f(l) + Vtf(4),

J -1

(5.3)

for which the equations (5.4a) (5.4b) (5.4c)

are inconsistent. It is, however, possible to obtain corrected midterm and trapezoid rules: r1

(5.5a) 0

f(x) dx = 4[f(0) +f(1)J +

1.

[f(0) -f(l)],

(5.5b)

which are exact for cubits. When these rules are compounded, the derivatives at the junctions cancel, so that only the derivatives at the end points

(5.2.1)

199

QUADRATURE BY DIFFERENTIATION

remain. The expressions are fb

f(x) dx

MN

(b-a)2 [J(a) -f(b)]

(5.6a)

and n

f

f(x)dx

TN +

a

(b2N)2 [f(a) -f(b)],

(5.6b)

where MN and TN are the values given by the compound midpoint and trapezoid rules with N panels. These expressions, like Simpson two-thirds rule, are exact for cubics. 5.2.1

Gregory and Sheppard Rules

Considerable work has been done on formulas where the derivatives are approximated by differences. The most common form (Gregory's) uses only values inside the range of integration, but more accurate results can be obtained by centering the differences and using nodes outside the range of

integration. The expressions required can be found by using the undetermined coefficient procedure of Section 4.2 to express the derivatives in terms of the values of the integrand. Sheppard has developed a refinement where the weights of the correction term depend on the number of panels. To illustrate the relation between the

two methods, a conventional difference approximation to (5.6a) uses f(a) = h

[f(a+')_f(a)]

f(b)

[1(b)

(5.7a)

and h

-f

`(b

11

2)J

(5.7b)

where h = (b-a)/N. This gives Parmentier's rule:

f:1xdx = MN+1-[f(a)-fla+2J-f(b-2) +f(b)].

(5.8)

Sheppard's expression makes

J:f(x)dx

^' MN +

A[f(a) - f(a + Z) - f(b - 2) +f(b)]

(5.9)

200

INTEGRATION FOR ENGINEERS AND SCIENTISTS

exact for f(x) = x2. The sum for MN can be evaluated analytically and A is given by (h/ 12) [2 N/(2 N - 1)], so Sheppard's expression reduces to the conventional difference form as N - oo. Sheppard's expressions become rather cumbersome as the number of differences are increased; for example, 1

b

h N(80N-177) 960(N-2)(N-3)

f(x)dx '= Mx a

I 1

-f(a+2J-f(b-2) (5.10a)

+

N(40N-57) 960(N-3)(N-4)

fa+\

h

//

+fb-521 \

) - f(b

- f(a

2

32

f.b

f(x) dx

TN +

h

)

N(15N-26) {f(a+h)+f(b_h)l

120(N-1)(N-2)

- f(a)-f(b) (5.10b)

N(5N-6) f(a+2h)+f(b-2h) 120 (N-2)(N-3) _ f(a+h) -f(b-h) h

,

but are relatively easy to program and warrant consideration. Simpson's Rule Corrections

5.5.2

The correction procedure can be applied to Simpson's two-thirds rule in two different ways. Lanczos developed the expression (' I JII

f(x)dx ^ - -

[7f(0) + 16f(j) + 7f(1)] +

.

[f(0) - f(1)],

(5.11)

0

which is exact for fifth-degree polynomials. Hamming and Pinkham have used difference approximations to 1 f(x) dx

J0

a tf(a) + 4f( j) +f(1)] + T 8I8 0 tf `''(0) -f `3)(1)],

(5.12)

(6.3.1)

QUADRATURE BY DIFFERENTIATION

201

which is also exact for fifth-degree polynomials. However, the error term is larger than for (5.11) and requires the third rather than the first derivative, which is harder to approximate by differences. Exercises 1

Derive end-point derivative correction terms for the two-point Gauss rule. (b) Show that compounding a Gaussian rule with end-point derivative corrections (a)

is analogous to the Ralston method (Section 4.7), except that a high-order derivative is needed instead of the function at the end point. 2

Derive the constants in

1f2 dx = W,f2(0)+ W2f2(f)+ W3f2(1)+C,f(0)[f(#)-f(0)l +

+ C2.f(1)[f(l) -f(f)). This is useful in calculating moments of inertia [H. K. Baker, "Numerical Integration and Moments of Area," Australian J. App/. Sci. 14(1963), 347-350). 3

Derive a Sheppard correction for the Bernstein rule.

5.3 SIMPLE QUADRATURE RULES USING DERIVATIVES

Chapter 4 presented simple quadrature rules that were made exact for polynomials of increasing degree by using more and more nodes. In this section rules are presented that are exact by taking more derivatives at the same nodes. 5.3.1

Generalized Midpoint Rule

The simplest example is a generalizaton of the midpoint rule b

Ja

f(x) dx

Wt f 0)

i=o

Cb + a

\2

(5.13)

where f ttl denotes the ith derivative. Expandingf(x) in Taylor series around the midpoint and integrating term by term gives the explicit formula

f(x)dx = E (b-a) 2i+, f(2,)(b+al i=o 4'(2i+1)! Ja

-2),

(5.14)

which involves only even derivatives. The odd derivatives cancel out because of the antisymmetry about the midpoint.

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202

To illustrate the power of this method compared to the compound midpoint rule, the integral J'o

exp(x) dx = exp(u) - 1 = 2 exp 2) sinh u

(5.15)

will be evaluated by both methods. The compound midpoint rule with N panels gives IN(U) =

U

N

Y ex

N i=1

[f2i_l p1u l 2N

=

(u12N) (e°-1). sinh( u/2N)

(5.16)

Therefore the accuracy of the result depends on (u/2 N)2 being small. Using Eq. (5.14) gives 1 (u/2)2'(5.17)

I,y (u) = 2 exp{ u }

\2 1=1 (2i-1)!

The summation is the first M terms of the power series expansion of sinh u/2,

which is a convergent series. While the number of derivatives required increases with u, the number required for a given accuracy is appreciably less than the number of panels for the compound midpoint rule. 5.3.2

Generalized Trapezoid Rules

The problem of generalized trapezoid rules M

iabf(x)dx ^-' Y WI[f'''(a)+(-1)'f °(b)]

(5.18)

=o

is more difficult and requires defining the Bernoulli numbers and Bernoulli polynomials. The Bernoulli polynomials can be defined by the generating function

tet e`-1

t° °° _ Y B,,(x) -

(5.19)

n!

a=o

The Bernoulli numbers B. are the values of the polynomials at x = 0. We have Bo

B1

_

B2

1

1

2

6

B4

_

1

30

(5.3.2)

203

QUADRATURE BY DIFFERENTIATION

are 0 except B, . All the odd Whittaker and Watson derive the quadrature formula

f

b

l(x) dx =

a

b2

a

[f(a) +f(b)]

- "-1 (-1)'Bj+,(b-a)z` [f (a) -I"'(b)] _

2N+ 1

+(b (2N)!

1

p

B2N(t)f2N[a +(b-a)t]dt,

(5.20)

which involves only odd-order derivatives. However, this formula is only an asymptotic approximation. If we consider the case of cos x dx = 0

(5.21)

0 Jn

the first term gives it and all the odd derivatives vanish, so the remainder integral is always -7C, no matter how large N is. An alternate approach requires weights that depend on N. The basic form b

f

f(x)dx = Y (b-a)r+'WjM[f("(a)+(-1)`l`i'(b)] i=o

a

(5.22)

can be made exact for polynomials of degree M - I by the method of undetermined coefficients. It is possible, however, to obtain an explicit expression WM =

for the weights. While Wo

(1+2M-i)!(1+M)! (1 +i)!(M-i)!(2+2M)!

(5.23)

for all M, the other weights depend on M.

For M = I

Jf(x)dx

[f(0) +I(1)] + -2 [f(o) -1(1)]

(5.24a)

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204

corresponds to (5.5b); the next case is

[f(o) -f(1)] +

Jo f(x)dx ^-' - [f(0) +f(1)] +

120 [f(0)+f(1)].

(5.24b)

Lanczos has shown that this class of formulas converges if there are no singularities in a narrow strip extending from

a - -}(2`12-1)(b-a) to b+1(2' 2-1)(b-a). To illustrate the effect of a singularity, I

dx

= 2[a"2 - (a-1)112]

(5.25)

Jo (a-x)''2

was evaluated for a number of values of a. The convergence criterion in this case is a> 1.207.1t can be seen in Table 5.1 that as M increases, the result oscillates around the correct value, and that for a = 1.10 the violent divergence is obvious; for a = 1.20 there is an initial improvement with increasing M before the divergence sets in.

This type of quadrature formula has found some application in the numerical solution of differential equations and Lotkin has reported good results from quadrature rules based on replacing the derivatives in (5.25) by differences. However, the most promising application to the approximate solution of eigenvalue and boundary-value problems will be discussed in more detail later. Table 5.1

a

Exact

M= 1

M=2 M=3

M=4 -0.287

1.466 1.297

2.057

0.776

2.500

1.20

1.575

1.141

1.407

1.211

1.25

1.236

1.447

1.143

1.287

1.206

1.40

1.100

1.213

1.073

1.109

2.00

0.8284

0.8535

0.8265

0.8286

1.099 0.8283

1.10

Exercises 1

2

Find the limiting value of W" defined by Eq. (5.23) as M-3 oo (Takeyama). Find a set of weights based on making Eq. (5.22) exact for exp(ik,rx) (- N<, k < + N) (Squire).

(5.4)

205

QUADRATURE BY DIFFERENTIATION

5.4 SUMMATION FORMULAS

In this section, two formulas for expressing a sum as an integral plus correction terms are presented. Both expressions are special cases of a general form k+ 1 __

i=k

f(i) _

.f(x)dx I-z

ICI

Bn(z) n=1

ji

[ f(n-1)(j-Z) - f(n-))(k+1-z)], (5.26)

which like (5.20) is not necessarily convergent, but we will derive them from the quadrature rules of the previous section.

If Eq. (5.22) is applied to the range j to j+ 1 , j+ 1 to j+2, ..., k- I to k and the results added, the derivative terms cancel at the intermediate points. The resulting expression, equivalent to (5.26) with z = 0, is i=k

k

E f(i) = I-j

fI

f(x) dx + f(j) +f(k)

-

2

2

+ ;' [f(3)(j) - f(3)(k)) _ 30

[f(j) - f(k)] + [f(s)(j) o

- f(5)(k)] + (5.27)

+ ....

This is the famous Euler-Maclaurin formula. It can be looked upon as a procedure for evaluating either sums or integrals. This expression is the basis of some results presented in the previous chapter. The proof by Davis in Section 4.13, that the trapezoid rule applied to a function with a k``'

derivative of bounded variation is of order of hk+', is based on the cancellation of derivatives in (5.27). Similarly, the accuracy of the GoodwinMoran procedures in Section 4.16 is also explained by the Euler-Maclaurin formula. The relatively large error is in cases where the derivatives do not vanish very rapidly If Eq. (5.14) is used for the ranges j-4 to j+l, j+4 to j+}, ..., k-1 to k+4, the resulting expression i=k

k+ 112

E f(i) =

I-1i2

'1

f(x) dx + za[f(j-1) -f(k+#)]

- 57 ir-0

[f(3)(j-i) -

f(3)(k+i)]

+ 9wrr 0 (f(S)(j-#) -f(5)(k+4)] + ...

(5.28)

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206

corresponds to (5.26) with z = This expression has been traced back to Maclaurin, but is not as widely known as the Euler-Maclaurin formula. This is unfortunate because since the correction terms are of opposite signs, the two expressions will frequently bracket the correct value of the sums. 2.

Equation (5.28) is used to prove the analog of Davis's result in Section 4.13

for the midpoint rule. Moran noted that the value given by Eq. (4.127) varied with 6 and appeared to have a period h. On the basis of (5.27) and (5.28) it would appear that the values given for d and 6+(h/2) bracket the correct value when termination error is negligible. Exercises

I

Evaluate

(2n+l)exp[--n(n+1)0-t] n=1

(which arises in the statistical mechanics of diatomic molecules) using both summation formulas. 2

It is easily seen that 00

-Zq

e- 2nq

l-e-Zq

n(-rl

Evaluate the sum by both summation formulas. (The function arises as a Laplace

transform with q = xs' in a heat conduction problem and cannot be inverted. The summation can be inverted term by term. I. Frank ["An Application of the Euler-Maclaurin Sum Formula to Operational Mathematics," Quart. App!. Math. 20(1962), 89-91] suggested using the Euler-Maclaurin sum formulas to obtain an approximation that could be inverted.)

5.5 THE GENERALIZED MIDPOINT RULE WITH A WEIGHT FUNCTION

The rule presented in this section is a combination of the midpoint rule with derivatives developed in section 5.3 with the method used in Section 4.16.

The general form is

fb Wi; ft`l(m;)

w(x)f(x)dx

(5.29a)

i=o 1=t

o

where Wit is given by the explicit expression 1

ICU = i!

+:

-s

tiw(mj+t)dt

(5.29b)

(6.5)

207

QUADRATURE BY DIFFERENTIATION

where

s=h-a and m;=a+(2j-1)s. 2N

The derivation is straightforward. As in Section 4.16, the range of integration is split into N equal parts, and by a linear change of variable N

J+s

j=I

-s

b w(x)f(x) dx = Ja

w(m;+t)f(mj+t)dt.

(5.30)

Expanding f(m; +t) in a series about m; and integrating term by term gives the desired result. It should be noted that since the derivation is based on a power series expansion, it is not applicable when the derivatives of f

are not continous. The use of this generalized form appreciably increases the accuracy of the method described in Section 4.16 for evaluating Fourier integrals. For M = I the expressions

2sinps N

fOx sinpxdx

E f(m;)sinpm;+

p

f""

1=I

+

2

a (sings - ps cos ps) P

f(m;)cospm; (5.31a) 1=I

and

f(x) cos pxdx Ja

2sinps

N

p

i=1

f(m) cos pm; N

2

- Z (sin ps - pscos ps)

f(m;)sin pm; (5.31b) j=2

P

are obtained. Applying (5.31 a) to J.' e" sin px dx gives the percentage errors shown in Table 5.2. These errors are appreciably better than those obtained in Section 4.16 using only the first sum. The method can also be applied to repeated integrals, such as 0.1

4

x

x

x

24dx

101010 (1+x)5

= 0.090909 x 10-3,

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208

by rewriting it as

' 4(0.1-x)3 dx (1 +x)5

Jo

applying (5.29a) with N = 1 (no compounding) gives N

=

0

10°. Integral = 0.78352

2

3

0.907896

0.908996

1

0.895459

4

5

0.909085

0.909091

Table 5.2 p

1

10

100 1000 10000

N = 10

N = 20

N = 40

0.0417 0.0400 0.1007 0.1021 0.2067

0.0104 0.0103

0.0026 0.0026 0.0012

-0.0133 0.0130 0.0906

-0.0092 0.0156

Exercises

I

Write a generalized midpoint rule for n

f

assuming

1(X) a

csin

os

[ pg(x)] dx

.f(x) =.f(m)+.f(m)(x-m), g(x)

g(m) + g(m)(x-m),

and use it to evaluate the form preceding Eq. 4.110 in Section 4.16.

5.6 LINEAR EIGENVALUE PROBLEMS

An important class of problems in applied mathematics involves the solution of a linear homogeneous differential equation with a parameter [LN + Aq (x)] y = 0

(5.32a)

(5.6)

209

QUADRATURE BY DIFFERENTIATION

subject to N conditions of the form (1-1)Ciky

i=1

`O

(5.32b)

at two different values of x. Nontrivial solutions usually exist only for special values of A that are called the eigenvalues or characteristic values of the problem. The corresponding solutions are called the eigenfunctions or

characteristic functions. In many cases it is particularly important to determine the smallest eigenvalue. A simple illustrative problem is

y+Ay=0

(5.33a)

y(0) = Al) = 0,

(5.33b)

subject to

which describes the vibration of a string. The general solution of the differential equation is y = A sin%112x + BcosA112x.

(5.34)

The boundary conditions give two homogeneous linear equations 0 = A sin O + B cos 0,

(5.35a)

0 = A sin A112 + B cos A112

(5.35b)

for A and B. The condition for A and B not to be 0 is that the determinant of the coefficients vanish. This gives sinA1,2 = 0

(5.36)

so that Ak = k2 n2

(5.37)

are the eigenvalues.

Whenever the differential equations can be solved analytically, an equation

can be obtained for the eigenvalues, though it is generally necessary to solve this equation numerically. In many important problems, however, the differential equation cannot be solved analytically. Many approximate methods have been developed for determining the eigenvalues in such cases, the best known being the Rayleigh-Ritz method.

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210

5.&2 Application of Quadrature by Differentiation

A very powerful method for attacking such problems can be based on the quadrature formula (5.22), which uses the derivatives at the end points of the range of integration. The basis of the method is that the differential equation determines all the derivatives at the two end points in terms of the parameter A and 2N values of y and the first N-1 derivatives. However, only N of these quantities are known. We therefore take the missing values

as unknowns and then express the higher derivatives in terms of these quantities and A. Then the quadrature rule is used to convert N integral identities y(!-1)(b) =

y(')dx

y(;-1)(a) +

(5.38)

1

a

into algebraic equations for these unknowns with ). in the coefficients. Setting the determinant of the coefficients equal to 0 gives an equation for A. As a first illustration for Eq. (5.33 a, b) y(O) and y(l) are designated by u and v, giving Table 5.3. The equations are

fl

0=

(M/2]

jydx = E WM.if(u+v)

(5.39a)

1=0

0

Table 5.3

x

Y

0

0

u

1

0

v

Y

y(s)

y(4)

0

-Au

0

d=u

0

-AV

0

A2v

Y

y(5)

where [M/2] means the integer part and I

v=u+

fo

ydx = u +

((M-1)/21

Y

f=0

(-1)`WWJ11+1(v-u).

(5.39b)

Because of the symmetry of the problem, the two equations separate. Equation (5.39 a) is automatically satisfied by eigenfunctions such as

sin ax, sin 3 nx, sin 5 nx, and so on, which are symmetrical about the midpoint and for which y (1) _ - y (0), whereas Eq. (5.39 b) is automatically satisfied for eigenfunctions such as sin 2nx, sin 411x, sin 611x, and so on,

(6.6.2)

211

QUADRATURE BY DIFFERENTIATION

which are antisymmetrical about x = -, so that y(1) = y(0). The eigenfunctions of the first type satisfy the equation [(M - 1)/2]

0=1+

Y-

(-1)i+' WM+1Ai+1.

(5.40)

i=0

Thus for M = 1

0=1-12A;

(5.41a)

0=1-1'-0,1;

(5.41b)

1

for M = 2 for M = 3

0= 1

22;

(5.41c)

and for M = 4

0=1-

336 j +

3024

3

A2

3O T4

(5.41 d)

It frequently happens that two equations of the same degree are obtained for two successive values of M. The higher value of M uses the information

that a higher derivative vanishes and is therefore more accurate. The sequence of approximations for the lowest eigenvalue is 12, 10, 9.875, 9.870, which yields a value quite close to the exact value n2 = 9.8696. The eigenfunctions of the second type are given by the solution of [M/2l

0=

Y-

(-1)'W2,Ai.

i=o

(5.42)

For M = 2 this gives A

0=1

;

(5.43a)

120

2

for M = 3

0=12

;

(5.43b)

84

forM4 I

42A

A2

2

3024

30240

0=-

(5.43c)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

212

The sequence 60, 42, 41.25 also approaches the correct value 4n2 = 39.48, though not as closely. The approach of the second root of the quadratic to the higher eigenvalues is not as good, and it is necessary to go to higherdegree equations to obtain accurate values. 5:6.1

A Specific Example

A more difficult example is Y(4)+y2

d

(xi)-ay

dx

(5.44)

subject to the boundary conditions u(0) = u(0) = ii(1) = 0,

(5.45a)

u(1)+y2ali(1)=0,

(545b)

which describes the vibrations of a prestressed beam. Using M = 4 gives a determinantal equation

[d;;] = 0 where for M = 4 typical elements are

d

1 - W3 A, a

a

(5.46a) 2

(5.46b)

d3i = d(4 W4 y2 + W3ay2 - Wi),

(5.46c)

d43 = 3 - W2 y2(a+1) + 3 W3 y2 + W4[). + y4(a+1)2],

(5.46d)

d44 = 1 - W14y2(a+1)+3W2y2+ W34 [A+y2(a+1)2]+8W4y4(a+1). (5.46e)

This is not as formidable as it looks, since the determinant elements are obtained by a systematic procedure. Lower-order approximations are obtained by replacing Wa by W? or Wi and taking the weight as 0 when the

superscript is smaller than the subscript. If it were necessary to multiply out the determinant and obtain the equation for ). explicitly, the method would not be very convenient when either the degree of the equation or the order of the approximation were large. However, this is not necessary as

213

QUADRATURE BY DIFFERENTIATION

(6.6.1)

there are computer programs that will evaluate the determinant for any specific value of A. It is therefore feasible to consider the determinant as the function and find the value of ). for which it vanishes by iterative schemes such as interval halving or the rule of false position, which do not require a derivative. Table 5.4 compares the result obtained by the present method for M = 4

with those obtained by Tu and Handelman by several methods. While in some cases there is a considerable error, it could be reduced by taking a higher-order approximation. The characteristic feature of the method was that it gave "reasonable" results over the entire range of y and a considered by Tu and Handelman, whereas all of the other methods failed for some combination of a and y. Table 5.4 Comparison of Elgenvalues of Eq. (5.44) Obtained by Present Method and by a Number of Other Methods Pertur-

bation Southwell Comparison

2

Schwarz

Iteration

Quadrature

U. B.

L. B.

(N = 4)

12.3647 10.4861 8.4998 6.6548 4.6621 2.6927 0.6639

12.3622 10.4841 8.4964 6.6515 4.6591 2.6881 0.6566

12.36236 10.48423 8.57840 6.64724 4.68427 2.68860 0.65760

12.3622 10.1782 7.6303 3.2191

12.36236 10.20305 7.90191 5.56534 3.26949 1.71418

a 12.3622 10.4857 1.3 8.5865 0.6 6.6645 0.9 4.7197 1.2 1.5 2.7521 0.7618 1.8 1 . 885 0 . 0

7.7748 6.5375 5.3001 4.0627 2.8253 1.5879 0.3506 0 .0

12.3622 9.6152 6.5896 3.5394 0.3649

0

12.3622 7.4676

12.3622

0

---

-

-

a = -0.25 5

10 15

20 23.5

12.3622 10.1727 7.6951 4.9295 1.8758

0.0

---

---

12.3647 10.1896 7.8496 5.7796

--

--

Exercise 1

(a) Solve the eigenvalue problem

Y=AZ(x2-1)y subject to y(0) = y(l) = 0 using the weights given by Eq. (5.23). (b) Derive a set of weights based on making the integration exact for cos[(2 n -1)n/21 and use these weights.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

214

5.7 BOUNDARY-VALUE PROBLEMS

In this section we consider second-order differential equations

y = f(x, y, .

)

that are not necessarily linear or homogeneous and that are subject to conditions at two points. The illustrative examples considered involve either y(0) = A

and

y(1) = 0

or

y(a) = y(b) and y(a) = Y(b), which implies a periodic solution if f(b, y, Y) =f(a, y, Y),

but the method is applicable to other conditions. It can also be directly generalized to higher-order equations and multipoint boundary conditions. Boundary-value problems are difficult to solve numerically because the standard methods for numerical solution of a differential require a complete set of initial conditions. The conventional methods of solution are iterative

and require repeated solutions of either the original system or related systems of differential equations. In general, the difficulty increases rapidly with the number of missing initial conditions. The method described here is

capable of providing surprisingly good values of the initial conditions without solving the differential equation. 6.7.1

A Linear Problem

We begin by considering the simple linear problem y + k2 y = f sin 2 nx.

(5.50)

The solution subject to (5.48) is

f

y = A [cos kx - cot k sin kx] + k2

- 4n2 sin 2 ax

provided k is not a multiple of it. The missing initial condition is

(5.51)

(5.7.1)

QUADRATURE BY DIFFERENTIATION

215

y(0) = kz2n41[Z - Akcotk.

(5.52)

For the periodic case the solution is

f sin2nx Y =

provided k

(5.53)

k2-41t2

21r. The initial conditons are y(0) = 0,

(0) =

(5.54a)

2 7rf k2

-

4n2.

(5.54b)

The method of determining the missing initial values is similar to that used in previous sections for eigenvalues, except that the equations obtained by approximating the integrals are solved. The essential distinction between the boundary-value problems of this section and the eigenvalue problems of the previous section is that in the boundary-value problems the combination

of boundary conditions and forcing function makes a unique solution possible except for some special cases, while for the eigenvalue problems generally there is no nontrivial solution except for special cases (the eigenvalues) where there is a nonunique solution. For Eq. (5.50) subject to (5.48) the value of the derivatives at 0 and 1 are given in Table 5.5. To express the solution compactly the groupings

p1(k2) = 1 - W," k2 + W3 k° - WS k6 + ,

(5.55a)

p2(k2) = # - W2 k2 + W4 k4 - W6 k6 + ...,

(5.55b)

p3(k2) = Wz - W; (k2 +4 n2) + W6 (k4 +4 7[2 k 2 + 16n4) +

(5.55c)

are introduced. We have two linear equations

0 = Ap, + (u+v)p2 + 4irfp3,

(5.56a)

0 = (v-u)p, + Ape + 4fp3.

(5.56b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

216

Table 5.5 x = 1

X = 0

Y(') Y(2)

Y(3) Y(4) Y(5)

A

0

U

V

-k2A 2nf-k2u

0

k4A

0

-2nf(4n2+k2)+k°u

-2nf(4n2+k2)+k`v

2nf-k2v

etc.

The solution gives 2

2

u = A P1- P2 - 2irf P3 . 2

P) P2

(5.57)

P2

Table 5.6 shows this to be good approximation to the exact value given by Eq. (5.52) when k is small. The approximation for (k2 -47r2)- t is very good for all k, as the zeros of p3 cancel those of P2 except at the first zero corresponding to k = 27r, but the approximation of cot k is only good for a limited range, as a finite polynomial cannot give all the poles of cot k. Within the range of applicability, however, the accuracy is remarkable. Increasing M from 4 to 6 increases the range of k, for which cot k is approximated. Table 5.6 Quadrature by Differentiation Approximation to Initial Values k

-kcotk

M=6

M =4

1.6

6.4 - 54.5382 12.8 - 53.7872 25.6 -50.7411

M=4

M=6

-54.751797

- 70.2092

- 54.6019

- 2.40807

103.74 5.730

-0.946089 -0.776972 -0.776972 0.0467392 -0.0467391

3.2 -54.7253

4n2)-1

-0.946089 -0.0254334 -0.0252684 -0.0254327 -0.776972 -0.0257477 -0.0255705 -0.0257471 -0.0467391 -0.0270868 -0.0268986 -0.0270860 -0.0342017 -0.0338960 -0.0342004 -54.7257

0.4 -0.946089 0.8

(k2 -

- 26.4466

0.674924 0.00804106 0.00162367

0.837444 0.00807031 0.00162275

0.675567 0.00804421 0.00162369

For the periodic boundary conditions we have Y

u

at both 0 and 1.

y(1)

y(2)

1, (1)

a

-k2u

2nf-k2t)

Y(2)

-21rf(4ir+k2)

(6.7.2)

217

QUADRATURE BY DIFFERENTIATION

The two equations in terms of the functions defined above are

0 = 2p2v+4103,

(5.58a) (5.58b)

0 = - 2 up2 .

Therefore u = 0, which agrees with (5.54a) and

v = -2nf P3,

(5.59)

P2

which is the same approximation as in Eq. (5.57).

5.7.2 A Nonlinear Problem

The solution of these problems was relatively simple because of their linearity. For nonlinear problems the practical application of the method depends on setting up a suitable iterative scheme for finding the missing initial values. The problem

y+.v+y2=0

(5.60)

y(0) = 1 and ,y(1) = 0

(5.61)

subject to

gives the pair of nonlinear equations N

0 = I + 0.5(u+v) + y-

W7[y(i+1)(0)

-

(, ,1)(1)],

i=1

(5.62a)

N

v = u -(u+v) + E WN[y(1+2)(0) _ y(i+2)(1)],

(5.62b)

i=1

where u = y (0) and v = y (1). If we designate the sums by a1(u, v) and 6r2(u, v) and solve (5.62 a, b) as though it were a pair of linear equations for u and v, the expressions

u = - 1.25 - 1.5a1(u, v) - 0.5a2(u, v),

(5.63a)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

218

(5.63b)

v = - 0.75 - 0.5or I (u, v) + 0.5a2(u, v)

are obtained. Starting with u = - 1.25 and v - 0.75 gives fast convergence. As N was increased, the following results were obtained.

N= _U =

1

2

3

4

1.33846

1.35085

1.35017

1.35013

Numerical integration of the differential equation showed that the correct value was between - 1.350122 and - 1.350123, so that the accuracy in this case is remarkable.

However, there is an important inherent limitation to the method since it is only valid for problems where the coefficients and forcing functions in the

differential equations are analytically defined functions with continuous derivatives of the requisite order. Unfortunately, in many engineering problems some of these functions are defined only by a table of values, so the present method is not applicable. The usual variational methods involving integrations can be used without any difficulty for such problems. Exercises 1

Solve the initial value problem

y+siny=0 subject to y(0) = A,

y(0) = 0,

which describes a nonlinear pendulum, by finding the period as a function of A. 2

Find the initial value of y(0) and y(0) for which the solution of

y=(1-v2)9-y+cost has a period 2n. (Warner gives y(O) = 1.190 and y(o)

1.448.]

(6)

QUADRATURE BY DIFFERENTIATION

219

BIBLIOGRAPHIC NOTES AND COMMENTS

The author's interest in this area was first stimulated by Chapter 6 of C. Lanczos' Applied Analysis (Prentice-Hall, Englewood Cliffs, New Jersey, 1956); as a result he wrote a master's thesis (Univ. of Buffalo, 1958). The essential parts appear in W. Squire, "Some Applications of Quadrature by Differentiation," SIAM J. 9(1961), 94-108. At that time the author was not

fully aware of the literature in this area, some of which is referenced in Section 5.3. SECTION 5.1

I. J. Schbnberg, "On Monosplines of Least Deviation and Best Quadrature Formulae," J. SIAM Numer. Anal. B 2(1965), 155-170. SECTION 5.2

The example Eq. (5.3) is taken from Mikeladze. We have a preprint of a beautiful analysis by A. Sharma and J. Prasad, "On Abel-Hermite-Birkhoff Interpolation," J. SIAM Numer. Anal 5(1968), 864-881. in which a method is developed for determining when a generalization of (5.3)involving derivative of arbitrary order has a solution. The Gregory and Sheppard rules are covered in Irwin- Equation (5.11) is from Lanczos, while (5.12) is from R. W. Hamming and R. S. Pinkham, "A Class of Integration Formulas,"

J. Assoc. Compui. Mach. 13(1966), 430-438. Algorithm 280 of the CACM performs Gregory rule intergration. SECTION 5.3

A classic reference is M. V. Obreschkoff, "Neue Quadraturformeln," Abhandi. Preuss. Akad. Wiss. Moth.-Naturiwiss. 4(1950),1-20. Other important references are H. Takeyama,

"Expressions for Interpolation and Numerical Integration of High Accuracy, "Tech. Repts. Tohoku Univ. 23(1958), 48-70, and "Formulae for Interpolation and Numerical Integration of Multiple Integrals," Tech. Repis. Tohoku Univ. 25(1961), 1-38.

P.C. Hammer and H.H. Wicke, "Quadrature Formulas Involving Derivatives of the Integrand," Math. Comp. 14(1960), 3-7.

J. D. Lambert and A. R. Mitchell, "The Use of Higher Derivatives in Quadrature Formulae,"Computer J. 5(1962/3),322-327.

J. D. Lambert and A. R. Mitchell, "Repeated Quadratures Using Derivatives of the Integrand," ZAMP 15(1964), 84-90.

1. V. Petukhov, "Contour Series and Their Use in Integral Methods," USSR Comp. Math. Math. Phys. 1(1961), 214-227. J. P. Imhof, "Remarks on Quadrature Formulas," SIAM J. 11(1963), 336-341, derives weights using a criteria analogous to the "best" weights in Chapter 4.

Some very general convergence theorems are stated without proof by V. 1. Krylov and T. Arljuk, "On the Convergence of the Quadrature Process Containing Values of the

Derivatives of the Integrated Function," Dokl. Akad. Nauk BSSR 1(1963), 721-723 (a translation is available as AD 651 243).

The application of this class of formulas to the numerical solution of differential equations is discussed in Chapter 7 of F. Ceschino and J. Kuntzan, Numerical Solution of Initial Value Problems (Prentice-Hall, Englewood Cliffs, New Jersey, 1966). M. Lotkin, "A New Integrating Procedure of High Accuracy," J. Math. Phys. 32(1952), 29-34.

220

INTEGRATION FOR ENGINEERS AND SCIENTISTS

SECTION 5.4

The history of these summation formulas is discussed in H. W. Gould and W. Squire, "Maclaurin's Second Formula and Its Generalization," Amer. Math. Monthly 70(1963), 44-52. However, we missed a footnote in Whittaker and Watson (p. 127 of the American edition), which states that Euler acknowledged Maclaurin's priority in a letter to Sterling. An interesting application to estimating the error in quadrature rules is given in J. H. Lyness and B. W. Ninham, "Numerical Quadrature and Asymptotic Expansions," Math. Comp. 21(1967), 162-178. SECTION 5.6

Eigenvalue and boundary-value problems are generally lumped together. There is an extensive literature on the subject. Some interesting references are L. Fox, The Numerical Solution of Two Point Boundary Value Problems in Ordinary Differential Equations (Oxford Univ. Press, London and New York, 1957). S. H. Gould, Variational Methods for Eigenvalue Problems (Univ. of Toronto Press, Toronto, Canada, 1957).

R. E. Langer (Ed.), Boundary Value Problems in Differential Equations (Univ. of Wisconsin Press, Madison, Wisconsin, 1960). R. E. Bellman and R. E. Kalaba, Quasilinearization and Non-linear Boundary-Value Problems (American Elsevier, New York, 1965). Equation (5.59) is treated by Y. O. Tu and G. H. Handelman, "Lateral Vibrations of a Beam under Initial Linear Axial Stress," SIAM J. 9(1961), 455-473. The calculations using quadrature by differentiation are from a West Virginia University master's thesis by George Debell. SECTION 5.7

Equation (5.60) and the second problem are from F. J. Warner, "On the Solution of `Jury' Problems with Many Degrees of Freedom," MTAC 11(1957), 268-71.

Chapter 6

INTEGRAL EQUATIONS

6.1

INTRODUCTION

This chapter presents a brief account of the theory of integral equations

and of numerical methods for their solution using the quadrature rules developed in the previous chapters. Inherently, integral equations are no more difficult than differential equations but most of the available treatments are on a much more advanced mathematical level or deal with applications requiring considerable background knowledge. Although it should be possible to present integral equations on a level comparable to the traditional first course in ordinary differential equations, this has not been done here. Since the book is intended for readers who have already studied differential equations, this background is assumed to condense the presentation. The integral equation formulation is generally considered more elegant

and compact than the differential equation because it does not require supplementary initial or boundary conditions. However, its application has been largely confined to theoretical aspects, such as existence theorems. There is also a growing body of calculations that indicate that in specific

cases that can be formulated as either integral or differential equations, a solution of the integral equation gives more accurate results than a solution of the differential equation using the same step size and degree of precision in the integration procedure. There is, however, very little material available on general numerical procedures for solving integral equations analogous to

the Runge-Kutta and predictor-corrector methods for the numerical solution of differential equations. Some of the material in this chapter represents a preliminary study of this aspect of the subject. The author hopes to do more work in this area in the future and thought it would be useful to summarize his present views of the subject in Sections 6.9-6.11. However, they are based on a very limited amount of experience and should be read with caution. With the improved mathematical training of engineers and physicists it is reasonable to expect integral equations to play a more important role in the future. Already in some fields, such as aeroelasticity, the standard texts use integral equations extensively. An interesting but little-known historical sidelight is that Lanczos formulated Heisenberg's matrix mechanics in terms

INTEGRATION FOR ENGINEERS AND SCIENTISTS

222

of an integral equation shortly before Schrodinger expressed it as a partial differential equation, but because integral equations were relatively unfamiliar to physicists this approach attracted little interest. It should be noted, however, that at that time matrices were also strange to physicists and Heisenberg rediscovered their properties for himself. 6.2 CLASSIFICATION OF INTEGRAL EQUATIONS

The general linear integral equation for the unknown function u(x) is 6(x)

K(x, t)u(t)dt.

g(x)u(x) = F(x) + , A.

(6.1)

a(x)

K(x, t) is called the kernel and A the parameter of the integral equation. If F(x) = 0, the equation is referred to as homogeneous. When g(x) = 0, the equation is of the first kind; otherwise, it is of the second kind. In equations of the second kind g(x) is generally brought into the kernel. This can be done by straightforward division. However, as equations with symmetrical kernels for which K(x, t) = K(t, x) have useful properties, it is sometimes convenient to eliminate g(x) in a manner that retains the symmetry. By introducing a new unknown v(x) = u(x)[g(x)]112

(6.2a)

Eq. (6.1) becomes an integral equation for v 6(x)

v(x)

[gx)]"2 + '1

a(x)

v(t)dt

(6.2b)

in which the kernel K(x, t)/[g(x)g(t)]'"2 retains its symmetry. In practice the limits encountered are either (1) a(x) and b(x) are constants or (2) a(x) is a constant and b(x) = x. The first type of equation is called a Fredholm equation and the second a Volterra equation. It is possible to consider a Volterra equation as a special case of a Fredholm equation with a kernel that vanishes when t > x. This permits a general theory, but generally Volterra equations are simpler to deal with, so it is useful to make the distinction. If the range of integration is infinite or if the kernel becomes infinite in the range of integration, the integral equation is called singular. The singularity of the kernel may be integrable but it is also possible to use integral equations based on the principal part of the integral.

(8.3)

223

INTEGRAL EQUATION

Nonlinear integral equations can take many forms but we will only consider

u(x) = f(x) + A J 6 K(x, t, u) dt

(6.3a)

a

and the corresponding Volterra equation

u(x) = f(x) + ).

K(x, t, u) dt

(ax

(6.3b)

where the nonlinearity is inside the integral. Some important physical problems give rise to systems of linear integral equations; in particular, mixed boundary-value problems generate what are

called dual integral equations. We will not consider these in detail but simply note that it is possible to convert a system of integral equations into a single integral equation. This is analogous to the equivalence of a system of differential equations to a single high-order differential equation. 6.3 CONVERSION OF DIFFERENTIAL TO INTEGRAL EQUATIONS

A linear differential equation subject to initial conditions can be converted to a Volterra equation of the second kind in several ways. The simplest way of converting N

y(N)(x) = f(x) + F^CJ(X)Y11-11(X) j=J

(6.4)

into an integral equation is to set y(N)(x) = u (x),

(6.5a)

from which it follows that x

y(N-')(X) = y(N-')(a) + J u(t)dt,

(6.5b)

a

-x

y(N-2)(x) = y(N-2)(a) + y(N-')(a)

- (x-a) +

(x-t)u(t)dt,

f 0

and so on.

(6.5c)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

224

Therefore (6.4) is equivalent to ('

u(x) = F(x) + I s K(x, t)u(t)dt

(6.6a)

a

where N

K(x, 1) _ Y_ Cj(x)( j=1

t

xj

1

(6.6b)

)

(1-l)!

and F(x) isf(x) plus a polynomial in (x - a) generated by the initial conditions.

After u(x) is found, y(x) is obtained by an explicit integral relation, Eq. (6.5a, b). When applied to nonlinear equations, this method does not lead to the form considered in this chapter. An alternate procedure illustrated by the second-order equation (6.7)

Y = p(x);Y + q(x)y +f(x) involves integrating to obtain

p(t)y(t)dt + a

s

s

s

(x) = (a) +

q(t)y(t)dt +

f(x) dx.

(6.8)

a

a

Applying integration by parts to the first integral on the right side gives

f.X p(()Y(t)dt = p(x)y(x) - p(a)y(a) -J P(t)y(t)dt.

(6.9)

a

Then a second integration gives the integral equation

y(x) = y(a) + [y(a) - p(a)y(a)](x-a) + f. (x-t)f(t)dt a

+J

x

{(x-t)[q(t) + P(t)] - p(t)} y(t)dt.

(6.10)

a

The generalization to higher-order linear equations is straightforward and in many cases the application to nonlinear equations will give the type of nonlinear integral equation considered in this chapter. A third procedure is based on the general solution of (6.7) when particular solutions of the homogeneous equation

y= p(x)y+q(x)y

(6.11)

(6.3.1)

226

INTEGRAL EQUATION

are known. If S(x) is the solution of (6.11) satisfying

S(a) = 0,

S(a) = 1,

(6.12a)

C(a) = 0,

(6.12b)

and C(x) is the solution satisfying

C(a) = 1, then the general solution of (6.7) is

y(x) = y(a)C(x) + y(a)S(x) +

0x

f(t)

S(C(xI) C()S(tt)

0

- S(t)

C(S(xt))

) - C`, )

A

(6.13)

This can be used to convert (6.7) into an integral equation by writing y = p(x) y + 4 (x) y +f(x) + r(x) y,

(6.14)

splitting q(x) into 4(x)+r(x) in such a way that the differential equation for 4" can be solved. For convenience we call the solutions S and C. Then in analogy to (6.13) y(x) = F(x) +

S(x) C(t) - S(t) C(x) r(t)y(t) dt ox C(t) s(t) - 0) S(t)

(6.15)

where F(x) is given by the right side of (6.13). If the division of q(x) can be made in such a way that r(x) is a small term, F(x) will be a good approximation to g, and the successive approximation technique discussed in Section 6.6 will converge very rapidly. This technique of obtaining an integral equation is often applicable to nonlinear problems. 6.3.1

Boundary-Value Problems and Green's Function

The derivation of the integral equation for a boundary-value problem is shown by considering the equation y + Ay = f sin 2 irx

(6.16a)

subject to the condition y(0) = y(1) = 0.

(6.16b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

226

We consider the related problem

G=0

(6.17)

subject to the same conditions. Then

G(x,t)=k(1-t)x,

O<x<, t,

= k(1-x)t,

t < x 1 1,

satisfies the differential equations and boundary conditions except at the point x = t, where the derivative of G is discontinuous. If k is set equal to 1, the change in G at t will be 1. It is easily seen that G(f sin2nx - ),y) = GY-yG =

d

dx

(Gy- yd).

Integration gives

1

t

G(x, t)(f sin 2 tx - Ay) dx = y(t)dt;l

o

For k = I we have noted that dG = 1, and since G(x, t) is symmetrical, a change in the variables gives the integral equation

y(x) = F(x) + A Iot G(x, t)y(t)dt J

where I

F(x) = f f G(x, t) sin2ntdt. 0

As shown in the preceding chapter, when f # 0, a solution exists only for A = nZ nZ, while if f = 0 a soluton exists for all other values of A. We show

later that analogous results hold for integral equations in general. The function G(x, t), which depends on both the differential operator and the boundary conditions, is called Green's function. The Green functions for the

more common differential operators and boundary conditions have been tabulated. Exercises

I

Convert the Bessel differential equation

+

x

+1

2 y=o VZ

x

into an integral equation by the three methods used in this chapter.

(6.4.1)

2

227

INTEGRAL EQUATION

Convert the boundary value problem 9+Y+ey2=1(x)

subject to the boundary conditions

y(0) = y(1) = 0 into an integral equation. (This can be done by using the Green function for y = 0 or the Green function for y + y = 0.) 6.4 DIRECT DERIVATION OF INTEGRAL EQUATIONS

As noted in the introduction, integral equations are more general than differential equations, so that in some cases the integral equation must be derived directly from the problem statement. Even when the problem can be formulated as a differential equation, the formulation as an integral equation, though less familiar, is more direct. 6.4.1

Bernoulli's Problem

A very simple example is a geometrical problem dating back to John Bernoulli: find the shape of a curve for which the area under the curve is a specified fraction k of the area of the circumscribed rectangle as shown in Fig. 6.1.

Y

x

Figure 6.1

The expression as an integral equation is direct

y(t)dt = kxy(x).

(6.22)

fox

The solution is easily shown to be

y = Cx' -k the constant C acting as a scale factor.

(6.23)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

228

6.4.2 Motion of a Particle

A more advanced problem is the derivation of the integral equation of motion of a particle with a single degree of freedom, corresponding to Newton's law m

d2x d r2

= F(x, t).

(6.24)

The integral equation is obtained from the kinematic relations

x(t) = x(0) + f vdt,

(6.25a)

0 o

(6.25b)

V (t) = v(0) + I a dt, ,,11110

and Newton's law

a=

F(x, t)

(6.25c)

,

in

which give

F(x, t) x(t) = x(0) + v(0)t + J (t - T) dT.

(6.26)

m

o

If the force is simply proportional to the mass, as in the case of a freely falling body, so that F/m = g, then (6.26) gives the familiar result x(t) = x(0) + v(0)t + jgt2.

(6.27)

If the force is due to a linear spring, so that

F = -kx,

(6.28)

the integral equation for simple harmonic motion

x(t) = x(0) + v(0)t is obtained.

k

M

f (t-T)x(T)dT o

(6.29)

(6.4.3)

229

INTEGRAL EQUATION

6.4.3 Abel's Problem

The earliest example of an integral equation that is not equivalent to a differential equation arose from Abel's problem of a bead sliding on a wire under the influence of gravity (Fig. 6.2a). If the shape of the wire is given as an equation S = f(y) where S is distance along the wire, then the velocity of the bead along the wire is dS/dt. By conservation of energy

ds

= [2g(h-y)]112

(6.30)

dt

where g is the acceleration of gravity and h is the point at which the bead starts with zero velocity. The time required for the bead to fall from an initial point y = h to y = 0 is given by the integral

T(h) = -(2g)-112$

S(y)dy

0 (h-y) In

Y

x

Figure 8.2a

Figure 6.2b

(6 . 31)

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230

If the shape of the curve is known, the time of descent can be calculated.

Abel, however, considered the inverse problem of finding a curve for which the time of descent is a specified function of the initial position. In particular, the curve for which the time of descent is constant is called a -S(2g)- 112 and changing the tautochrone (equal time). On setting u = variables from h to x and y to t, the integral equation becomes c

u(t)dt

(6.32)

o (x-t)'iz

a singular Volterra equation of the first kind. Recalling the definition of fractional integration in Chapter 1, this can be interpreted as a differential equation of fractional order. The solution can be written as

u(x) = f()

d

dx

f(t) di. o (x-t)'/z

(6.33)

In many practical applications, f is an experimentally determined function, so that numerical methods such as were discussed in Chapter 4 must be used to evaluate the "exact" solution. For the case f(x) = const., the solution of the integral equation is equivalent to ky-"2. (6.34) S(y) = It can then be shown that the tautochrone is a portion of a cycloid generated by a circle rolling on the y axis (see Fig. 6.2b). The parametric equations of the cycloid are (6.35a) y = R(9 + sing), x = R(1 - cosO).

A simple calculation/ shows that

dy = 1 + I dy

2]112 I

- L +( S cos9)z] 1

(6.35b)

[1

192

.

(6 . 36)

t

6.4.4 Influence Coefficients

A very powerful technique for deriving Fredholm integral equations is the method of influence functions. The influence function of a structure K(xl, x2) is defined as the deflection produced at x, by a unit load at X2For simple systems it can be derived analytically, while for a complex

(6.6.1)

231

INTEGRAL EQUATION

structure such as an airplane wing it may be necessary to resort to measurements. The deflection D is expressed in terms of the loading by

D(x) = J 6 K(x, t)L(t)dt.

(6.37)

a

When the deflection is specified, this is a Fredholm equation of the first kind for the load. In many applications, however, the loading is related to the deflection. For example, the aerodynamic loading of a wing depends on its angle relative to the flow that is altered by the deflection. When the relation between the loading and deflection is linear, a Fredholm equation of the second kind is obtained relating the deflection and load. For linear conservative structures there is an important principle due to

Lord Rayleigh, known as the reciprocity principle, which states that the deflection at x; due to a unit load at xj is the same as the deflection at xf caused by a unit load at xi. This means that for such a system the kernel is symmetrical. Exercise 1

Derive the integral equation for the nonlinear pendulum for which it is not valid to make the simplifying assumption sin 0 ^-- 0.

6.5 EXACT SOLUTION OF INTEGRAL EQUATIONS

In this section some exact analytical solutions of integral equations are presented. Both general methods that are applicable to large classes of integral equations and some isolated particular solutions are shown. 6.6.1 Fredholm with Degenerate Kernel

A general class of Fredholm equations of the second kind that can be solved are those having kernels of the form N

K(x, t) _ Y f1(x)gi(t). i=I

(6.38)

This type of kernel is variously referred to in the literature as a produc kernel, a degenerate kernel, or a Pincherle-Goursat kernel. Let us write the equation as "b

u(x) = fo(x) + A I=i

f;(x) J g1(t)u(t)dt.

(6.39)

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232

Since the integrals on the right side do not depend on x, the solution must be of the form N

u(x) =.fo(x) + I CIIi(x) i=I

(6.40)

Substituting this into the equation gives a set of N linear equations: f.b

N

fo(t)g;(t)dt =

N

r

(' b

C;1 bij - A J

I=1

g.(t)f;(t)dt

(6.41)

0

for the C 's. The determinant of the coefficients is equivalent to an Nthdegree polynomial equation for A. Therefore there are at most N characteris-

tic values of A for which the equation cannot be solved but for which the homogeneous equation fo(x) = 0 may have a solution. The underlying principle of the method can be applied to some nonlinear cases. For example, the solution of

u(x) = a + A J0I u2(t)dt

(6.42)

must be of the form u(x) = const. On setting u2(t)dt = AC2,

A

(6.43a)

0

the algebraic relation

a+AC2 =a + A(a+AC2)2

(6.43b)

is obtained. Some straightforward algebra then gives u(x) = 1 ±(1-4aA)Ii2

(6.44)

2

Examination shows that real solutions only exist for A < 1/4a. There are two such real solutions except for A = 1/4a, which is referred to as a bifurca-

tion point. The point A = 0 is a singular point. For a = 0 the equation becomes homogeneous and the solution is (6.45)

(6.5.1)

233

INTEGRAL EQUATION

A somewhat more complex problem is the homogeneous equation

AJI

u(x) _

(x-t)u2(t)dt.

(6.46)

0

It can be seen that the necessary form is

u(x) = px+q,

(6.41)

where p and q are constants. Their value can be determined by the set of simultaneous quadratic equations

P(3+Pq+q2

(6.48a)

z

q=-.l(3+2pq+jq2.

(6.48b)

Since all the terms on the right side are second degree in p or q, or both, it can be seen that if new variables p = Ap and q = Aq are introduced, a general solution independent of). can be obtained. The set of equations has four solutions: (a) a trivial solution

u(x) = 0; (b) a real solution U(X) _

(12x-6)

A

(c) and two imaginary solutions

u(x)=

-(6x-3+3i)

and u(x)=

-(6-3x-3i)

The method can be extended to 6

u(x) = c + dx + I [xf(t) + g(t)]u2(t)dt. a

(6.49)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

234

6.5.2

Convolution Kernels

The second important class of integral equations that can be solved are linear Volterra equations with kernels depending only on the difference (x- t). Such kernels are referred to as difference kernels, convolution kernels, or Faltung kernels. The solution of

u(x) = F(x) + A foX K(x-t)u(t)dt

(6.50)

is easily obtained by use of the convolution theorem for Laplace transforms, which was discussed in Chapter 3. If Laplace transforms are designated by bars, applying the convolution theorem gives

u = F + AKu =

F

1-AK

=F+

AK

1-AK

F.

( 6 . 51 )

On inverting the transforms, the expression

u(x) = F(x) + A J X R(x-t)F(t)dt,

(6.52)

0

where R(x) is the inverse transform of K((1-AK), is obtained.

It is also possible to solve nonlinear Volterra equations of the form u(x) = F(x) + A f u(x-t)u(t)dt

(6.53)

0o

by use of the convolution theorem. Applying the transform and solving the

resulting quadratic for u gives

1-(1-4AF) 2A

2F 1 +(I - 4AF)l12 '

(6.54)

the sign being determined by the condition that u(x) -> F(x) as A -+ 0. Fredholm equations with convolution kernels over the range - oo to 0o can be solved by using the convolution theorem for either the bilateral Laplace transform or the Fourier transform. When the range is 0 to oo, the equation is known as a Wiener-Hopf equation. Such equations can frequently be solved analytically, but the method involves some complex variable methods beyond the scope of this book. A number of physical applications give rise to integral equations with kernels of the form (x-t)-', where the integral must be interpreted as the

235

INTEGRAL EQUATION

(1.6.3)

Cauchy principal value discussed in Chapter 1. The solution of

u(x) = F(x) + 2

u(t)

-00 t-x f '0

dt

(6.55a)

is

u(x) = F(x) + A

F(t) dt.

J-00 t-x

(6.55b)

Frequently, F(x) is experimentally determined, so that the methods discussed in Chapter 4 must be used to evaluate the integral in (6.55 b) numerically.

6.6.3. Miscellaneous Examples

The equation

u(x) = 2

"O exp( - Ix - 11) u (t) dt J1

(6.56)

o

can be solved by reducing it to a differential equation. After rewriting it as

u(x) _ 2I J s exp[-(x-t)]u(t)dt + f exp[-(t-x)]u(t)dt}, o

(6.57a)

'0

differentiation gives

I-exp(-x)

!

x

I

0

exp(t)u(t)dt + exp(x) f 'O exp( - t) u (t) dt];

.

(6.57b)

then a second differentiation and a simple substitution give

u = 22u-u.

(6.58)

By setting x = 0 in Eq. (6.57 a) it can be seen that the initial condition is 4(0) = u(0) = A f exp( - t) u (t) dt.

(6.59)

0

It is easily shown that when 2 > 1,

u(x) = C{sin[(22-1)"2x] + (2A-1)1'2cos[(2A-1)112x]}

(6.60)

and that when 0 < 2 <#, the trigonometric functions become hyperbolic

INTEGRATION FOR ENGINEERS AND SCIENTISTS

236

functions. However, if A <0, the integral in (6.59) is divergent and no solution is possible. The solution is of some theoretical interest because the characteristic values are continuous rather than a discrete set. The integral equation

° exp[-(x-t)] - exp{-[x-r(t)]} dt

1=

u(t)

(6.61)

arises in a theoretical analysis of forest fires. Actually it is a linear equation

in 1/u, but is written in the foregoing form because u is the physically significant variable. It is easily shown by differentiation with respect to x that

u(x) = A(1 - exp(-[x-r(x)]}).

(6.62)

The nonlinear integral equation `I

J`0

f

exp[ -Au (t) x] dt = exp(- kx)

(6.63)

arises in a biological problem relating to circulation. It can easily be seen that (6.64)

is a solution. The question, however, arises whether it is unique. It can be shown that it is the only real solution by the following simple argument. If

u(t) = k + v(1),

(6.65a)

A

then v satisfies the integral equation

I exp[-Av(t)x]dt = 1. 0

Expanding in a power series shows that

v"dt=0

forn> 1.

0

If we consider the case of n = 2, it can be seen that v cannot be real.

(6.65b)

(6.5.3)

237

INTEGRAL EQUATION

Exercises 1

Solve (a)

U (X) = x + A f t xtu (t) dt; 0 (b)

u(x) = x + A 2

Carry out the analysis of Eq. (6.49).

3

Solve

1

(xt+t2)u(t)dt.

(a)

u(x) = 1 -) I x (x-t)u(t)dt; Jo

(b)

u(x) = 0.5sinx + 0.5 f u(x-t)u(t)dt. 0ox

4

Verify that (6.55 a) does satisfy (6.55b).

3

Solve +00

u(x) _

J 6

exp(-Ix-tl)u(t)dt.

I

- 00

Verify that (a) u=CxI/3

satisfies

0 = fox [t2 + 3t(t-x) - (t-x)2]u(t)dt; (b)

u = Cx"-t

satisfies

u(x)

=5 txu(t)dt 0

(Lovitt). 7

Solve the generalized Abel equation

F(x) =

" u (t) dt

o (x-t)"

a<1.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

238 8

Show that C3

U(X)

4 satisfies ox

dt [u(x) - u(t)]112

f

[M. L. Juncosa, "Solution of an Ablation Problem," RAND Corp. Memo. RM-4067PR (April 1964), AD 602647.]

6.6 LIOUVILLE-NEUMANN THEORY

A natural approach to the equation of the second kind

u(x) = F(x) + A

K(x, t)u(t)dt

b fa

(6.66)

is the method of successive substitutions. Starting with

uo(x) = F(x)

(6.67a)

gives

u1(x) = F(x) + .t

F(t)K(x, t)dt,

(6.67b)

F(t)K(x, t) J b K(t, t1)dtt dt,

(6.67c)

D

fa

u2(x) = ut(x) + A.2

b

J

"

and so on. This can be written as e

uk(x) = F(x) +

K. (x, t)F(t)dt

All

(6.68a)

I

where K1 (x, 1) = K(x, 1)

(6.68b)

and f.b

I) =

K(x,

t)dz.

(6.68c

(6.6)

239

INTEGRAL EQUATION

The series always converges for Volterra equations but for Fredholm equations the process is only convergent for small A. The condition for convergence can be obtained by applying the ratio test to the series. For constant limits of integration A" f 6

K,, (x, t)F(I)dt <, )"(b-a)"MKMF

(6.69)

a

where MF and MK are the maximum values of F and K. The ratio of two consecutive terms is less than 1, if

<

1

(6.70)

MK(b-a)

This, however, is a sufficient but not necessary condition. It can be shown

that for a linear Fredholm equation the Liouville-Neumann procedure converges if A is less than the smallest eigenvalue of the homogeneous equation (see Section 6.8). However, when the upper limit is the variable x All

X

t)F(t)dt

"

a

(x-a)"MKMF.

(6.71)

1t!

Because of the factorial term this series is always convergent.

H. T. Davis has proven similar results for nonlinear equations using the following assumptions. 1. F(x) is bounded. 2. K(x, t) is integrable and bounded. 3. K(x, t, u) satisfies a Lipschitz condition with respect to u; that is, there is a finite constant C such that

IK(x, t, u) - K(x, t, u`)I < Clu - u'J.

Examination of Eq. (6.68 a, b, c) shows that the solution of a linear equation of the second kind can be written as

u(x) = F(x) + Iab F(t)R(x, (; A) dt

(6.72a)

where R(x, t;A), which is called the resolvent of the kernel K(x, t), is given by

R(x, t; A) =

t). n=1

(6.72b)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

240

For Fredholm equations this representation is only convergent when (6.70) is satisfied, but the resolvent exists and can be found by other methods.

The next section presents the convergent Fredholm representation of the resolvent. But before describing that method, a technique for improving the convergence of the Liouville-Neumann series is noted. 6.6.1

The Euler Transformation

A simple method, equivalent to the Euler transformation, is to introduce a

parameter e = A/(I + A) and substitute A = e/(1- e) into (6.72b) so that R(x, t; A)

n=1 (I-e)"

K"(x, t).

(6.73)

Expanding (1- e)-" by the binomial theorem and collecting terms with the same power of a gives R(x, t; A) = sK, + g2 (K, +K2) + e3(K, +2 K2+K3) +

+ e4(KI +3 K2+3K3+K4) +

,

(6.74)

which will generally converge more rapidly. A number of similar devices are described by Kantorovich and Krylov. 6.6.2 Aitken's 62 Method

The application of Aitken's b2 method, described in Chapter 3, to three successive approximations, appears to improve the convergence considerably and in many cases will cause an apparently divergent sequence to converge. However, the conditions under which it works have not been established. If the process is applied to (6.67 a, b, c) the resulting expression b

F(t)K(x, t)dt

A

ua(x) = F(x) +

f b fb

fo

(6.75)

F(t,)K(x, t)K(t, t,)dtdt,

1-A o

a

b

f

F(t)K(x, t)dt

a

can be shown to be exact for simple product kernels K(x, t) = g, (x)g2(t). In general, its application to a linear integral equation converts the power series in A to a rational fraction in a. If the rational fraction is divided out, the first terms will agree with the power series. It therefore is related to the Pade approximant, which is the subject of considerable research at present.

241

INTEGRAL EQUATION

(6.7)

Exercises 1

Examine the successive approximation solution of

= k-(k-1)ex+k

u(x)

(which is theoretically convergent) for k

f.o u(1)dt o

10.

(a) Apply successive approximations to the two equations of the first exercise of the preceding section. (b) Investigate the effect of the Aitken J2 method on the convergence.

2

6.7 FREDHOLM THEORY

Fredholm derived a representation of the resolvent 1

A"d"(x, I)

(6.76a)

t)dt

1 + Y-nC= I (-1)"(A"/n!) f a

w here d"(x, t) is defined recursively by

d1(x, t) = K(x, t)

(6.76b)

and

d"(x, t) =

K(x, t)

6

Pb

d"_,(t, t)dt + J K(x, tt)d"_(t1, t)dt,

.

(6.76c)

fa This representation converges for all A except those special values for which

the denominator vanishes. An alternate representation of the d"(x, t) in terms of determinants

-

d2(x, t)

1

2!

s:s:

b

K (x, 1)

K (x, t 1) I

fa K(tt, t) K(tt, t1)

d11

i

K(x, t) K(x, t1) K(x, t2) K(t1, t) K(t1, t1) K(t1, t2) dt1 dt2, K(12, t)

K(12, 11)

(6.77a)

(6.77b)

K(t2, 12)

and so on is more usual. Neither is easy to use, except for product kernels where the series terminates. However, the method described in Section 6.4 is generally simpler for these equations.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

242

A rigorous derivation of the Fredholm form of the resolvent is difficult. We will simply indicate its genesis from the analogy with a system of linear

equations and state the important results on the existence of solutions. Approximating the integral by a suitable quadrature formula gives the set of equations x

F, _ Y [bi; - AW, K,; u;)

(6.78)

=I

where

F,=F(x;), uj=u(x;), Kj1=K(x;,xj), the x's and W's being a set of weights and nodes for a quadrature formula. The term u is then expressed as the ratio of two determinants, and when these determinants are expanded in powers of 1., the coefficients are interpreted as integrals to give (6.76). For this reason the integrals in the numerator

are often referred to as Fredholm's minors. The analogy with a system of linear algebraic equations explains many of the basic features of linear integral equations. For example, a Volterra equation corresponds to a system of equations with 0's above the diagonal, so that the solution is straightforward, though as we will see there are practical difficulties. The resolvent can be interpreted as the inverse of the matrix; therefore, if the same equation is to be solved for different F's, it is advantageous to compute the resolvent, whereas for a single solution, other methods may involve less work. The basic existence theorem for linear Fredholm equations, known as the Fredholm alternative theorem, is analogous to the conditions for solution of algebraic equations. This theorem states: If ,. is not a zero of the denominator of (6.76), the nonhomogeneous equation has a unique nontrivial solution and the homogeneous equation has only a trivial solution.

If A is a zero, the homogeneous equation has nontrivial solutions. The number of such solutions does not exceed the multiplicity of A. For these

values of A the nonhomogeneous equation has no nontrivial solutions except for F's satisfying the condition fb

F(x)uk(x)dx = 0

(6.79)

a

where uk is the solution of the homogeneous equation. Exercise

I

Apply the Fredholm approach to the two equations in the first exercise of Section 6.5.

(8.8)

243

INTEGRAL EQUATION

6.8 HILBERT-SCHMIDT THEORY

It was noted in Section 6.2 that there was an important simplification of the theory for symmetrical kernels. This simplification is due to the orthogonality of solutions of the homogeneous equation when the kernel is sym-

metrical. Let uj(x) and uk(x) be solutions of b

uj(x) = Aj f K(x, t)uj(t)dt

(6.80a)

JO

and

Uk(X) = Ak I

b

,Ja

K(x, t)uk(t)dt,

(6.80b)

respectively. Multipying (6.80a) by ).kuk(x) and (6.80b) by ).juj(x), subtracting, and then integrating from a to b gives (Ak - Aj) Ja b u j(x)Uk(x) dx

= Aj)k (b Jb K(x, t)[uk(x)uj(() - uk(t)uj(x)]dtdx.

(6.81)

When K(x, t) = K(t, x), it is easily shown that b

' K(x,

fas

fb

()uk(x)uj(t)dtdx =

Jsb

K(x, ()uk(t)uj(x)dtdx,

(6.82)

so that the right side of (6.81) vanishes. Therefore

b

u j(x)uk(x)dx = 0

(6.83)

A

if Aj # .Zk.

This orthogonality property makes it possible to represent functions by a series in the u's g(x) = Y "=I

and to determine the

by the integral relation

(6.84a)

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244

g(x)un(x)dx

6

C°

(6.84b)

f.b u (x) dx

It is possible to take the u,,'s so that the integral in the denominator is unity. Such a set of solutions is referred to as an orthonormal set. The solution of the nonhomogeneous Fredhoim equation of the second kind b

u(x) = F(x) + A fa K(x, t)u(t)dt

(6.85)

with a symmetrical kernel can be expressed in terms of the orthonormal solutions of the homogeneous equation. Introducing v(x) =

u(x)

- F(x) = Y_ Cnun(x), n=1

(6.86)

A

the coefficients Cn are given by Cn = fa b un(x)v(x)dx.

(6.87

From the integral equation, the relation 1 °b

un(x) J°b K(x, t)u(t)dtdx

J'ab v(x)un(x)dx

r

=

b

f

u(t)b K(t, x)un(x)dxdi

1

J

fa

b

u(x)un(x)dx

(6.88)

n

A.

is obtained by using the symmetry of the kernel and the definition of u,,. A similar manipulation of the integral equation gives

Ibuxuhxdx

An

= A

I"Fxu0xdx. a

(6.89)

(9.9.1)

245

INTEGRAL EQUATION

This determines the Cn's and gives b Go

u(x) = F(x) +

f°

F(x)un(x)dx

n=1-/ln

(6.90)

when A is not one of the eigenvalues 2n. When A is one of the eigenvalues, in accordance with the general theory of the preceding section, there' is no solution unless b

f°

F(x)uk(x)dx = 0.

If this condition holds, the solution is f.b 00

F (x) un (x) dx

u(x) = F(x) + Cuk(x) + E n=1 n*k

Ak-An

where C is an arbitrary constant. Exercise 1

Show that a homogeneous linear equation with an antisymmetrical kernel K(x, t) K(t, x) has purely imaginary eigenvalues.

6.9 NUMERICAL SOLUTION OF VOLTERRA EQUATIONS

In this section we first consider methods for solving linear Volterra equations by reduction to linear algebraic equations and then take up methods, such as successive approximation combined with numerical evaluation of the integral, that can be applied to nonlinear problems also. 6.9.1

Linear Volterra Equations

We first use the Bernstein quadrature rule to approximate

u(x) = F(x) + f X K(x, t)u(t)dt.

(6.93)

0

Essentially similar results can be obtained by using the compound trapezoid or midpoint rules. The sequence of algebraic equations

u(0) = F(0),

(6.94a)

INTEGRATION FOR ENGINEERS AND SCIENTISTS

246

u(h) = F(h) + (K (h, 0)u(0) + K(h, h)u(h)],

(6.94b)

2

u(2h) = F(2h) + h [K(2h, 0)u(0) + K(2h, h)u(h) + K(2h, 2h)u(2h)]

;

(6.94c)

is obtained. The general terms are

u(Nh) = F(Nh) +

Nh

N

Y K(Nh,jh)u(jh),

N+1 J=o

(6.95a)

which gives

u(Nh) =

F(Nh) + [Nh/(N+1)] J=o K(Nh,jh)u(jh) 1 - [Nh/(N+1)]K(Nh, Nh)

(6.95b)

In the general case the sum must be recalculated at each step because of

the change in the first argument of K. However, for product kernels M

K(x, t) _ E Gk(x)Hk(1)

(6.96)

k=1

there is an important simplification because M

N-I

N-1

Z K(Nh,jh)u(jh) = Y Fk(Nh) Y Gk(jh)u(jh). k=1

J=0

(6.97)

j=0

When N is increased, the inner sums simply add an additional term. Shucker apparently was the first to point out that Volterra equations corresponding

to linear differential equations are of this form and to develop numerical procedures utilizing the simplification. 8.8.2 Equations of the First Kind'

Applying the same procedure to

F(x) = J x K(x, t)u(t)dt

(6.98a)

0

P. Linz, in University of Wisconsin MRC Tech. Summary Rep. 825 (Nov. 1967) "The

Numerical Solution of Volterra Integral Equations by Finite Difference Methods," has obtained some very important results for Volterra equations of the first kind. He shows that procedures based on low-order quadrature rules (such as the midpoint rule) are stable, whereas the methods based on higher-order rules (such as Simpson's rule) are unstable. Unfortunately, the author learned of this work too late for incorporation into the text.

(6.9.3)

QUADRATURE BY DIFFERENTIATION

247

gives

F(Nh) - [Nh/(N+1)]"-' K(Nh,jh)u(jh)

u(Nh)

0

[Nh/(N+1)]K(Nh, Nh)

(6.98b)

Aside from the special case where K(x, x) = 0, this form is difficult to handle. As h becomes small, it is indeterminate, since the numerator becomes

the difference of two almost equal terms and the denominator becomes small.

The easiest way to deal with a Volterra equation of the first kind is to convert it to an equation of the second kind by differentiating with respect to x and dividing by K(x, x). This converts (6.98a) into

u(x) -

F(x)

aK(x, t)/ax

K(x,x)

jo

K(x,x)

u(t)dt

(6.99)

providing K(x, x) does not vanish. In some cases where K(x, x) does vanish, repeated applications finally succeed. 6.9.3 Higher-Order Methods

The use of the Bernstein rule, trapezoid rule, or midpoint rule is not very precise. One way of increasing the precision is to obtain a set of starting values by a series expansion or by the methods of the previous section with a very small step size and then add an additional value by using a high-order

quadrature rule to evaluate the integral. This is analogous to well-known methods for solving differential equations and is used by Shucker. In this section we develop a procedure using an N-point quadrature formula and some auxiliary rules that require solving a set of simultaneous equations at each step but give N values at a time. We will work out the method for the two-point Gauss rule

Jo

f(x)dx =

[l(ath)+f(a2h)]

(6.100)

2

where a, = 0.21132487 and a2 = 0.78867513. The method then requires weights for the auxiliary rules: f0a,h

j(x)dx =

(6.lOla)

f(x) dx = hC21 f(a, h) + hC22 f(a2 h).

(6.101b)

Jazh

0

INTEGRATION FOR ENGINEERS AND SCIENTISTS

248

By the methods developed in Chapter 4 C11 = C22 = 0.25, C21 = 0.53867513

and C12 = -0.03865713.

Then at each step we obtain two values of u by solving the system consisting of

{1 - hC,, K[(N+a,)h, (N+a1 h)]}u[(N+al)h] - hC12K[(N+al)h, (N+a2)h] u[(N+a2)h] h N-1

= F[(N+a,)h] + - Y K[(N+a,)h, (j+a,)h]u[(j+a,)h] 2 j=o

+ K[(N+al)h, (j+a2)h]u[(j+a2)h]

(6.102)

and another similar equation. 6.9.4 Product Integration

The methods in the preceding section approximate K(x, t)u(t) in an interval by a polynomial in t. Obviously it would be better to approximate u(t) alone by a polynomial. The difficulty with this procedure is that it is necessary to calculate a set of weights for each x. However, it is claimed that the gain in accuracy justifies this. For the generalized midpoint rule developed in Section 5.5, which can be considered a one-node product integration formula, the weights can be expressed explicitly by (j+1)h

('(j+I)h

K(x, t)u(t)dt = u[(j+})h] J jh

K[(j+1)h, t]dt.

(6.103)

jh

This gives an analog to (6.95)

u[(N-F,1)h]

F[(N+Ih)h] +Y;=1' Kj - u[(j-#)h]

1-K,

(6.104a)

where

Kj =

j

(f.'

- 1)h

K[(j-1)h, t]dt.

(6.104b)

(6.9.6)

INTEGRAL EQUATION

6.9.5

Euler's Method

249

A very simple general procedure that is the analog of Euler's method for integrating an ordinary differential equation is based on the quadrature rule ti

f

f(x) dx = hf(0).

(6.105)

0

Applying this to the nonlinear integral equation

fx u(x) = F(x) +

K(x, t, u)dt

(6.106a)

0 gives

N-a

u(Nh) = F(Nh) + h Y K[Nh, jh, u(jh)].

(6.106b)

j=o

This procedure is not very accurate, as the error tends to accumulate rapidly, but it can be used to obtain a first approximation. 6.9.6

Successive Approximation

It was shown in Section 6.6 that successive approximations converged. Especially with nonlinear equations the required integrations can become impossible to evaluate analytically. We can, however, resort to the procedures developed in Section 4.18.

It should, however, be noted that difficulties can be encountered in practice. As we noted in Section 6.6, Exercise 1, the equation

u(x) = k+(1-k)e + k J u(t)dt

(6.107)

0

is difficult to handle for large k. While analytical successive approximations will utimately converge, in a numerical scheme it would not be possible to evaluate the integrals accurately unless very small steps were taken. As a result of the error introduced in this way, the process may not converge. Exercises

I

Apply the methods of this section to

u(x)=k+(k-1)ex+kI 0

u(t)dt

INTEGRATION FOR ENGINEERS AND SCIENTISTS

250

and to

u(x) _

ir2J2 x +

210

(x-2t)u(t)dt.

2

Develop a method similar to that in (6. 95a, b) based on Dufton's rule.

3

Try to solve

(x-t)u(t)dt = 0.5x2

I

numerically by using the analog of (6.98b) based on the midpoint rule.

6.10 NUMERICAL SOLUTION OF FREDHOLM EQUATIONS

The numerical procedures for Fredholm equations involve solving a system of simultaneous equations. However, even the solution of a set of linear equations that is straightforward in principle can involve considerable difficulty in practice. Since these problems are beyond the scope of this presentation, however, the reader is referred to the literature. 6.10.1

Linear Fredholm Equations of the First Kind

It appears characteristic of linear Fredholm equations of the first kind b

J

K(x, t)u(t)dt = F(x)

(6.108)

that the approximating system of algebraic equations is poorly conditioned; that is, the determinant of the system is small compared to the individual coefficients, so that roundoff introduces difficulties. An interesting discussion is given by McKelvey.

Practically, the larger the approximating system, the more serious the difficulty. Frequently, a reasonable result is obtained by using a small number of points, but when more points are used to increase the accuracy, the solution shows violent oscillations. A number of techniques have been

developed for obtaining "smooth" solutions but apparently no single method or combination of methods can be relied on. There is one interesting general result for when K(x, t) is positive symmetrical. In this case the sequence b

U11+1(X) = un(x) + e[F(x) - fa K(x, t)un(t)dt]

(6.109)

(8.10.3)

251

INTEGRAL EQUATION

converges when s is less than twice the smallest eigenvalue of the homogeneous equation. 6.10.2 Homogeneous Linear Equation The homogeneous linear equation

u(x) = A f K(x, t)u(t)dt

(6.110a)

a Q

is converted to an approximating algebraic system AAu = u

(6.11 Ob)

where u is a vector with elements u; and A a square matrix with elements aj j. We are therefore dealing with the problem of finding the eigenvalues and eigenvectors of a matrix, which has an extensive literature. However, since the matrix eigenvalue problem is conventionally formulated as

det[a;; - uS) = 0,

(6.111)

the largest eigenvalue and corresponding eigenvector of the approximating algebraic system correspond to the smallest eigenvalue and corresponding eigenfunction of the integral equation. Obviously an N by N matrix can only give N eigenvalues and eigenvectors. Those corresponding to smaller eigenvalues will not be as good approximations to the higher eigenfunctions. This is because the higher eigenfunctions oscillate, so that an N-point quadrature does not give a good approximation. 6.10.3

Linear Fredholm of the Second Kind

For the nonhomogeneous equation we have to solve the corresponding approximating system of linear equations. We will not discuss the problem of solving a system of linear equations except to note that the work involved in an exact solution increases very rapidly with the size of the system. As computer hardware and programming improve, the size of the system that can be dealt with increases; but there are limitations that can be exceeded, particularly when dealing with a multidimensional integral equation. The use of an effective quadrature formula can save considerable work by decreasing the size of the approximating algebraic systems. In cases where the kernel is continuous, the use of Gaussian quadrature can save considerable work. However, when the kernel is discontinous or has discontinuous derivatives, the accuracy of Gaussian quadrature is reduced. Stroud and

INTEGRATION FOR ENGINEERS AND SCIENTISTS

252

Robinson report that reasonably good results can be obtained by using a sufficiently

high-order Gaussian quadrature and that each additional

significant figures doubles the number of nodes. The author has found, in

a particular case where the first derivative was discontinuous, that the compound midpoint rule gave better results than a Gaussian quadrature with the same number of points. A similar result would be expected when the kernel and forcing function were periodic.

It is claimed that product integration rules are effective and worth the effort of calculating the required weights, since it is easier to solve a number of small systems than a large system. 6.10.4

Kantorovich's Method of Treating Discontinuities

In cases where F(x) has a discontinuity it is easily shown that

v(x) = u(x) - F(x)

(6.112a)

satisfies the integral equation

v(x) = F(x) +

16

K(x, t)v(x)dt

(6.112b)

u

where 'b

F(x) = ,lj K(x, t)F(t)dt

(6.112c)

a

is continuous because of the smoothing effect of integration.

Generally, when the kernel or its derivatives are discontinuous, the discontinuity is at x = t. This can be eliminated by using the equivalent equation b

u(x)[1 -,1 fa K( x, t)dt] = F(x) + A J b K(x, t)[u(1) - u(x)]dt (6.113) L

and approximating this by an algebraic system instead of the original system. 6.10.5 Successive Approximation

Before the advent of the computer, iterative methods for solving systems of linear equations were widely used and these methods are still being used. While it would seem preferable to use an exact procedure, there are, as we have pointed out, practical difficulties and in many cases iterative methods are faster. In the case of an approximating algebraic system there is a factor

(6.10.6)

253

INTEGRAL EQUATION

that facilitates the use of iterative methods considerably. Since the solution of the algebraic system approximates a continuous function, a small system can be solved and the solution of a larger system estimated by interpolation. The existence of a good first approximation facilitates the use of successive approximations. In the case of a nonlinear system it should be possible to solve a small approximating system by the method described in Section 4.3. Alternately, we could use the sequence of linear equations

un+l(x) = F(x) + A

K[x, t( )n(t)] un+l(t)dt

f

(6.114)

n

starting with u, (x) = F(x) and solve the resulting algebraic equation for a small N.

After obtaining a reasonable first approximation, the sequence

I = F(x) + ).

{

6

K [x, t, un(t)] dt

(6.115)

a

could be used with numerical evaluation of the integral. It is true that we

have shown that such a sequence diverges for large A if we start with ul = F(x), but it will frequently improve a good approximation. The use of Aikten's extrapolation on three successive approximations will frequently accelerate the convergence. 8.10.6 Approximation by a Product Kernel

It was shown in Section 6.5.1 that linear Fredholm equations with degenerate kernels can be solved explicitly. It is possible to approximate a general kernel by a product kernel and thus obtain an approximate solution. If we designate the difference between the exact kernel and the product kernel by S(x, t) and the difference between the exact and approximate solutions by e(x), then the latter satisfies the integral equation b

a (x) = A fa S(x, t)F(t)dt + A fab K(x, t)e(t)dt.

(6.116)

This has the same kernel as the original integral equation but a different forcing function. A relatively crude solution of (6.116) will indicate the error with sufficient

precision for practical applications. It is also simple to obtain an upper limit for the error

INTEGRATION FOR ENGINEERS AND SCIENTISTS

254

fba F(t) dt

aamax

Emax '

(6.117)

-,if.b

K(x, t)dt

1

which shows that the permissible deviation between the exact and product kernel decreases very rapidly with increasing A. Exercises

I

Test some of the methods of this section on the integral equation

u(x) = x° - 2x3 + x + A f K(x, t)u(t)dt, o

K( x,t)=t(1-x),

0
=x(1-t),

x
which is used in the deflection of a horizontal beam. (The eigenvalues area = n2 n2.) 2

Develop a procedure for solving

u(x) =

+(x2-b2)

- AJ

b

K(x, t)

a

u(t) dt [1 + u(t)]3

where

K(x, t) = (x-t) lnlx-il - (b- I) lnIb-ti + (b-x) lnlb-xI, which occurs in the theory of elastohydrodynamic lubrication (K. Herrebrugh, Trans ASME, J. Lubrication Technol. 90, Ser. F (1968), 262-270).

6.11

PRACTICAL EXAMPLE

Heat conduction in a slab is governed by the partial differential equation OT

a2 T

at

ax2

(6.118)

An extensive literature of exact solutions is available for linear boundary conditions of the form ax

+ bT = f(t)

(6.119)

(6.10.6)

255

INTEGRAL EQUATION

where b is a constant. However, even linear problems with b dependent on t

cannot be solved in closed form. Many practical problems involve nonlinear boundary conditions. For example, at a surface losing heat by radiation

aT= ax

(6.120)

-CTS.

For such boundary conditions the superposition principle does not apply, so an analytical approach is difficult, though some approximate analytical methods (such as the heat balance integral) can be used. It is possible to obtain numerical solutions by using finite difference approximations to the partial differential equation, but the alternate approach described below appears preferable. If the surface temperature were known as a function of time, then the complete solution could be obtained by the methods used for linear problems. It is possible to derive a nonlinear integral equation for the surface tempera-

ture, and although this cannot be solved analytically since it is equivalent to the original problem, a numerical solution of the integral equation that only involves one independent variable appears inherently simpler than numerical solution of the partial differential equation with two independent variables.

For the general case of a slab of length L subject to the boundary conditions

aT

TS

ax

at x = 0

(6.121a)

and

aT=0

atx=L

(6.12(b)

att=0

(6.121c)

Ox

and the initial condition

T= l

the surface temperature satisfies the integral equation

TS (t) = I -

Jo

TS (T) 1

+2

J=1

exp[-m2L2/(t-t)]

in (t -'r)] 112

dT.

(6.122)

The kernel is a theta function, which frequently arises in heat conduction

INhEGRATION FOR ENGINEERS AND SCIENTISTS

256

problems. For the limiting case L -+ oo this reduces to

Ts(1) = 1 -

T4(r)dT

(6.123)

o [n(t-T)) 112'

Both (6.122) and (6.123) are nonlinear Volterra equations with a singularity in the kernel as T -+ I. Abarbanel obtained a solution of (6.123) by a successive approximation technique in which the singularity was handled by a subtraction technique,

the iteration being (n+ 1)

t\ Ts(t) = 1 - 2(7)

1/2 (n)

t

Ts(t) - Jo

(n)T S(T) - (n)Ts(t) dT

[n(t-T)]1/2

(6.124)

in which the integral is well behaved. This procedure has two drawbacks. First, it requires a computer with a large memory to store the solution. This is no longer serious because of the increase in the capacity of computer memories. The second drawback is that it is difficult to extend this procedure to the finite case. In a West Virginia University thesis by Brant, carried out under the author's supervision, a step-by-step procedure was developed. The account here shows the difficulties encountered and how they were

circumvented by the use of some of the quadrature rules described in Chapters 4 and 5. The calculation was started by computing a number of values of T by the series

/3/2

Ts(t)=1-2+4t-321-) n n

12

256

n

15

+1212--(L)5/2.

(6.125) a

Then additional values of t were obtained by a step-by-step procedure in which the integral from 0 to t -AT was evaluated by the compound trapezoid rule and the integration from t -A T to t by the relation

Ts(r)dr t-A, (t-t)1/2

24TTs ((-AT).

(6.126)

It was necessary to use very small dT (- 0.0001) at the beginning of the calculation when the surface temperature changes rapidly, but it was essential to increase the step size as t increased and the temperature changed

less rapidly. The goal was to carry out the calculation to t = 100, so that it was impractical to store all the values. Therefore, when the change in T, was less than a specified value, the step size was doubled and the inter-

(8.10.6)

257

INTEGRAL EQUATION

mediate values of Ts(i), which were no longer needed for the integration, were discarded. The first calculation gave satisfactory results up to t = 1.2288

at which time the calculated value of the temperature increased and then began to oscillate.

The first attempt to remedy this involved using the quadrature rule

Ts(')2

t-at (t-T)

2zr''2[z Ts(t-AT) - +Ts(t-24T)]

Figure 6.3a 90

Typical Integrond at Small Value of Time so

Ts4(r) 112

70

(t- r )

(Calculated at

I/2 vs Time Steps t

0. 0006, A r + 0.0001)

60

Ts4 (T)

*

1/2

(t-r)

1/2

50

40

30

20

10

01

0

_..

I I

I.

..

2

3

Time

I

I

4

5

Steps

instead of (6.126). This worked for t < 1.8432. Therefore, the behavior of the integrand was examined and found to be as shown in Fig. 6.3 a for small values oft but became of the form shown in Fig. 6.3b as t increased. Tie difficulty was in the region between T = 0 and T = AT, where the trapezoid rule was inaccurate. The program was therefore modified to keep the values of Ts(t) in this region, so that a compound trapezoid rule could be used. Using eight panels extended the stability until t = 23.7568. Using

6

INTEGRATION FOR ENGINEERS AND SCIENTISTS

258

Figure 6.3b 0.9

0.71-

Ts4(T)

* I(

0.21-

0. i1-

0.0L

0

2

3

4

5

Tim

6

7

6

9

10

II

12

Steps

Simpson's rule extended the stability to 58.9824 and Boole's rule to 60.6208.

Therefore the integrand was reexamined and found to be as shown in Fig. 6.3c. The difficulty now was the curvature at the ends of the region AT to t-AT, rendering the trapezoid rule inaccurate. Therefore the program

for this region was modified to use the Shephard correction terms. This extended the region of stability of the calculation to t = 101.5808, which was our goal. Figure 6.3d shows the calculated surface temperature and the onset of instability for the different versions. On the basis of hindsight, the author now believes that it would be better to handle region 1 by making it include several points, say 0 to 4 AT instead

of 0-4T, and to use a special formula that would give exact results for t°, t112, t, t3l2, and t2. The program described above was easily adapted to the finite case, where

it gave satisfactory results.

(6.10.6)

INTEGRAL EQUATION

259

Figure 6.3c 0.10 Typical Inlegrand at Large Value 0.09

Ts4 (r)

1/20_r)1/2 vs

of Time

Time Steps

0.08 32.7680, At-1.63041

(Calculated oft 0.01

T T1

IT) (t-r)1/

0.06E 0.05 0.04 0.03

0.02

0.01

0

.

0

2

4

I

I

6

8

Time

Figure 6.3d

to

10

I

I

I

12

14

16

Steps

1

18

I

20

260

INTEGRATION FOR ENGINEERS AND SCIENTISTS

BIBLIOGRAPHIC NOTES AND COMMENTS

Smith [7] has a short but good introductory chapter on integral equatoins, while Whittaker and Watson [8] give a more advanced presentation. Volume IV of Smirnov [6] has a more extensive treatment. More comprehensive references on the theory are the following. W, V. Lovitt, Linear Integral Equations (Dover, New York, 1950) was originally published in 1924 and was the standard textbook for a considerable period. F. G. Tricomi, Integral Equations[Wiley (Interscience), New York, 1957] is representative of a number of modern texts. F. D. Hildebrand, Methods of Applied Mathematics, 2nd ed. (Prentice-Hall, Englewood Cliffs, New Jersey, 1965) has an excellent 100-page presentation that includes an extensive set of exercises.

W. Pogorzelski, integral Equations and Their Applications [Macmillan (Pergamon), New York, 1966] is a very complete treatment of the theoretical aspects. The early history is covered by H. T. Davis, "The Present Status of Integral Equations,"

Indiana Univ. Studies 13(70)(1926), and "A Survey of Methods for the Inversion of Integrals of the Volterra Type," Indiana Univ. Studies 14(76, 77)(1927). There is as yet no single complete treatment of numerical techniques. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [Wiley

(Interscience), New York, 1958)] has a chapter on methods for Fredhoim equations. This book is a classic of pre-computer methods. L. Collatz, Numerical Treatment of Differential Equations, 2nd ed. (Springer, Berlin, 1966) has a good chapter with interesting examples. Z. Kopal, Numerical Analysis, 2nd ed. (Wiley, New York, 1961) has a good chapter on

the numerical solution of integral and integro-differential equations. A number of interesting papers appear in Symposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integra-Differential Equations (Birkhhuser, Basel, 1960). They are

referred to here by author and title and "PICC Symposium" (Provisional International Computation Center, the organizer of the symposium). A. Walther and B. Dejon, in "General Report on the Numerical Treatment of Integral and Integro- Differential Equations" (PICC Symposium, pp. 645-671), classify methods of solution and list more than 100 references. S. G. Mikhlin and K. L. Smolitskiy, Approximate Methods jor Solutions of Differential and Integral Equations (American Elsevier, New York, 1967) has only a short chapter on integral equations but stresses semi-analytical methods not treated in this book. The doctoral dissertation by S. Shucker, "Application of Integral Equations to Approximate Solution of Electrical Engineering Problems," Univ. of Pennsylvania, 1961 (Order No. 61-3555) has some interesting derivations as well as numerical solutions. SECTION 6.1

Some examples of the superiority of integral equations are shown in: G. J. Makinson and A. Young, "The Stability of Solutions of Differential and Integral Equations," PICC Symposium, pp. 499-509. R. L. Bisplinghoff, H. Ashley, and R. L. Halfman; use integral equations extensively in Aeroelasticity (Addison-Wesley, Reading, Massachusetts, 1955). J. L. Hess and A. M. O. Smith, "Calculation of Potential Flow about Arbitrary Bodies," Progr. Aeronaut. Sci. 8(1967), 1-138, is an extensive review of the application of integral

(6)

261

INTEGRAL EQUATION

equations to potential theory. The authors have developed some very useful computer programs. M. Jammer, Conceptual Foundations of Quantum Mechanics (McGraw-Hill, New York, 1966), describes Lanczos' work in its historical setting. SECTION 6.2

Dual integral equations are treated in Chapter 8 of C. J. Tranter, Integral Transforms in Mathematical Physics (Methuen, London, 1956). SECTION 6.3

A tabulation of Green's functions is given in H. Margenau and G. M. Murphy, Mathematics of Physics and Chemistry, Vol. 1, 2nd ed. (Van Nostrand, Princeton, New Jersey, 1956). SECTION 6.5

Equation (6.55 a, b) is discussed by E. H. Bareiss and C. P. Neumann in "Singular Integrals and Singular Integral Equations with a Cauchy Kernel and the Method of Symmetric Pairing,"Argonne Nat. Lab. Rept. 6988 (January 1965). Equation (6.61) is from an analysis by H. Emmons, "Fire in the Forest," in Air Space and Instruments, S. Lees, ed. (McGraw-Hill, New York, 1963). Equation (6.63) is from B.A. Hills, "Distribution of Circulation Rates within a Single rissueType," Science 157(1967), 942-943. SECTION 6.6

H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (U. S. Atomic Energy Commission, Washington, D.C., 1960). SECTION 6.8

The Hilbert-Schmidt theory is extended to multidimensional equations by C. L. Best in "An Integral Equation Approach to Vibrating Plates" Virginia Polytechnic Institute, 1962 (Order No. 65-544). SECTION 6.9

Some recent relevant references are the following.

M. K. Jain and K. D. Sharma, "Numerical Solution of Linear Differential Equations

and Volterra's Integral Equation Using Lobatto's Quadrature Formula," Comp. J. 10(1967), 101-107.

B. A. Beltyukov, "An Analog of the Runge-Kutta Method for Solutions of Nonlinear Volterra Type Integral Equations," Du . Equation 1(1965), 417-426. The PICC Symposium has a number of papers. A particularly interesting example is given by J. Douglas, Jr., "Mathematical Programming and Integral Equations," pp.269274. He shows that for a Volterra equation of the first kind

(x-t)It (t)dt=0.5x2, f 0o

which has the exact solution u(x) = 1, the approximating algebraic system gives

u(jh) = 1 +(- )f.

INTEGRATION FOR ENGINEERS AND SCIENTISTS

262 SECTION 6.10

Current computational methods are reviewed by G. E. Forsythe in "Today's Computational Methods of Linear Algebra," SIAM Rev. 9(1967), 489-515. Comparing this with an earlier paper, G. E. Forsythe's "Solving Linear Algebraic Equations 'can be Interesting," Bull. Amer. Math. Soc. 59(1953), 299-329, we see the tremendous progress that has been

made in recent years. The best detailed reference is J. H. Wilkinson's The Algebraic Eigenvalue Problem (Oxford (Clarendon Press), London and New York, 19651.

R. W. McKelvey, "An Analysis of Approximative Methods for Fredholm's Integral

Equations of the First Kind," David Taylor Model Basin, Appl. Math. Tech. Rept. 21 (December 1956) (AD 650530).

A discussion of methods for linear Fredholm equations of the first kind with a bibliography is presented in R. E. Bellman, R. E. Kalaba, and J. Lockett, Numerical Inversion of Laplace Transforms (American Elsevier, New York, 1966).

Some references for Fredholm equations in addition to the PICC Symposium are A. Young, "The Application of Product Integration to the Numerical Solution of Integral Equations" Proc. Roy. Soc. London A224(1954), 561-573. D. Elliot et a!., "Algorithms for the Numerical Solution of Linear Integral Equations of the Second Kind," Mathematics Dept. Rept. 7, Univ. of Tasmania, April 1965 (AD 621358) uses Chebyshev series methods.

The error estimate for the degenerate kernel approximation is from P. Moon and D. E. Spencer, "Errors in the Solution of Integral Equations," J. Franklin Inst. 264(1957), 29-41. Kantorovich and Krylov also treat this problem. S. M. Robinson and A. H. Stroud, "The Approximate Solution of an Integral Equation Using High Order Gaussian Quadrature," Math. Comp. 15(1961), 286-288. SecrtoN 6.11

The integral equation formulation has been treated by S. S. Abarbanel in "On Some Problems in Radiative Heat Transfer," MIT Fluid Dynamics Res. Group Rept. No. 59-1 (AFSRTN 59-531), April 1959, and by P. L. Chambre in "Nonlinear Heat Transfer Problems," J. App!. Phys. 30(1959), 1683-1688. A survey of the direct solutions of the partial differential equation is given by B. Gay, "Comparison of Methods for Solution of the Heat Conduction Equation with a Radiation Boundary Condition," Int. J. Heat Mass Transfer 8(1965), 507-508.

Problems from the Current Literature We list the following equations from the current literature for the reader to test his skill in developing numerical techniques. The original references should be consulted for reasonable values of the parameters and numerical results. 0

u(x) = 0.25 exp[a(x-x*)] + 0.5 f u(t)K(jx-tI)dt 0o

where

exp(-x) dt

K(x) = f 100

x

(6)

263

INTEGRAL EQUATION

(Y. S. Kuznetsov and B. V. Ovchinsky, "Numerical Solution of an Integral Equation

in the Theory of Light Scattering in the Atmosphere," NASA TTF 372 (September 1965).] 2

u (x)

6n

fon In

sin2[(x+t)/2] sin2[(x- t)/2]

sin u(t)

A

1 + A f0 sin u(v)dv

(H. Buckner, "An Iterative Method for Solving Nonlinear Integral Equations," PI CC Symposium, pp. 613-642.) 3

u(x) =

n-1/2 exp[-(t2

+ Ax

t/

di + An-tn I "0 K(!x-t))u (t)dt Jo

where

K(x)

exp[-(t2 + )ll

di t

has a logarithmic singularity as x--a 0. [K. S. Nagaraja and J. P. Hudson, "On an Alternative Method of Numerically Evaluating the Linearized Problem of an Impulsively Moving Flat Plate in Rarefied Gas Flow," Aerospace Res. Lab. Rept. 67-0124 (June 1967) (AD 658458).]

Appendixes

Appendix 1

LIST OF DOCTORAL DISSERTATIONS ON INTEGRATION AND INTEGRAL EQUATIONS Group A General Theory of Integration Garth William Warner, Jr., "Quasi Additive Set Functions and Non-Linear Integration Over a Variety." The University of Michigan (1966), Order No. 66-14611.

Morteza Anvari, "Singular Set-Functions in Abstract Space and Derivation and Integration in Banach Spaces." University of Illinois (1962), Order No. 62-6097.

Edward Takashi Kobayashi, "Integration of Subspaces Derived from a Linear Transformation Field." University of Washington (1959), Order No. 59-5465.

Jesus Gil de Lamadrid, "Topology of Mappings in Locally Convex Topological Vector Spaces, Their Differentiation and Integration, and Application to Gradient Mappings." The University of Michigan (1956), Order No. 18604. Charles Carpenter Buck, "The Algebraic Aspect of Integration in Space." The University of Michigan (1954), Order No. 7618.

Richard Brian Darst, "On Measure and Integration." Louisiana State University (1960), Order No. 60-5905. Samuel Henry Coleman, "Integration in Infinite Product Spaces." University of Virginia (1960), Order No. 60-4597.

Robert Vernon Mendenhall, "On Lebesgue Measure and Integration in an Abstract Space." The Ohio State University (1952), Order No. 24507.

Erling Stormer, "Point Measure in Two-Sided Non-Commutative Integration Theory." Columbia University (1963), Order No. 64-3132.

James Donald Kuelbs, "Integration on Spaces of Continuous Functions." University of Minnesota (1965), Order No. 65-15203.

Group B

Numerical Quadrature

Charles Samuel Frady, "Accuracy in Interpolatory Numerical Integration." Auburn University (1966), Order No. 66-15044.

Vemuri Lakshmi Narayana Sarma, "Eberlein Measure and Mechanical Quadrature Formulae." The University of Rochester (1962), Order No. 62-6652. Richard George Hetherington, "Numerical Integration over Hypershells." The University of Wisconsin (1961), Order No. 61-5939.

268

INTEGRATION FOR ENGINEERS AND SCIENTISTS

Robert Ellis Barnhill, "Numerical Contour Integration." The University of Wisconsin (1964), Order No. 64-10206.

William Hollis Peirce, "Numerical Integration over Planar Regions." The University of Wisconsin (1956), Order No. 19126.

Robert Henry Wassmuth, "Some Complex Numerical Integrations of the SchwarzChristoffel Transformation and Application to Potential Flow." Oregon State University (1966), Order No. 66-3044.

Edward Benoit Anders, "An Extension of Romberg Integration Procedures to N-Variables." Auburn University (1965), Order No. 65-9632.

George William Tyler, "The Experimental Evaluation of Definite Integrals." Virginia Polytechnic Institute (1949), Order No. 5340.

Group C

Integral Equations

George Andrew Sormani, "Three Singular Integral Equations Containing Jacobi Elliptic Functions." New York University (1965), Order No. 67-4840.

David Lee Skoug, "Generalized llstow and Complex Weiner igtegrals and a Related Integral Equation." University of Minnesota (1966), Order No. 67-880. Merlynd Keith Nestell, "The Convergence of the Discrete Ordinates Method for Integral

Equations of Anisotropic Radiative Transfer." Oregon State University (1966), Order No. 67-734.

Bennie Burns Williams, "An Integral Equation and the Related Differential Equation Boundary Problem." The University of Texas (1966), Order No. 67-3367.

Leon Yuda Bahar, "On an Elastostatic Problem in Nonhomogeneous Media Leading to Coupled Dual Integral Equations." Lehigh University (1963), Order No. 63-7856.

James Robert Dorroh, "Integral Equations in Normed Abelian Groups." The University of Texas (1962), Order No. 62-4835.

Satyanarayan Rao Channapragada, "Singular Integral Equations with an Infinity of Intersecting Arcs." University of Illinois (1961), Order No. 61-4273.

Sidney Shucker, "Application of Integral Equations to Approximate Solutions of Electrical Engineering Problems." University of Pennsylvania (1961), Order No. 61-3555. Kenneth Gerald Grossman, "A Finite Difference Solution for Singular Integral Equations of the First Kind, with a Cauchy Type Kernel." New York University (1964), Order No. 64.10030.

Ashfaq Ahmad Khan, "A Comparison of the Radial Distribution Functions of Fluid Argon Computed with the Percus-Yevick and CHNC Integral Equations Using (1) The Guggenheim McGlashan and (2) The Lennard-Jones Potentials." The University of Florida (1963), Order No. 64-5612.

A.D. Stewart, "On the Abel Integral Equation in N-Dimensions, N GE 2."The University of Texas (1964), Order No. 64-8035.

Appendix 1. DISSERTATIONS ON INTEGRATION AND INTEGRAL EQUATIONS

269

Ben Clarence Johnson, "Integral Equations Involving Special Functions." Oregon State University (1964), Order No. 64-7421.

Don Barker Hinton, "Two Stieltjes-Volterra Integral Equations." The University of Tennessee (1963), Order No. 64-4881.

James Hampton Fithian, "An Investigation of a Certain Singular Integral Equation and Its Solution by an Approximation Method." New York University (1963), Order No. 64-1743.

Arthur D. Wirshup, "Application of the Puiseux Polygon to the Solution of Nonlinear Integral Equations." Oregon State University (1963), Order No. 64-2387.

Georges Washington Batten, Jr., "Iterative Solution of Integral Equations of First Kind with Applications to Continuation Problems." Rice University (1963), Order No. 63-7151.

Mosur Kalyanaraman Sundaresan, "Study of Momentum Space Integral Equations." Cornell University (1955), Order No. 15432.

Joyce White Williams, "Singular Integral Equations with Symmetric Complex-Valued Kernels of Class I." University of Illinois (1954), Order No. 9163.

William Alexander Michael, Jr., "Singular Integral Equations with Normal Kernels." University of Illinois (1954), Order No. 9110.

Samuel Eli Benesch, "Theory of Non-Linear Integral Equations with Complex-Valued Singular Kernels." University of Illinois (1953), Order No. 5942.

Fred Alvin Hinchey, "Solution to a Class of Singular Integral Equations Occurring in Mathematical Physics." New Mexico State University (1960), Order No. 60-5209. Charles Gladstone Costley, "Singular Nonlinear Integral Equation with Complex-Valued Kernels of Type N." University of Illinois (1960), Order No. 60-3895.

Dale William Swann, "Applications and Extensions of the Method of Wiener and Hopf for the Solution of Singular and Non-Singular Integral and Integro-Differential Equations." Stanford University (1960), Order No. 60-3837. James Edward McFarland, "Iterative Solution of Nonlinear Integral Equations." Oregon State University (1960), Order No. 60-3347.

Patricia James Wells, "The Solution of the Integral Equation for the Prolate Spheroidal

Transmitting Antenna." Michigan State University of Agriculture and Applied Science (1955), Order No. 58-5723.

Raymond Peter Polivka, "A Generalization of Singular Integral Equations with Complex Kernels." University of Illinois (1958), Order No. 58-5477.

Edward Allen Newburg, "Potential Representations in Integral Equations." University of Illinois (1958), Order No. 58-5470.

John Hilzman, "Application of the Frechet Differential to the Approximate Solution of the Volterra Integral Equation." Oregon State University (1958), Order No. 58-3808.

270

INTEGRATION FOR ENGINEERS AND SCIENTISTS

Evelyn Kendrick Kinney, "Integral Equations in Representation Theory of Functions of a Complex Variable," University of Illinois (1958), Order No. 58-1711.

John William Neuberger, "Continuous Products and Nonlinear Integral Equations." The University of Texas (1957), Order No. 25186.

Hideo Yoshihara, "Rigorous Solutions of the Prandtl Integral Equation for a Certain Class of Airfoils." The University of Michigan (1954), Order No. 1180.

Martin Marc Pincus, "Gaussian Processes and Hammerstein Integral Equations." New York University (1965), Order No. 66-5745. Gunnar-Arnvid Brosamler, "Potential Theoretic Analysis of a Certain Integral Equation." University of Illinois (1966), Order No. 66-7719.

John Myron Tomlinson, "A Method of Solving the Integral Equations for the Vertical

Propagation of Time-Harmonic Electromagnetic Plane Waves in Anisotropic Vertically-Inhomogeneous Media." The Pennsylvania State University (1965), Order No. 66-8762.

Michael Bernkopf, "The Development of Function Spaces with Particular Reference to Their Origins in Integral Equation Theory." New York University (1965), Order No. 66-5646.

Robert Gerald Cawley, "An Integral Equation for the Anomalous Deuteron Vertex and a Comparison of Two Normalization Conditions." University of Illinois (1965), Order No. 66-4154. Ivan ltkin, "Approximate Methods for Solving Linear and Non-Linear Multidimensional Integral Equations and Boundary Value Problems." University of Pittsburgh (1964), Order No. 65-10508. Elmer Lawrence Johansen, "The Solution of Wiener-Hopf Problems Using Dual Integral Equations." The University of Michigan (1964), Order No. 65-5325.

James Allen Reneke, "Integral Equations in Linear Spaces." The University of North Carolina at Chapel Hill (1964), Order No. 65-9050. James Victor Herod, "Solving Integral Equations by Iteration." The University of North Carolina at Chapel Hill (1964), Order No. 65-9017.

John Tunstall Welch, Jr., "Numerical Treatment of an Integral Equation Arising in a Mixed-Boundary Elasticity Problem." North Carolina State University at Raleigh (1964), Order No. 65-2854. Elmer Eugene Lewis, "A Boltzmann integral Equation of Resonance absorbtion in Reactor Lattices." University of Illinois (1965), Order No. 65-7125.

Bruce E. Goodwin, "On the Theory and Application of Fredholm Integral Equations and on Integral Equations Whose Kernels Depend on the Eigenvalue Parameter." Rensselaer Polytechnic Institute (1963), Order No. 65-6446.

Dario Castellanos, "On a Class of Integral Equations and Its Applications to the Theory of Linear Antennas." The University of Michigan (1964), Order No. 65-5281. William Henry Somerville, "An Integral Equation in a Certain Semi-Lattice. " The University of North Carolina at Chapel Hill (1964), Order No. 65-4035.

Appendix 1. DISSERTATIONS ON INTEGRATION AND INTEGRAL EQUATIONS

271

Frank Joseph Rizzo, "Some Integral Equation Methods for Plane Problems of Classical Elastostatics." University of Illinois (1964), Order No. 65-3662.

Charles Luther Best, "An Integral Equation Approach to Vibrating Plates." Virginia Polytechnic Institute (1962), Order No. 65-544.

Raymond Howard Cox, "Integral Equations in Certain Topological Rings." The University of North Carolina at Chapel Hill (1963), Order No. 64-9410.

Charles S. Kahane, "Analyticity of Solutions of Mildly Singular Integral Equations. Extension of a Theorem of Lewy." New York University (1963), Order No. 63-7220.

Group D Applications of Integrals John Robert Ehrman, "Functional Integration Techniques in Quantum Field Theory." University of Illinois (1963), Order No. 64-2875.

Walter Eugene Strimling, "Solution of a Linear, Second Order, Partial Differential Equation in Three Variables by Wiener Integration." University of Minnesota (1953), Order No. 6399.

Vernon Valentine Neis, "A Study of a Multiple Integral Representation of the Constitutive

Equation of a Nonlinear Viscoelastic Solid." University of California, Berkeley (1966), Order No. 67-5128.

Howard Emery Bethel, "On the Convergence and Exactness of Solutions of the Laminar Boundary-Layer Equations Using the N-Parameter Integral Formulation of GalerkinKantorovich-Dorodnitsyn." Purdue University (1966), Order No. 66-13173.

Kenneth Laurel Deckert, "Solutions for Nonlinear Diffusion Equations with Integral Type Boundary Conditions. " Iowa State University of Science and Technology (1963), Order No. 63-7250.

Stephen Esrael Speilberg, "Solutions to Certain Non-Homogeneous Second Order Partial Differential Equation Expressed inTerms of Weiner Integrals." University of Minnesota (1963), Order No. 64-10883.

John Alfred Beekman, "Solution to Generalized Schroedinger Equations Via Feynman Integrals Connected with Gaussian Markov Stochastic Processes." University of Minnesota (1963), Order No. 64-7225.

Raymond Hugh Rolwing, "On a System of Quadratic Equations and Its Integral Analogue." University of Cincinnati (1963), Order No. 64-4673. Edward Olaf Nelson, "A Solution of the Generalized Heat Flow Equation in a Bounded Region as a Wiener Integral." University of Minnesota (1959), Order No. 59-6033.

Richard Lynn Saylor, "A Generalized Boundary-Integral Representation for Solutions of Elliptic Partial Differential Equations." Rice University (1966), Order No. 66-10375

Ralph Arthur Kallman, "On Using the llstow Integral to Solve a Certain Partial Differential Equation." University of Minnesota (1965), Order No. 66-8901.

Howard John Deacon, Jr., "Application of the Method of Integral Relations to the Nonequilibrium Hypersonic Boundary-Layer Equations." Purdue University (1965), Order No. 66-5258.

272

INTEGRATION FOR ENGINEERS AND SCIENTISTS

John Patrick Maloney, "On Certain Nonlinear Integral Operators with Applications to Ordinary and Partial Differential Equations." Georgetown University (1965), Order No. 65-12512.

John Melvin Holt, "Integral Inequalities Related to Non-Oscillation Theorems for Differential Equations." State University of Iowa (1964), Order No. 65-468. The foregoing dissertation list was selected from the titles listed by Datrix, a computerized information retrieval system operated by Dissertation Abstracts. It is apparently based on the appearance of key words in the title and therefore failed to turn up some relevant references such as the Struble and Richardson dissertations cited in the text. However, it is an interesting tool and will be improved with experience.

Appendix 2

INTEGRATION FUNCTIONS AND SUBROUTINES GENW8S (D)

Single and double precision subroutines for calculating weights for approximating a linear operator using specified nodes can be used for integra-

tion, differentiation, or interpolation. WLFW8S (D)

Single and double precision subroutines for calculating "best" weights corresponding to specified nodes for evaluating a definite integral. QCHEB

Single precision function for evaluating definite integrals by Chebyshev

quadrature (N = 1-7 or 9). KRAB, KRAIN, KRINZ

A group of single precision subroutines using 20-point Gaussian and 41-point Kronrod rules to obtain two estimates of f f(x)dx, f f(x) dx, and Jo f(x)dx, respectively.

;

TRAMID

A single precision function for evaluating a definite integral to a specified tolerance by repeated applications of trapezoid and midpoint rules. SIMPER

A single precision function for evaluating a definite integral to a specified tolerance by use of Simpson's rule and halving step size up to six times. QMGEN

A single precision function for evaluating JQ w(x)f(x)dx when j w(x)dx can be expressed analytically.

274

INTEGRATION FOR ENGINEERS AND SCIENTISTS

QGEOM

A single precision function for evaluating J f(x)dx using points in a geometrical progression. QINDEF

A single precision subroutine for calculating the indefinite integral of a tabulated function. The formula used has an error of order hs. QSHEP

A single precision function using the trapezoid rule and end point corrections (when at least seven points are available) to evaluate the definite integral of a tabulated function.

Appendix 2. INTEGRATION FUNCTIONS AND SUBROUTINES

275

SUBROUTINE GENW8S (X.W.N) C C

C C C C C C

C C C C

CALCULATES WEIGHTS FOR APPROXIMATING A LINEAR OPERATOR USING SPECIFIED NODES XsSINGLY SUBSCRIPTED ARRAY CONTAINING NODES WIISINGLY SUBSCRIPTED ARRAY ORIGINALLY W(l) CONTAINS THE RESULT OF APPLYING THE OPERATOR TO X**(I-1).THIS IS REPLACED DURING EXECUTION BY THE WEIGHT CORRESPONDING TO X111. N%NUMBER OF NODES.THE USE OF GSMELS(D).A SUBROUTINE FOR SOLVING A SET OF LINEAR EQUATIONS-LIMITS N TO 25. FOR DOUBLE PRECISION VERSION ADO DOUBLE PRECISION X.w.RECT DIMENSION X(I) , W(I) . RECT(25.27)

C

C

C C C

C

GENERATE SYSTEM OF LINEAR EQUATIONS DO 10 J-I.N RECT(1.J) 2 1. RECT(J.N+1) a W(J) 00 10 122.N RECT(19J) - RECT(I-I.J)*X(J) 10 SOLVE SYSTEM OF LINEAR EQUATIONS AND PUT RESULT IN W CALL GSNELS (N.RECT.W) FOR DOUBLE PRECISION VERSION SUBSTITUTE CALL GSNELD (N.RECT.W)

C

RETURN END

276

INTEGRATION FOR ENGINEERS AND SCIENTISTS

SUBROUTINE WLFW8D (X.W.N) C

CALCULATES BEST UUAORATURm wt1GhTS (RANG= OTO 1) FOR SPECWFitU NODES ACCORDING TO WILF CRITERION XsSINGLY SUBSCRIPTED ARRAY CONTAINING NOD=S W=SINGLY SUBSCRIPTED ARRAY IN WHICH THE wEIGhTS ARE PUT N-NUMBER OF NOUES.TNE USE OF GSNELS(D).A SUBROUTINE FOR SOLVING

C

C

C C C

C

A SET OF LINEAR ECUATIONS.LI--"ITs N TO 25

DOUBLE PRECISION X.W.RECT FOR SINGLE PRECISION OMIT ABOVE

C

C

DIMENSION X(1)

.

r(1)

. RtCT(2t).47)

C C

C C

10

GENERATE SYSTEM OF LINEAR EQUATIONS DO 10 I=I.N RECT(I.N+)) n -OLCG(l.-x(1))/x(I) FOR SINGLE PRECISION SUSSTITUTE RECT(I.N+)) _ -ALOG(1.-X(I))/x(I) DO 10 J=1.N RECT(I.J) = 1./(1.-X(1)*X(J))

C

C

SOLVE SYSTEM OF LINEAR EQUATIONS AND PUT RESULT

C

CALL GSNELD (N.RE.CT.W) FOR SINGLE PRECISION SUfi:TITUTE CALL GSNELS (N.RECT.W)

C C

IN W

277

Appendix 2. INTEGRATION FUNCTIONS AND SUBROUTINES FUNCTION CCHEE (A.b.F.,N.N) LVALJATION OF DEFINITE INTEGRAL oY CHC=YSHLV RVLc. A=LONCR LI("IT OF INTEGRAL B=UPPER LIMIT OF INTtC.RAL FUN=EXTERNAL F UNCTION DEFINING INTEGR4NC N=ORDER OF ROLE. ONLY 1-7 AND 9 A@E AL-o.ED. INT=GRAT ION TS EXACT FOR POLYNOMIALS OF DEGREE V.

C

C C C C C

S=B-A OCHER=O.0

EXITS FOR IMPROPER N IF(N LE. 0 OR. N GT. 9) OC TO 8 HANDLES CENTRAL NODE FOR UDU N. IF(2*(N/2) NE. N) GCHEB-FUN(A+.S*S) USES COMPUTED GO TO STATEMENT TO REAC-1 CORRECT F3RMULA GO TO (1.2.3.4.5.6.7.8.9).N

C C

C C

OCHEB=FUN(A+.78867513*S)+FUN)B-.78867513*S) GO TO UCHEb=FON(A+.59379624*S)+FUN(B-.59:i79624*S) I +FUN(A+.89732724*5)+FUNIB-.89732724*S) GO TO GCHEB=FUN(A+.63331770*S)+FUN(9-.6333I77O*S) +FUN(A+.71125933*5)+FUNIB-.71125933*S)

2

1

4

1

6

I

2

+FUN(A+.93312341*S)+FUN(6-.93312:.(41*S) GO TO EXIT FOR IMPROPER N PRINT 100,N FOQMAT)16.SIH IS AN IMPROPER VALUE OF N.ONLY TO 7 AND 9 ARE OK) RETURN OCHER=FUN(A+.85355339*S)+FUNIB-.853bb339*S) + OCHEO GO TO 1 I

C 8 103)

1

3 5

GCHEB=FUN(A+.68727O7U*S)+FUN(8-.6877070*S) + OCHEb +FUNIA+.91624774*S)+FUN(B-.91624874*S) GO TO GCHEB=FUNIA+.66195b91*S)+FUN(b-.66795591*S) + OCHtkb I +FUNIA+.76482839*S)+FUN(8-.76482839*S) +FUN(A+.94193085*S)+FUNIB-.94193085*S) 2 GO TO OCHER +FUN(A+.76438089*5)+FU4(8-.76438089*S) 1 +FUN(A+.800b0933*S)+FUN(d-.8C050933*S) 2 1

1

7

1

9

3 C C I

OBTAIN FINAL VALUE BY MULTIPLYING BY 4ANGL OVER NUM9MR OF POINTS OCHEB=S*GCHEB/FLOAT(N) RETURN END

SENTRY SIBSYS

278

INTEGRATION FOR ENGINEERS AND SCIENTISTS

SUBROUTINE KRAB( C C

SUBROUTINE FOR KRONROD OUADRATURE (N=20) OF AN INTEGRAL FROM A TO B. A=LOWER LIMIT OF INTEGRAL 5-UPPER LIMIT OF INTEGRAL GRAND=EXTERNAL FUNCTION DEFINING INTEGRAND. VK=VALUE OF INTEGRAL USING 41 NODES VG=VALUE OF INTEGRAL USING 20 NODES

C

C C C C C

DIMENSION A8B(2O).WG8(10).'WKB(20) DATA WGB / .41638371E-1 .31336024E-1 .71048055E-1 .65844319E-1 a DATA WKB / .73130846E-2 1 .10194187E-1 2 .18300085E-1 .20834437E-1

.88070036E-2 .50965060E-1 .74586493E-1 .15367919E-2 .12941067E-1 .23217411E-1 .31326619E-1 .36515345E-1

1

3

27597553E-1

29555700E-1

.34324336E-1 .35527212E-1 .38188934E-1 / .37852249E-1 DATA ABB / 0.5704821E-3 . 0.34357004E-2 .92460613E-2 .18014036E-1 . .29588683E-1 2 .60861594E-1 .80441514E-1 .10247929 3 .15338117 .18197315 .21242977 4 .27820341 .31314696 .34918607 5 .42369726 .46173574 / 4

. .

. .

.20300715E-1 . .59097266E-1 . .76376694E-1 / .43001349E-2 .15643653E-1 .25472287E-1 .32917299E-1 .37291438E-1

5 I

.

C 5

S-B-A VG=0.0 VK=.38300356E-I*GRAND(A+.5*S)

C I

15

10

DO 10 J=1.20 ADD=S*ABB(J) ADO=GRAND(A+ADD)+GRAND(6-ADO) VK=VK+WKB(J)+ADD JJ-J/2 IF(J.EO.2*JJ) VG=VG+WGB(JJ)*ADO CONTINUE

C

VK-VK*S VG=VG*S RETURN END

. .

.43882786E-1 .12683405 .24456650 .38610707

279

Appendix 2. INTEGRATION FUNCTIONS AND SUBROUTINES

C C C C

C C C

SUBROUTINE KRAIN (A.GRAND.VK.VG) SUBROUTINE FOR KRONROO OUADRATURE (N=20) OF AN INTEGRAL FROM A TO 1NFINITY.A MUST BE POSITIVE. A-LOWER LIMIT OF INTEGRAL GRAND-EXTERNAL FUNCTION DEFINING INTEGRAND.DGFINt IT AS ZERO FOR ARGUMENTS EXCEEDING SCOPE OF LIBR/RY FUNCTIONS VG-VALUE OF INTEGRAL USING 20 NODES VK-VALUE OF INTEGRAL USING 41 NODES

C

DIMENSION ABB(20).WGB(10).WKB(20) DATA WGB / .88070036E-2 .41638371E-1 .50965060E-1 1 .31336024E-1 .71048055E-1 .74586493E-1 2 .65844319E-1 DATA WKB / 15367919E-2 .10194187E-1 .73130846E-2 . .12941067E-1 I .20834437E-1 2 .18300085E-1 .23217411E-1 3 .27597553E-1 .29555700E-1 .31326619E-1 4 .34324336E-1 .35527212E-1 .36515345E-1 .38188934E-1 / 5 .37852249E-1 DATA ABB / 0.5704821E-3 0.34357004E-2 .18014036E-1 1 .92460613E-2 .29588683E-1 2 .60861594E-1 .80441514E-1 . .10247929 .18197315 3 .21242977 15338117 4 27820341 .34918607 .31314696 .46173674 / 5 .42369726

.

.

.

.

.

.

.

.20300715E-1 .59097266E-1 .76376694E-1 / .43001349E-2 .15643653E-1 .25472287E-1 .32917299E-1 .37291438E-1

.

.

-

C

vG - 0.0 VK - .15320142*GRAND(2.*A) C

10

DO 10 J-1.20 TJ - ABB(J) TI - I.-TJ ADD - GRAND(A/TJ)/TJ**2 + GRANO(A/T1)/TI**2 VK - VK + WKB(J)*ADO JJ - J/2 IF iJ E0. 2*JJ) VG - VG + WGB(JJ)*ADD

C

VK VG

VK*A VG*A RETURN END

.43882786E-1 .12683405 .24456650 .38610707

INTEGRATION FOR ENGINEERS AND SCIENTISTS

280

SUBROUTINE KRINZ (S.GRAND.VK.VG) SUBROUTINE rOR KRONROD QUADPATUPE IN=20) OF AN INTEGRAL FROM ZERO TO INFINITY. S=SCALE FACTOR.MUST BE POSITIVE. GRAND=EXTERNAL FUNCTION DEFINING INTEGRAND.DEFINE IT AS ZERO FOR ARGUMENTS EXCEEDING SCOPE OF LIBRARY FUNCTIONS. VK=VALUE OF INTEGRAL USING 82 NODES VG=VALVE OF INTEGRAL USING 40 NODES

C C C C

C C C

C

DIMENSION ABB(20).WGB(10).WK8(20) DATA WGB / .31336024E-1 . .41638371E-1 .710480'55E-1 2 .65844319E-1

.88070036E-2 . .50965060E-1 .74586493E-1 .15367919E-2 . .12941067E-1 . .23217411E-1 .31326619E-1 . .36515345E-1 .

I

DATA 81(8

/

.73130846E-2 .10194187E-1 .20834437E-1 2 .18300085E-1 .29555700E-1 3 .27597553E-1 4 .34324336E-1 .35527212E-1 5 .37852249E-1 . .38188934E-1 / DATA A68 / 3.5704821E-3 . 0.34357004E-2 .18014036E-1 . .29588683E-1 .92460613E-2 .80441514E-1 2 .6C861594E-1 .10247929 3 .15338117 .21242977 .18197315 4 .27820341 .31314696 .34918607 5 .42369726 .46173674 / .

1

. . .

.

.

I

.

. . .

.20300715E-1 . .59097266E-1 . .76376694E-1 / .43001349E-2 .15643653E-t

25472287E-1 .32917299E-1 .37291438E-1 .43882786E-1 .12683405 .24456650 .38610707

C

VG = 0.3 VK = .38300356E-I * C

DO 10 J=1.20 Ti - ABB(J) TI - I.-TJ ADO - GRANO(S*TJ)+GRAND(S*TI)+GRAND(S/TJI/TJ**2+GRAND(S/TI)/TI**2 VK = VK+WKB(J)*ADD JJ = J/2 10

IF (J E0. 2*JJ) VG = VG + WGB(JJ)*ADD

C

VK = VK * S VG - VG * S RETURN END

Appendix 2. INTEGRATION FUNCTIONS AND SUBROUTINES

281

FUNCTION TRAM IO(A.b.FUN.N.NSF) C C

ADAPTIVE EVALUATION OF DEFINITE INTEGRAL USING MIDPOINT AND TRAPEZOID RULES ANDS HALVING STEP SIZE A=LOWER LIMIT OF INTEGRAL B=UPPER LIMIT OF INTEGRAL FUN=EXTERNAL FUNCTION DEFINING INTEGRAND N=INITIAL NUMBER OF DIVISIONS NSF=NUMBER OF SIGNIFICANT FIGURES DESIRED

C.

C C C C C

C

REAL MIDP H=(B-A)/FLOAT(N) NUP=N-1 TRAP=.5*(FUN(A)+FUN(B) 00 10 J=1.NUP TRAP = TRAP + FUN(A+FLOAT(J)*H) TRAP=H*TRAP

10

C

SETS TOLERANCE USING NSF AND VALUE CALCULATED BY TRAPEZOID RULE WITH N DIVISIONS TOL = .1**NSF*ABS(TRAP) IF ESTIMATED VALUE IS SMALL RELATIVE TO TOLERANCE EXIT WITH WARNING THAT VALUE MAY BE ZERO. IF (TOL .LT. .01**NSF) GOTO 40

C C C C C

DO 33 NC = 1.6 MIDP = FUN(A+.5*H) DO 20 J=1.NUP 20 MIDP = MIOP + FUN(A+(.5+ FLOAT(J))*H) MIOP=H*MIDP DEL=ABS(MIDP-TRAP) IF(DEL.LT.TOL) GO TO 30 H=.5*H NUP=2*NUP+1 TRAP=.5*(TRAP+MIOPI 33 CONTINUE C C

REPORTS TOLERANCE WAS NOT MET WRITE(6.27) FORMAT(5X,37HTOLERANCE NOT MET WITH 64*N DIVISIONS/)

25 27 C

C

30

C

MAKES RESULT SIMPSONS RULE TRAMID = (2.*TRAP + MIDP)/3. RETURN

EXIT FOR NEAR ZERO RESULT 40 TRAMID = TRAP WRITE(6.100) IOC FORMAT( IX,I9HINTEGPAL MAY VANISH/) RETURN END

282

INTEGRATION FOR ENGINEERS AND SCIENTISTS

FUNCTION SIMPER(A.B.FUN.N.NSF) C

ADAPTIVE EVALUATION OF DEFINITE INTEGRAL USING RULE AND HALVING STEP SIZE A-LOWER LIMIT OF INTEGRAL B-UPPER LIMIT OF INTEGRAL FUN=EXTERNAL FUNCTION DEFINING INTEGRAND N=!NITIAL NUMBER OF DIVISIONS NSF=NUMBER OF SIGNIFICANT FIGURES DESIRED

C C C C.

C C C C

H=.S*(B-A)/FLOAT(N) NMI-N-I NUP-N SEND=FUN(A)+FUN(B) SOD-FUN(A+H) SEVEN=O. DO 10 J=1.NMI SOD + FUN( A+H*FLOAT(2*J+I)) SOD C 10

C C

SEVEN - SEVEN +FUN(A+H*FLOAT(2*J)) PREVAL=H*(SEND+4.*SOD+2.*SEVEN)/3.

SETS TOLERANCE USING NSF AND VALUE CALCULATED WITH N DIVISIONS TOL = .1**NSF*ABS(PREVAL) EXITS FROM DO LOOP IF TOLERANCE IS MET IF (TOL LT. .01**NSF) GOTO 40

C C

20

30

DO 33 NC = 1.6 H=.5*H SEVEN=SEVEN+500 SOD=0. NUP=2*NUP DO 30 J=1.NUP SOD = SOD + FUN(A+H*FLOAT(2*J-1)) VAL=H*(SEND+4.*SOD+2.*SEVEN)/3. CORR=(VAL-PREVAL)/15. ACOR=ABS(CORR)

C.

IF (ACOR.LT. TOL) GO TO 35 PREVAL - VAL 33 CONTINUE C

REPORTS TOLERANCE WAS NOT MET WRITE(6.51) 51 FORMAT(2X.37HTOLERANCE NOT MET WITH 64*N DIVISIONS/) C

50 C C

ADDS CORRECTION TERM WHICH MAKES RESULT HIGHER ORDER 35 SIMPER=VAL+CORR RETURN

C C

EXIT FOR NEAR ZERO RESULT 40 SIMPER = PREVAL WRITE(6i100) 100 FORMAT(1X419HINTEGRAL MAY VANISH/) RETURN END

Appendix 2. INTEGRATION FUNCTIONS AND SUBROUTINES

283

FUNCTION OMGEN( A,6,FUN.WINT.NI C

C

C C C

C C C

C C C

GENERALIZE MIDPOINT RULE FOR DEFINITE INTEGRALS WITH INTEGRABLE WEIGHT FUNCTION LIMIT OF INTEGRAL B=UPPER LIMIT OF INTEGRAL FUN-EXTERNAL FUNCTION DEFINING INTEGRAND WINT=EXTERNAL FUNCTION DEFINING INDEFINITE INTEGRAL OF WEIGHT FUNCTION. CONSTANT OF INTEGRATION IS ARBRITRARY AS ONLY DIFFERENCE OF TWO VALUES ARE USED. N.NUMBER OF NODES TO BE USED 5=.5*(B-A)/FLOAT(N) GMGEN=O.O

C

00 10 J.I.N ARG=FLOAT(2 J-1)*S WU=WINT(ARG+S) GMGEN=OMGEN+(WU-WL)*FUN(ARG) 0

SQL-WU

C

RETURN END

284

INTEGRATION FOR ENGINEERS AND SCIENTISTS

FUNCTION OGEOM (A.GRAND.S.RAT.N) C

EVALUATION OF INTEGRAL OVER SEMI-INFINITE RANGE BY GOODwIN-MORAN METHOD USING POINTS IN A GEOMETRIC PROGRESSION A.LOWER LIMIT OF INTEGRAL.IT CAN BE PCSITIVE.NEGATIVE OR ZERO. GRAND=EXTERNAL FUNCTION SPECIFYING INTEGRAND S=POSITIVE NUMBER SERVING AS SCALE FACTOR RAT=POSITIVE NUMBER (NOT I.) USED AS FACTOR IN GEOMETRIC PRRCRESSION N-NUMBER OF NODES (ACTUALLY 2*N+1 NODES ARE USED) THE FUNCTION GRAND IS EVALUATE AT POINTS BETWEEN A+5*RAT**N ANQ A+S/RAT**N.SELECT VALUES TO COVER SIGNIFICANT RANGE.DEFINE GRAND TO GUARD AGAINST UNDERFLOWS.

C C

C C C C C C C C C

RP = S RM = S OGEOM = S*GRANO(A+SI C

30

DO 30 J=I.N RP s RP*RAT RM = RM/RAT OGEOM - OGEOM + RP*GRAND(A+RP) + PM*GRAND(A+RM) OGEOM = OGEOM * ASS(ALCG(RAT11

C

RETURN END

285

Appendix 2. INTEGRATION FUNCTIONS AND SUBROUTINES SUBROUTINE OINDEF(RUN.RINT.N.H) C C C C C C C C

SUBROUTINE FOR CALCULATING INDEFINITE INTEGRAL OF EQUALLY SPACED DATA.ERROR IS OF ORDER H**S.PEOUIRES 4 OR MORE POINTS. RUN=SINGLY SUBSCRIPTED ARRAY CONTAINING DATA PINT=SINGLY SUBSCRIPTED ARRAY IN WHICH VALUES OF INDEFINITE INTEGRAL ARE PUT.RINT(l) MUST HAVE INTEGRATION CONSTANT N=NUMBER OF DATA POINTS H=SPACING

C

DIMENSION RUN(N).RINT(N) S=H/24. T-13.*S C C

SPECIAL FORM FOR FIRST INTERVAL RINT(2).S*(9.*RUN(I1+19.*RUN(2)-5.*RUN(3)+RUN(41)

+PINT(1)

C

NM-N-I 10 C C

DO 10 J=3.NM RINT(J)=T*(RUN(J)+RUN(J-1))-5*(PUN(J-2)+PUN(J+l))

+RINTIJ-I)

SPECIAL FORM FOR LAST INTERVAL R!NT(N)=S*(9.*RUN(NI+19.*RUN(N-I1-5.*RUN(N-2)+RUN(N-3))+RINT(N-I)

C

RETURN END

286

INTEGRATION FOR ENGINEERS AND SCIENTISTS

FUNCTION OSHEP(RAY.N.H) C

C C C C

C C

EVALUATE INTEGRAL OF A TASULATEU FUNCTION BY TRAPEZOID RULE AND ENDPOINT CORRECTION WHEN MOPE THAN 7 POINTS ARE GIVEN RAY=SINGLY SUBSCRIPTED VARIABLE CONTAINING DATA N-NUMBER OF POINTS H=SPACING

C

DIMENSION RAY(I) OSHEP-.5*(RAY(1)+RAY(N)) M=N-) IF(N.LT.7) GO TO 15 C C

CORRECTION TERMS OSHEP.0SHEP+FLOAT (M*(763*M*M-3444*M+3636))/FLOAT((M-))*(M-2)*(M-3) 1*5040)*(RAY(2)-KAY(l)-RAYIN)+RAY(N-1)) OSHEP=OSHEP-FLOAT(M*(119*M*M-504*M+432))/FLOAT((M-2)*(M-3)*(M-4) 1x1260)*(PAY(3)-RAY(2)-WAY(N-I)+RAY(N-2)) OSHEP=OSHEP+FLOAT(M*(= 13*M*M-462*M+36O))/FLOAT((M-3)*(M-4)*(M-5) 1*5040)*TRAY(4)-RAY(3)-RAY(N-2)+RAY(N-3))

C 15 10

DO 10 J=2.M OSHEP-OSHEP+RAY (J)

C

OSHEP=M*OSHEP RETURN END SENTRY

Appendix 3

SUBROUTINES FOR SOLVING INTEGRAL EQUATIONS VOLTLI

Solution of linear Volterra equations of the second kind

u(x) = f(x) + J; G(x, t)u(t)dt by an explicit procedure using one-point product integration VOLTL2

Solution of linear Volterra equations of the second kind using two-point Gaussian quadrature. VOLTGI

Solution of general Volterra equations of the second kind

u(x) = f(x) + J K(x, t, u(t))dt by a one-step procedure analogous to Euler's method for differential equations. FREDGI

Solution of general Fredholm equations of the second kind

u(x) = f(x) + A I' K(x, t, u(t))dt a by succesive approximations and Aitken extrapolation; integration is by compound midpoint rule.

288

INTEGRATION FOR ENGINEERS AND SCIENTISTS

SUBROUTINE VOLTLI (A.B.FOR.FINT.NP.U.X) C C

aU6ROOTINE FOR SOLVING LINEAk VOLTERRA EOUATION (SECOND KIND) USING PRODUCT INTEGRATION A=LOtiER LIMIT OF INTEGRAL b=uaPEP VALUE OF INDEPENDENT VARIABLE FOR CALCULATION FOR=EXTERNAL FUNCTION(ONE AkGUMENT) DEFINING FORCING FUNCTION FINT=EXTERNAL FUNCTION(TWO ARGUMENTS).INDEFINITE INTEGRAL OF KERNEL WITH RESPECT TO DUMMY VARIABLE U=SINGLY SUBSCWIPTED VARIABLE FOR SOLUTION X=SINGLY SUBSCRIPT_U VARIABLE FOR CORRESPONDING VALUES OF INDEPENDENT VARIABLE

C C C C

C C C

C C C

DIMENSION U(1).X(1) S=(A-A)/FLOAT (2*NP) C

1U

DO 10 I=1.NP X(I)=FLOAT(2*I-1)*S U(I)=FOR(X(I)) U(1)=U(1)/(1.-FlNT(A+b.A+S+S)+FINT(A+S.A)l

C

CO 30 I=2.NP N=I-1 20 3U

00 20 J=1.N U(I)=U(I)+U(J)*(FINT(X(I).X(J)+S)-FINT(X(I).X(J)-S)) Ut;)=UII)/(1.- F INT(X(j).Xtl)+S)+F INT(X(I).X(I)-S))

C

RETURN END SENTRY 51BSYS

Appendix 3. SUBROUTINES FOR SOLVING INTEGRAL EQUATIONS

SUBROUTINE VOLTL2 (A.B.FOR.FGPEEN.NP.U.X) C

SOLUTION LINEAR INTEGRAL EOUATION(SECOND KIND) BY TWO.STEP METHOD USING GAUSSIAN NODES A=LOWER LIMIT OF INTEGRAL B=MAXIMUM VALUE OF INDEPENDENT VARIABLE FOR=EXTERNAL FUNCTION (1 ARGUMENT) DEFINING FORCING FUNCTION FGREEN=EXTERNAL FUNCTION (2 ARGUMENTS) DEFINING KERNEL NP=NUMBER OF SOLUTION POINTS.SHOULD BE EVEN X-SINGLY SUBSCRIPTED ARRAY FOR INDEPENDENT VARIABLE DIMENSION U(1) X(1) H - 2.*(B-A)/FLOAT(NP) wII = .25*H w21 = .53867513*H W12 =-.03867513*H NU - NP/2

C C C

C C

C C

C

.

C

DO 10 J=1.NU XODD = H*(FLOAT(J)-.78867513) XEVE = H*(FLOAT(J)-.21132487) X(2*J-1) = XODD X(2*J) = XEVE Fl = FOR(XODD) F2 = FOR(XEVE) SUMI = 0.0 SUM2 = 0.0 IF (J EO. 1) GO TO 15 C C

SUM PREVIOUS VALUES NSUM = 2*(J-1) DO 12 K=1.NSUM

C

SUM) = SUMI + FGREEN( XODD.X(K))*U(K) SUM2 = SUM2 + FGREEN(XEVE.X(K))*U(K) C SET UP AND SOLVE 2 LINEAR EQUATIONS Fl - FI + .5*H*SUM1 15 F2 = F2 + .5*H*SUM2 Cil - W1)*FGREEN(XOOD9XOOD) C12 2 W12*FGREEN(XODD.XEVE) C21 = W2)*FGREEN(XEVE.XODD) C22 = W11*FGREEN(XEVE.XEVE) D - (1.-Cll)*(1.-C22) + C12*C21 U(2*J-I) = (FI*(I.-C22 + C12*F2)/O U(2*J) (F2*(1.-CI! 10 + C21*Fl)/D RETURN END 12

) )

289

INTEGRATION FOR ENGINEERS AND SCIENTISTS

290

SUBROUTINE VOLTGI (A.b.FOR.FKERN.NP.U.X) C C C C C C C

SOLUTION OF GENERAL VOLTERRA EQUATION (SECOND KIND) BY A ONE STEP METHOD ANALAGOUS TO THE SIMPLE EULER METHOD A=LOWER LIMIT CF INTEGRAL B=MAXIMUM VALUE OF INDEPENDENT VARIABLE FOR=EXTERNAL FUNCTION () ARGUMENT) DEFINING FORCING FUNCTION FKERN=EXTERNAL FUNCTIUN(3ARGUMENTS) DEFINING INTEGRAND NP=NUMBER OF SOLUTION PO)NTS.SHJULD BE EVEN U=SINGLY SUBSCRIPTED ARRAY FOR SOLUTION X=SINGLY SUBSCRIPTED ARRAY FOP INDEPENDENT VARIABLE

C C C C

DIMENSION U(NP).X(NP) H=(B-A)/FLOAT(NP-1) X(I)=A

u(l)=FOR(A) C

00 20 J=2.NP U(J)=FKERN(X(J).A.U11)) IF(J.EO.2) GO TO 20 JJ=J-I 25 20

00 25 K=2.JJ U(J)=U(J)+FKERN(X(J).X(K).0(K)) U(J)=FOR(X(J))+H+U(J)

C

RETURN END

Appendix 3. SUBROUTINES FOR SOLVING INTEGRAL EQUATIONS

291

SUBROUTINE FREDGI (A.E.AMBA.FOR.FKERN.NP.NI.IPRINT.U) C C

SOLUTION OF INTEGRAL EQUATION BY MIDPOINT RULE AND AITKEN EXTRAPOLATION A-LOWER LIMIT OF INTEGRAL B-UPPER LIMIT OF INTEGRAL AMBAsPARAMETER OF THE INTEGRAL EQUATION FOR=EXTERNAL FUNCTION 11 ARGUMENT) DEFINING FORCING FUNCTION FKERN=EXTERNAL FUNCTION I3ARGUMENTS) DEFINING INTEGRAND NP=NUMBER OF DIVISIONS TO BE USED NI=NUMBER OF ITERATIVE CYCLES IPRINT=OPTION CONTROL.IF NOT 0 INTERMEDIATE RESULTS ARE PRINTED U=DOUBLY SUBSCRIPTED VARIABLEtNP.3).AFTER EXECUTION SOLUTION IS IN FIRSTCOLUMN.NODES IN SECOND

C C

C C

C C C C

C C C

DIMENSION U(NP.3) X(I) IS FUNCTION DEFINING NODES X(I).A+S*(FLOAT(I)-.5)

C

C C

BEGIN EXECUTION BY CALCULATING SPACING AND WEIGHT S-(B-A)/FLOAT(NP) W AMBA*S C SET INITIAL APPROXIMATION IN FIRST COLUMN OF U DO 20 J=).NP U(J.1) = FOR(XIJ)) 20 C C

DO LOOP FOP NUMBER OF ITERATIONS DO 60 N-I.N) CALCULATE SECOND AND THIRD COLUMN BY NUMERICAL INTEGRATION 00 50 J=2.3 00 50 I=I.NP U(I.J)-FOR(X(I)) 00 50 K=I.NP

C

50

102 60 C C

40

U IF IPRINT 15 NOT IMPROVED VALUE IN FIRST COLUMN 00 60 I=1.NP IF)IPRINT.EO.O) GO TO 60 WRITE(6.102) FORMAT(3E20.6) U11.i)=U(I.1)-lU(I.ll-U(1.2))**2/(U(I.I)-2.*Vt1.21+Utt.3))

PUT VALUES OF NODES IN SECOND COLUMN OF U DO 40 lml.NP U11.2)-X(I)

C

RETURN END

AUTHOR INDEX

Numbers set in italics designate those page numbers on which the complete literature citations are given.

Abarbanel, S. S., 256, 262 Abel, N., 18, 35, 130

Abramowitz, M., 75, 76, 111, 120, 121, 169, 185

Agnew, R. P., 7, 34 Anders, E. B., 268 Anderson, D. G., 194 Anvari, M., 267 Archimedes, 1 Arljuk, T., 219 Ashley, H., 260 Bahar, L. Y., 268 Baker, H. K., 201 Bareiss, E. H., 261 Barnhill, R. E., 268 Batten, G. W. jr., 269 Beard, R. E., 192 Beckman, J. A., 271 Bellman, R., xi, 193, 194, 220, 262 Benesch, S. E., 267 Bernkopf, M., 268 Best, C. L., 261, 271 Bethel, H.E., 271 Bisplinghoff, R. L., 260 Beltyukov, B. A., 261 Berg, L., 122 Bernkopf, M., 270 Bernstein, S. N., 139, 152, 190, 191 Bernoulli, J., 227 Bierens de Haan, D., 78, 120 Bird, R. B., 36 Blackman, R. B., 193 Boas, R. P., jr., 122 Boley, B. A., 112, 113, 123 Boley, S. R., 112, 113, 123 Boyer, C. B., 34

Brant, D. N., 256

Bronshtein, I. N., 34 Brosamler, G. A., 270 Buck, C. C., 267 Buckingham, E., 123 Buckner, H., 263 Bushkovitch, A. V., 35 Cawley, R. G., 270 Ceschino, F., 219 Chambre, P. L., 262 Channapragada, S. R., 268 Chartier, J., 17 Chebyshev, P. L., 22, 55, 57, 132, 190 Childs, D. R., 88, 121 Clenshaw, C. W., 127, 190 Coleman, S. H., 267 Collatz, L., 260 Cook, W. J., 189 Costeldanos, D., 270 Costley, C. G., 269 Cox, R. H., 271 Curtiss, A. R., 127, 190 Curtiss, C. F., 36

Darst, R. B., 267 Davids, N., xi Davis, H. T., 35, 121, 239, 260, 261

Davis, P. J., ix, 158, 161, 162, 189, 192, 194, 200

Deacon, H. J. jr., 271 DeBruijn, N. G., 111, 122 Decell, H. P., 157, 192 Deckert, K. L., 271 Dejon, B., 260 de Lamadrid, J.G., 267 Diaz, J. B., 37 Dirichlet, P. G. L., 17, 20, 82 Ditkin, V. A., 122

AUTHOR INDEX

294

Dorrak, J. R., 268 Douglas, J. jr., 26! Dufton, A. F., 135, 190 Edwards, J., 62, 75, 121, 122 Ehrman, J. R., 271 Eisner, E., 181, 193 Elliot, D., 262 Elliot, E. B., 103, 105, 106, 122 Eminton, E., 194 Emmons, H., 261 Engels, H., 193 Erdelyi, A., 122 Esnault-Pelterie, R., 123 Euclid, 2 Euler, L., 55, 79, 121, 172, 220

Farrel, O. J., 121 Felderman, E. J., 189 Fillipi, S., 193 Filon, L. N. G., 170, 171, 192 Fithian, J. H., 269 Fodor, G., 122 Forsythe, G. E., 262 Fox, L., 220 Frady, C. S., 267 Frank, I., 206 Franklin, F., 35 Franklin, P., 34 Fuller, F. B., 35

Garth, W. W. jr., 267 Gate, L. D. jr., 195 Gautschi, G., 192 Gay, B., 262 Geller, M., 121 Giddens, D. P., 168 Glaisher, J. W. L., 96 Goodwin, B. E., 270 Goodwin, E. T., 193 Gould, H. W., 122, 220 Gould, S. H., 220 Goursat, E., 58 Gradshteyn, I. S., 120 Gray, P., 76 Grobner, W., 75, 120 Grossman, K. G., 266 Hadamard, J., 35 Halfman, R. L., 260 Hall, H. S., 121

Hammer, P. C., 219 Hammersly, H. M., 191 Hamming, R. W., 36, 190, 200, 219 Handelman, G. H., 213,220 Handscomb, D. C., 191

Hardy, G. H., i, 48, 51, 52, 75, 101-102, 121, 122

Harper, M. J., 76 Hart, J. F., 76, 194 Hastings, C., 76 Heaslet, M. A., 35 Hedrich, 0., 58 Heisenberg, W., 221-222 Henstock, R., 35 Hermite, C., 50, 62, 75 Herod, J. V., 210 Herrebrugh, K., 254 Hershey, H. C., 194 Hess, J. L., 260 Hetherington, R. G., 267 Hildebrand, F. D., 260 Hildebrandt, T. H., 35 Hills, B. A., 261 Hilzman, J., 269 Hinchey, F. A., 269 Hinton, D. B., 269 Hippocrates, 2 Hirschfelder, J. O., 36 Hofreiter, M., 75, 120 Holt, J. M., 272 Howerton, R. J., 194 Huang, A. B., 168 Hudson, J. P., 263

Igoe, W. B., 194 Imhof, J. P., 219 Irwin, J. 0., 189, 193 Isayama, S., 35 Itkin, 1., 270

Jackson, D., 192 Jain, M. K., 261 Jammer, M., 261 Jeffreys, B., 34 Jeffreys, H., 34 Johansen, E. L., 270 Johnson, B. C., 269 Juncosa, M. L., 238 Kahane, C. S., 271 Kahn, H., 191

AUTHOR INDEX

296

Kalaba, R. E., 193, 220, 262 Kallman, R. A., 271 Kantorovich, L. V. 162, 192, 240, 260, 262 Khan, A. A., 268 Kharadze, A. K., 35 King, L. V., 76 Kinney, E. K., 270 Klamkin, M., 34, 36, 43 Klein, F., 122 Knight, S. R., 121 Kobayashi, E. T., 267 Kopal, Z., 192, 260

Martin, W. A., 76 May, K. 0., 3S McFarland, J. E., 269 McKelvey, R. W., 250, 262 McShane, E. J., 35, 36 Meir, A., 190 Mellers, H. A., 96, 122 Mendenhall, R. V., 267 Menger, K., 36 Merchant, C. F., 190 Metcalf, F. T., 37 Michael, W. A. jr., 267

Kronrod, A. S., 136-138, 154, 190

Mikeladze, Sh. E., 148, 179, 189, 191, 193,

Krylov, V. I., ix, 158, 180, 189, 193, 219, 240, 260, 262

Kuelbs, J. D., 267 Kuntzam, J., 219 Kuznetsov, Y. S., 263

Lambert, J. D., 219 Lanczos, C., 142, 191, 197, 200, 204, 219, 221, 261

Langer, R. E., 220 Laplace, P. S., 45 Lass, H., 34 Lea, R. N., 157, 192 Lebesgue, H., 3, 35 Lees, S., 261 Legendre, A. M., 35

Leibniz, G. W., I Lewis, E. E., 270 LindelOf, E. L., 121 Lindman, C. F., 120 Linz, P., 246 Liouville, J., ix, 40-42, 46, 63, 75, 122 Lockett, J., 193, 262 Lomax, H., 35 Lommel, E., 69 Longman, 1. M., 173, 193 Lorentz, G., 191 Lotkin, M., 219 Lovitt, W. V., 237, 260 Luke, Y. L., 120 Lyness, J. N., 191, 220 Mackenzie, J. K., 123 Maclaurin, C., 206, 220 Macneish, H. F., 75 Macoll, H., 36 Makinson, G. J., 260 Maloney, J. P., 272 Margenaw, H., 261

219

Mikhlin, S. G., 190,260 Miller, J. C. P., 190 Minding, F., 181 Mineur, H., 192 Monacella, V. J., 161, 192 Mitchell, A. R., 219 Moon, P., 31, 36, 262 Moran, P. A. P., 177, 193, 206 Mordukhay-Boltovskoi, D., 75 Morgenstern, 0., 190 Morris, W. L., 12-13, 35 Morteza, A., 267 Moses, J., 67, 123 Murphy, G. M., 261

Nagaraja, K. S., 263 Neis, V. V., 271 Nelson, E. 0., 271 Nestell, M. K., 268 Neuberger, J. W., 270 Neumann, C. P., 261 Newburg, E. A., 269 Newman, D. J., 34 Newman, 1., 34 Newton, 1., 1, 125 Ninham, B. W., 220 Norden, H. V., 178, 193 Nikolskii, S. M., 189, 191 Obreschkopf, M. V., 219 Ostragradsky, M., 50, 75 Ostrowski, A., 75, 100, 122 Ovchinsky, B. V., 263 Ower, E., 136, 190 Page, C. H., 35 Pankhurst, R. C.. 190 Peirce, W. H., 268

AUTHOR INDEX

296

Petit Bois, G., 75 Petukhov, I. V., 219 Pincus, M. M., 270 Pinkham, R. S., 200, 219 Pogorzelski, W., 260 Polivka, R. P., 269 Polya, G., 139, 191 Prasad, J., 219 Price, J. F., 173, 182, 186, 193 Price, P. C., 136, 190 Prudnikov, A. P., 122 Pukk, R. A., 191 Rabinowitz, P., ix, 161, 162, 189, 192, 194 Rail, L. B., 157, 192 Ralston, A., 136, 146, 191 Rasulov, M. L., 121 Rayleigh, Lord, 231 Reneke, J. A., 270 Richardson, W., 123 Riemann, B., 4 Risch, R. H., 49, 75 Ritt, J. F., ix, 40, 46, 75 Rizzo, F. J., 271 Rolwing, R. H., 271 Romberg, W., 149, 191 Robinson, S. M., 252, 262 Rosenberg, L., 191 Ross, B., 121 Rufener, E., 121 Runge, C., 139, 190 Ryzhik, I. M., 120 Sard, A., 191 Sarma, V. L. N., 267 Saylor, R. L., 271 Schdnberg, I. J., 197, 219 Schoenfeld, L., 122 Schrodinger, E., 222 Secrest, D., 157, 190, 192 Semendyayev, K. M., 34 Sharma, A., 190, 219 Sharma, K. D., 261 Sheldon, E. W., 120 Sheppard, W. F., 199 Shisha, 0., 37 Shohat, J. A., 35 Shucker, S., 246-247, 260, 268 Simha, R., 194 Skoug, D. K., 268 Slagle, J., 63, 76 Slobin, H. L., 95

Smirnov, V., 34, 260 Smith, A. M. 0., 171, 193, 260 Smith, J. J., 35 Smith, L. P., 34, 121, 260 Smolitskiy, K. L., 190, 260 Sneddon, 1. N., 122 Somerville, W. H., 270 Sormani, G. A., 268 Speilberg, S. E., 271 Spencer, D. E., 31, 26, 262 Squire, W., 76, 181, 193, 204, 219, 220 Steffensen, J. F., 127 Stegun, I. A., 75, 76, 120, 169, 185 Stewart, A. D., 268 Stormer, E., 267 Strimling, W. E., 271 Stroud, A. H., 189, 190, 192, 193, 251, 262 Struble, G. W., 167, 192 Sweigert, P., 192 Sundresan, M. K., 269 Swann, D. W., 296 Synge, J. L., ix Szeqo, G., 192

Takeyama, H., 204, 219 Talbot, W, R., 194 Toeplitz, 0., 34 Tomlinson, J. M., 270 Tranter, C. J., 122, 261 Traub, J. F., 190 Tricomi, F. G., 260 Tu, Y. 0., 213, 220 Tuck, E. 0., 193 Tukey, J. W., 193 Tyler, G. W., 268

Ullman, J. L., 166, 192 Van de Vooren, A. 1., 192 Van der Pol, B., 76 Van Linde, H. J., 192 Vaschy, A., 123 von Hold, R. E., 194 Walther, A., 260 Warner, F. J., 218, 220 Wassmuth, R. H., 268 Watson, G. N., 34, 35, 121, 203, 220, 260 Weber, E., 99, 122

Welch, J. T. jr., 270 Wells, P. J., 269

AUTHOR INDEX

Whittaker, E. T., 34, 35, 121, 203, 220, 260 Wicke, H. H., 219 Widder, D., 34, 35 Wilf, H., 154, 155, 191 Wilkinson, J. H., 262 Williams, B. B., 268 Williams, J. D., 77, 121 Williams, J. W., 269 Willis, H. F., 105, 122 Wilson, H. K., 37

297

Wirshup, A. D., 269 Wolstenholme, J., 95

Yakolev, N. N., 190 Yokota, S., 165-166, 192 Yoshihara, H., 270 Young, A., 192, 260, 262 Yuan, H. F., 193

Zakin, J. L., 194

SUBJECT INDEX

Abel's problem, 229, 230, 237 aeroelasticity, 221, 260

Aitken extrapolation, 151, 173, 240, 253, see also extrapolation and Richerson's deferred limiting process

completing the square, 89, 116 complex variable, 28, 29, 40, 41,59, 60, 93, 121, 122, 133

133, 134, 136, 155, 156, 188, 198,

computer x, 41, 49, 63-68, 78. 118, 125, 126, 168, 190, 192 convergence, 17-19, 138, 139, 202, 203, 219, 239, 240, 249, 261

232, 233, 242, 245, 248, 250, 251,

cubature, 8, 189, see also integrals, multiple

see also systems of equations antiderivative, 1, 25 applications, ablation, 238 circulation, 236 diode detector, 117 forest fires, 236 heat transfer, 98, 189, 206, 254-258, 262 light scattering, 262, 263 lubrication, 254 particle mechanics, 228 rarefied gas flow, 263 statistical mechanics, 36, 119, 206 sonic boom, 186, 194 thermal explosion, 72-74 wave drag, 186, 194 vibration, 69, 209, 212, 220 approximation, of functions, 72, 76, 194 of indefinite integrals, 70-74, 77 of definite integrals, 108-118, 121 asymptotic approximations, 70, 72, 105, 203

and quadrature rules for multiple

algebraie equations, 51, 85, 86, 128, 129,

Bernoulli numbers and polynomials, 202, 203

Bernoulli's problem, 227 Bernstein polynomials, 152, 153, see also quadrature rule, Bernstein bifurcation 232 Boley's method, 112, 113, 115, 123

boundary value problems 214-218, 220, 225-227, see also eigenvalues and eigenvalue problems center of gravity, 83, 185

integrals cycloid, 230

Darboux sum, 4, 5, 90, 91 definite integral, 25-27, 77, 78 classical calculus methods for, 84-90 continuity of, 26, 27 differentiation of, 16, 25-27, 84 Laplace's method, 108-111 ; see also approximation methods for definite integrals; integrals design of experiments, 136, 190 difference equation, 180 differential equation, 27, 47, 48, 210-220,

235

conversion to integral equations, 223-227 definition of functions by, 44, 68, 97 evaluation of integrals by, 68-70, 85, 123 numerical solution of, 134, 195, 214, 219, 221

partial, 15, 73, 254, 262 differentiation, 16, 25-27, 63, 84 fractional, 28, 29, 35, 230 numerical, 186-188, 193, 194 partial, 41, 51, 52

with respect to a parameter, 84 see also integro-differential operators ; quadrature by differentiation dimensional analysis, 32, 116, 117, 123 Dirichlet's reduction, 82-84 eigenvalue and eigenvalue problems, 208-

213, 220, 245, 251, 262, see also boundary value problems

SUBJECT INDEX

300

error integral, 77, 78, 114 expansion, Boley's, 112, 113, 115 Laplace-Winckler, 113-115 Willis, 105-107 exponential integral, 99, 112, 113 extrapolation, 148-152, 173, 240, 253

Euler-Maclaurin formula, see summation methods Euler's constant, 82, 102 Fourier series and integrals, 158, 170, 207 Fredholm equation, see integral equation function, algebraic, 42, 43, 52, 122 Bessel, 69, 97, 120 beta, 80 convex, 37 definition of elementary, 41 gamma, 28, 44, 79, 83 hypertranscendental, 44 increasing or decreasing, 22 of bounded variation, 6 rational, 41 theta, 255

transcendental, 43, 44, 47, 121

see also approximation of functions, differential equation ; functions defined by functional equation, 86, 87, 117 geometry, 2, 3, 29, 36, 77, 185, 227 Green function, 225-227, 261 history, 1, 34, 125, 222, 260, 261

Archimedean, 7 bounds for, 23, 24, 35, 77, 78, 119 Cauchy principal value, 15, 92, 115, 235 Chebyshev's, 55, 57 Christmas tree, 57 definitions of, 4-8 elliptic and pseudoelliptic, 57, 58, 76 finite part, 15, 16, 19, 35, 161 Frullanian, 99-105, 115-117, 122 improper, 14-16, 100, 160 infinite, 14, 16, 20, 168, 174-178 iterated, 9, 10

Lebesgue, 6, 7, 35, 36 line, 29, 30 Lommel, 69 multiple, 8-13, 36, 82, 103-105, 181-186 properties of, 3, 4, 7-9 Riemann, 4, 5, 7, 8, 36 Stieltjes, 5, 6, 23 surface, 31 volume, 31

see also definite integral ; indefinite integral; quadrature Integral equation, xi, 1 analytical solution, 231-238 classification, 222, 223 derivation, 223-231 dual, 223, 261 error estimate, 253, 254, 262

existence and uniqueness, 221, 233, 236, 239, 242 Fredholm theory, 241, 242 Hilbert Schmidt theory, 243-245, 261 homogeneous, 25 Liouville-Neumann theory, 238-240 nonlinear, 223-225, 232, 233, 239, 256, 261, 263 numerical solution, 221, 246-258, 260-263 relation to algebraic equations, 232 singular, 222, 256, 261 successive approximations, 238-240, 249, 252, 253, 263 Wiener-Hopf, 234

indefinite integral, 25, 39, 178-181 inequalities, 35-37 Chelyshev's, 22-24, 35 Holder's, 21, 24 Minkowski's, 21, 24 Schwartz-Bunyakowsky, 21 Wirtingers, 37

integral representation, 88, 89

influence coefficients, 230, 231

integral transformn, 96-99, 122, see also

ill conditioned equations, 168, 250 instability, of indefinite integration rules, 179-181 of numerical solutions, 246, 250 see also Runge Phenomenon integral,

Laplace transform integration, as inverse of differentiation, 1, 39 by parts, 28, 39, 70, 111-114. computer programs for symbolic, 63-68, 76

301

SUBJECT INDEX

in finite terms, 40, 45-49, 75 interchange of order, 13, 88, 89, 98 numerical, see quadrature fractional, 28, 29, 35, 230 general forms for, 52-55, 71, 72, 94, 95 of algebraic functions, 52-58 of inverse functions, 59-62 of rational functions, 49-52

use of differential equations for, 68-70, 76, 85

with respect to a parameter, 84, 85, 89 see also quadrature integro-differential operators, 1, 186, 188,

189, 260

interpolation, 128, 189, see also Bernstein polynomial; quadrature rules, interpolatory; Runge phenomenon Jacobian, 11, 13

Jury problems, see boundary value problems kernel, 222

convolution (difference or Faltung), 234 degenerate (or product), 231, 241, 246,

253

symmetric, 222, 243

Lagrange interpolation formula, 128, 129, 167, see also interpolation, Runge phenomenon Laplace transform, 97, 105, 106, 122, 234, see also integral transform least squares, 187 l'Hospital's rule, 40, 43, 61 linear operation, 129-186-189, 193, 194 Lipschitz condition, 239 lunes of Hippocrates, 2, 34

product representation, 81, 82 proof, ix, x

quadrature, adaptive, 147, 148, 191 by differentiation, 197, 210-220 convergence of, 138, 139, 191 Monte Carlo, 153, 154, 191 Romberg, 149-151, 191 using Riemann sum, 157, 191 quadrature rules, Bernstein, 152-154, 201, 245, 247 best, 154-157, 191

Boole's, 130, 258 Chebyshev, 132-135, 139, 144, 145, 151, 162, 163, 165, 166, 169, 190 Clenshaw-Curtiss, 127, 128, 132, 190, 193 closed, 127, 131, 145 compound, 141, 142, 144, 150, 153, 170, 171, 198-201

counter examples, 143, 160 criteria for comparing, 126, 190 Dufton's, 135, 141, 144, 145, 150, 190 error estimates, 139-144, 183 for indefinite integrals, 178-180, 193 Filon type, 170-172, 192, 193 for infinite range, 168, 174-178 for multiple integrals, 181-186, 193 for periodic functions, 158-160 Gaussian, 135-139, 145, 159, 160, 166-169 181, 182, 184, 190, 201, 248, 251

generalized midpoint, 201, 202, 206-208 generalized trapezoid, 202-204 Goodwin-Moran, 177, 178, 132 indeterminacy, 174 interpolatory, 165, 184 Kronrod, 137, 138, 148, 190 Lobatto, 136, 137, 159, 173, 261 midpoint, 131, 141, 142, 145, 147, 150, 153, 159, 160, 162, 170, 198, 245, 247

method of statistical trials, see quadrature, Monte Carlo moment of inertia, 5, 6, 13, 83, 201 Monte Carlo method, 125, 126

Newton-Cotes, 127, 130, 131, 145, 148, 150

open, 127, 131 Parmentier's, 199

quasi Chabyshev and quasi Gaussian, orthogonal functions and polynomials, 136, 166-168, 184, 192, 244 orthogonality, 243 paradox, 14, 18, 19, 39, 78, 100 parity, 31, see also symmetry product integration; 164-173, 206-208, 248

164, 165, 169

Radeau, 136, 137 Ralston's compound, 146-148, 191, 201 remainder, 140-144 Sheppard, 199-201, 219, 258 Simpson's, 130, 142, 147, 150, 161, 182,

SUBJECT INDEX

302 183, 195, 200, 258 Steffensen's, 131, 133, 142, 145

trapezoid, 141, 142, 145, 147, 149, 153, 198, 245, 247

using correction terms, 198-201, 258 using specified nodes, 127-132 with weight function, see product

integration, see also extrapolation rationalization, 55-57 reciprocity principle, 231 resolvent, 239-242

Richardson's deferred limiting process, 150, 151

roundoff error, 139, 140, 183 Runge phenomenon, 139, 190

SAINT, see symbol manipulation scale factor, 174, 176, 178, 227 series, 17, 70, 71, 90-92, 121, 201, 202 series transposition, 92, 95 sets, 6, 7, 190

singularity, 14, 15, 117, 138, 160-165, 168, 188, 192, 204, see also integral

equations, singular; subtraction of singularities Stirling's approximation, 110 Sturm-Liouville system, 68, 123 substraction of singularities, 162, 181, 192, 252, 256 summability, Abel, 18, 98 Cesaro, 18, 20 summation methods, 91, 121, 172, 173, 205, 206, 220 superposition principle, 16,255 symbol manipulation, 63-67, 76, 118, 123,

of functions of complex arguments, 60 of integration formulas, 53, 59, 64, 65 of orthogonal polynomials, 168 of transformations for infinite integrals, 175

of weights and nodes for quadrature, 155, 156, 168, 190

tautochrone, 230 termination error, 174 theorem, Chartiers, 17, 35 convolution, 98, 99, 234 for convergence of quadrature, 138, 139, 191, 204, 219

for Laplace's method, 109 for Laplace transforms, 96-99 Fredholm alternative, 292 Fubini's, 13 mean value, 20, 26, 94 Mikeladze's, 148, 149 on integration in finite terms, 45, 47 on Riemann integrability, 7 pi, 116, 123

Weierstrass approximation, 139, 186 transformation, Euler, 55-57, 172, 173, 240 Morris, 12, 13, 154 of quadrature rules, 131,

164,

165,

174-178

of variables of integration, 10, 11, 65, 66 tree, 66

truncation error, 140, see also quadrature rules, error estimates undetermined coefficients, 187-189, 198

symmetry, 31-33, 94, 95, 129, 146, 147, 155, 156, 158, 201

Vandermonde determinant, 128, 132, 190 variational methods, 209, 218, 220 vector analysis, 29, 30, 36 Volterra equation, see integral equation 'Volume of a sphere, 10-12

tables, Bierens de Haan's, 78, 118, 120

weight function, see product integration

192, 197

symmetric pairing, 161, 261