Nine Introductions in Complex Analysis

NINE INTRODUCTIONS IN COMPLEX ANALYSIS REVISED EDITION

NORTH-HOLLAND MATHEMATICS STUDIES 208 (Continuation of the Notas de Matemática)

Editor: Jan van Mill Faculteit der Exacte Wetenschappen Amsterdam, The Netherlands

Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo

NINE INTRODUCTIONS IN COMPLEX ANALYSIS REVISED EDITION

SANFORD L. SEGAL University of Rochester Rochester, USA

Amsterdam – Boston – Heidelberg – London – New York – Oxford Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK

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Printed in Hungary 07 08 09 10 11

10 9 8 7 6 5 4 3 2 1

Foreword The content of a one-semester course in complex analysis is reasonably certainexcept that one would like to include the Riemann Mapping Theorem, but often does not manage to have the time. What a second course should contain is far less clear. In this book we try to give introductions to several (but certainly not all!) of the many topics which naturally present themselves when a first one-semester course has been completed. (There is a presupposition of working familiarity with the I'- function, Weierstrass Products, and the elements of analytic continuation. Students whose first course failed to include these topics may easily acquire the requisite knowledge from any of the standard texts, e.g. Conway's Introduction to Complex analysis or Stein and Shakarchi Complex Analysis.) In a few places the concept of Lebesgue measure is used, but not in a significant way. This book is much too large for all topics to be treated in one semester; however, an instructor and students may reasonably select various material to examine or decide on a longer course. It is to be stressed that all chapters are introductions, -indeed the material introduced in almost every chapter has been the subject of book-length presentations (often more than one). Each topic involved may be pursued further in the more specialized books and research papers listed in the bibliography. Students are encouraged to do this. Nevertheless, the subject is so vast that no pretense of a complete bibliography is claimed. The text consists of two interspersed parts: Theorems with Proofs, and Notes. Each is numbered consecutively in each chapter; so that a Note 3.5 say, means that there is a Note 3.4 preceding it within section three of that chapter. The Notes consist of glosses on the text, references to the literature and further extensions, historical remarks, and the like. They are more informal in style than the Theorems and Proofs. With only a few exceptions there are no references to Notes other than in other Notes. Thus, by and large, a reader may in fact read just the Definitions, Theorems, and Proofs, skip all the Notes and acquire a coherent presentation of the material; however, such a reader would be very ill-advised to do so, for the Notes contain a context in whch the material should be seen. On the other hand, I have felt free to include occasional mention in the Notes of concepts which are not defined in the text and may be unfamiliar to a reader-in such a case pursuit of the topic will require preliminarily some consultation of an elementary textbook, or at the very least a good mathematical dictionary. Internal textual references have

vi

Foreword

three preceding numerals if and only if the reference is not to the chapter in which it appears; thus a reference in Chapter 4 to Theorem 3.1 would refer to the first theorem of section 3 of that chapter, while a reference to Theorem 3.3.1 would be to the first theorem of section 3 of Chapter 3. Above all the attempt has been made to provide a book which can be read for profit and not be just a shelf adornment. The material in this book is not original, but it is basic to various areas of one complex variable; I hope it may stimulate students to pursue further such topics. Chapter 1 discusses some material on conformal mapping and fills the frequent lacuna of the Riemann Mapping Theorem. The proof using normal families is given, and the topic (usually not discussed) of construction of the mapping for general regions is treated. The Schwarz-Christoffel formula for polygons is also discussed. Chapter 2 deals with Picard's Theorem, both from the Bloch-Landau point of view, and using the elliptic modular function (constructed in an ad hoc manner). The problem of Bloch's and Landau's constants and the Ahlfors-Grunsky bounds for them is also discussed. Chapter 3 presents an introduction to the basic ideas of entire function theory: order, type, the Phragm6n-Lindelof indicator, and the relationships elementary in that theory. Chapter 4 presents an introduction to Nevalinna theory including some of the more initially startling standard applications such as the identity of two functions which assume five distinct values at the same points in the complex plane, or the existence of fixed points of order n. As Nevanlinna Theory may be regarded as a far-reaching deepening of Picard's Theorem, it follows naturally after Chapters 2 and 3. On the other hand, Chapter 4, a t least as regards the proof of Nevanlinna's Second Fundamental Theorem, which is the kernel of all later developments, is probably more difficult than any of the preceding material, and seems inevitably to involve a somewhat denser style of exposition. It is possible (although not necessarily recommended) for a reader to omit Section 4.2 and "take on faith" the "second version" of Nevanlinna's Second Fundamental theorem which appears as Theorem 4.3.1, and which is the version often used in applications. Chapter 5 returns to entire functions from a slightly different point of view and presents results on asymptotic values; in particular, Julia's Theorem which deepens Picard's Theorem in a different direction, and the Denjoy-Carleman-Ahlfors Theorem limiting the number of asymptotic values an entire function of finite order can have. Chapter 6 is a change of pace in that it is concerned with functions represented by power series with a finite radius of convergence. Here we discuss some problems of analytic continuation and the many seemingly different kinds of conditions which produce natural boundaries. The Hadamard and Fabry Gap Theorems, overconvergence, the P6lya-Carlson Theorem on power series with integral coefficients, and P6lya's converse of Fabry's Gap Theorem are among the several topics discussed. Nevertheless, a certain continuity of ideas with some of those in other chapters should be apparent.

Foreword

vii

Chapter 7 provides an introduction to what became the classic problem in the theory of functions univalent in a disk: the so-called Bieberbach conjecture. This was solved in 1984 by Louis De Branges. The first edition of this book (in 1982) contained a discussion of some of the prior attempts at its solution. These have been omitted in favor of De Branges' solution, this has necessitated also an introduction of the Loewner differential equation, omitted in the first edition. De Branges actually proves a conjecture of Milin that implies the Bieberbach conjecture. The material concerning De Branges' solution, like the material in Chapter 4 about Nevanlinna's Second Fundamental theorem, is somewhat denser than the rest of the book's exposition. The first section is concerned with distortion theorems in general, even if they are not used for the solution of the Bieberbach conjecture. In Chapter 8 elliptic functions are discussed both from Weierstrass's and Jacobi's point of view. Throughout the emphasis is on the structure of this area of analysis. Because of the antiquity of the subject of elliptic functions, and the way the subject grew, it often seems in its classical analytic aspect like a welter of intriguing but incoherently linked formulas, while, paradoxically, abstract algebraic versions of some of these analytic ideas are in the forefront of contemporary research. The treatment in the rather lengthy Chapter 8, which, nevertheless, hardly contains all the relevant details, is both "classical" and, I hope, coherent. Chapter 9 first presents a classical proof of the Prime Number Theorem as an example of using complex analysis and as motivation for discussing the Riemann Zeta-function. It concludes with a discussion of Riemann's famous unproved hypothesis concerning the Riemann Zeta-function. The prime number theorem was one of Hadamard's chief motivations in creating entire function theory and so questions solved and unsolved which are related to it seem especially appropriate to a book of this sort. Nevertheless, the chapter is an introduction to the Riemann Zeta- function, and not to the theory of prime numbers, let alone analytic number theory-thus, there is, for example, no mention of sieve methods, nor even of L-functions. The notes do contain relevant information about results in prime number theory which seem related, but here again a very great deal has been omitted without mention; how much can be seen by referring to some of the standard works cited in the chapter. The book concludes with an Appendix in which proofs are given of some of those standard tools which rarely find their way into a first course: The Area Theorem, the Borel-Carathbodory Lemma, The Schwarz Reflection Principle, Hadamard's Three Circles Formula; as well as a special case of the Osgood-Carathbodory Theorem which finds application in Chapter 2, the Carathbodory Convergence Theorem that finds application in Chapter 7, and a special case of the Fourier Integral Theorem used in Chapter 9. It also contains brief discussions of Farey Series and Bernoulli numbers. Throughout this book multiple proofs of the same major result are frequently given in the belief that the demonstration of different points of view can only serve to elucidate a subject. A consequence, of course, is that a subject matter cannot be followed in detail to the same depth that it might otherwise be.

...

vlll

Foreword

The various chapters are largely independent, though appropriate cross- references are usually given. Some basic ideas that appear throughout the book such as the growth of entire functions, normal families, univalence, are not always crossreferenced after their first introduction and definition, as to do so would be excessive. The reader in doubt should be able to use the index and Table of Contents to find appropriate definitions if they are not known or the chapters are not being read in sequence. The first four chapters form a natural sequence, and might be considered as a unit for a one-semester course, with, perhaps, some additional material selected from one of the later chapters. Every effort has been made to eliminate errors, typographical and otherwise; nevertheless it is too much to hope especially in a book this size, that all have been found. Although several colleagues have made suggestions about one point or another, needless to say all such errors are my own. I can only hope that they are neither too frequent nor egregious, and welcome any corrections from readers. The ( ~ a r t i a l )index generally lists the first occurrence of a word or phrase in a particular context, usually a definition, but it does not list all occurrences. There has been an attempt to list all such phrases that might be of interest to readers of this book, but there is no guarantee that this has been achieved. Nevertheless, it is hoped that this partial index, together with the various chapter headings and divisions, will allow readers to find their way quickly to topics of interest to them. I also wish to thank Joan Robinson of the University of Rochester for her retyping the content of the first edition, and Alessandro Rosa for the diagrams. The whole electronic restyling of this edition also features new computer generated pictures of the behavior of functions, processed after scanning bounded complex regions or the whole Riemann sphere. The function f ( z ) is calculated at each complex point associating, via a one-to-one-painting algorithm, one different color for drawing each screen point. The original version works with the whole color spectrum; here, for print reasons, there is a gray-scaled version, which still achieves the original tasks: darker shades still indicate values close to zero, so that each polynomial root is painted black; lighter shades indicate values which are getting larger and larger, and poles are painted white. This allows an easy illustration of the concepts and examples throughout these pages.

Foreword

A NOTE ON NOTATIONAL CONVENTIONS

We list here a few notations used throughout this book usually without explicit definition. A region always refers to an open connected set in the plane. C denotes the complex plane, and C, the usual "extended plane", C U {co)which maps onto the Riemann sphere under stereographic projection. If X is a set, BdX. denotes the boundary of X and the closure of X . B ( a , r ) denotes the open disk with center a and radius r , that is the set {z : lz - a1 < r) and C(a, r ) = BdB(a, r) the circle with center a and radius r , namely the set {z : lz - a1 = r). Thus B(a, r ) = B(a, r ) U C(a, r ) = {z : Iz - a1 r). All contour integrals are assumed to be taken in the positive (counterclockwise) direction unless explicitly mentioned otherwise. [y] invariably refers to the greatest integer 5 y. C' indicates a summation in which the term corresponding to 0 has been omitted. M ( r , f ) (or M ( r ) if there is no danger of confusion) indicates the maximum modulus of the function f in B(0, r ) . The Bachmann-Landau 0,o notation for error terms is used; namely

<

f (x) = O(g (x)) as x means that

+a

1 # 1 is bounded as x + a, and f (x) #

= o(g(x)) as x

-+ a means that

+ 1 as x + a. Curves, unless explicitly stated otherwise, are means that assumed to be rectifiable and without self-crossings. Any other notation should either be familiar as standard, or defined at appropriate places in each chapter. Some of the chapters contain a summary of special notational conventions of their own.

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Contents Foreword

v

Contents

xi

1 Conformal Mapping and the Riemann Mapping 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.2 Linear fractional transformations . . . . . . . . . 1.3 Univalent Functions . . . . . . . . . . . . . . . . 1.4 Normal Families . . . . . . . . . . . . . . . . . . 1.5 The Riemann Mapping Theorem . . . . . . . . . 2 Picard's Theorems 2.1 Introduction . . . . . . . . . . . . . . 2.2 The Bloch-Landau Approach . . . . 2.3 The Elliptic Modular Function . . . 2.4 Introduction. . . . . . . . . . . . . . 2.5 The Constants of Bloch and Landau

3 An 3.1 3.2 3.3 3.4

. . . . .

. . . . .

Theorem

.... .... .... .... ....

. . . . .

. . . . .

................ . . . .

. . . .

... ... ... . .. .

. . . .

Introduction to Entire Functions Growth, Order, and Zeros . . . . . . . . . . . . . . Growth, Coefficients, and Type . . . . . . . . . . . The Phragmkn-Lindelof Indicator . . . . . . . . . . Composition of entire functions . . . . . . . . . . .

4 Introduction to 4.1 Nevanlinna's 4.2 Nevanlinna's 4.3 Nevanlinna's

..... ..... ..... ..... .....

... ... ... ... . . . .

... .. . ... ...

. . . .

...... ...... . ..... ......

. . . .

. . . .

. . . .

1 1 5 10 13 17 35 35 36 45 45 56

67 . . . 67 . . . 80 . . . 89 . . . 102

Meromorphic finctions 107 Characteristic and its Elementary Properties . . . . . . 108 Second Fundamental Theorem . . . . . . . . . . . . . . 124 Second Fundamental Theorem: Some Applications . . . 137

5 Asymptotic Values 155 5.1 Julia's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 The Denjoy-Carleman-Ahlfors Theorem . . . . . . . . . . . . . . . . 169

xii

Contents

6 Natural Boundaries 6.1 Natural Boundaries-Some Examples . . . . . . . . . . . . . . . . . 6.2 The Hadamard Gap Theorem and Over-convergence . . . . . . . . . 6.3 The Hadamard Multiplication Theorem . . . . . . . . . . . . . . . . 6.4 The Fabry Gap Theorem . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Pblya-Carlson Theorem . . . . . . . . . . . . . . . . . . . . . . .

189 189 201 215 221 243

7 The Bieberbach Conjecture 257 7.1 Elementary Area and Distortion Theorems . . . . . . . . . . . . . . 258 7.2 Some Coefficient Theorems . . . . . . . . . . . . . . . . . . . . . . . 273 8 Elliptic Functions 8.1 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Weierstrass'p-function . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Weierstrass' C- and a-functions . . . . . . . . . . . . . . . . . . . . . 8.4 Jacobi's Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Modular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 301 306 317 330 352 373

9 Introduction to the Riemann Zeta-Function 397 9.1 Prime Numbers and [(s) . . . . . . . . . . . . . . . . . . . . . . . . . 398 9.2 Ordinary Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . 403 9.3 The Functional Equation, the Prime Number Theorem. and De La Vallbe-Poussin's Estimate . . . . . . . . . . . . . . . . . . . . . . . . 412 9.4 The Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 437 Appendix

451

Bibliography

473

Index

485

Chapter 1

Conformal Mapping and the Riemann Mapping Theorem 1.1 Introduction Without question, the basic theorem in the theory of conformal mapping is Riemann's mapping theorem. It is one of those results one would like to present in a one-semester introductory course in complex variable, but often does not for lack of sufficient time. Indeed, depending on the time available and the text used, elementary conformal mapping in general is a subject which in an introductory course may not be adequately treated. This chapter begins, therefore, with an introduction to some basic results on conformal mapping especially those involving univalent functions. The concept of normal families is then introduced and developed far enough to be able to give the well-known elegant existence proof of the Riemann Mapping Theorem resulting from the reworking of ideas of Caratheodory and Koebe by Fejkr and F. Riesz. We then turn to the related construction of the mapping, following the work of Koebe and Ostrowski. Finally the Schwarz-Christoffel Theorem giving explicit mappings of polygonal regions is treated. Examples are given throughout.

1.1.1

Review

We begin with a semi-informal review of the basic mapping properties of an analytic function j a t a point zo where j1(zo) # 0. Suppose C1 and C2 are two continuous curves intersecting in a point zo, and such that each has definite tangents a t zo (i.e. they represent functions differentiable a t zo). An "angle between C1 and C2" is an angle formed by the tangents a t zo. Let f (2) be an analytic function in a region D of the z-plane. Let zo be an interior point of D and CI, C2, two continuous curves passing through zo which

I. Conformal Mapping and the Riemann Mapping Theorem

2

have definite tangents there. Suppose fl(zo) # 0. Then an angle between f(C1) and f (C2) at zo exists, since well-defined tangents exist. If the tangent to C1 at zo makes the angle a1 with the real axis and the tangent to C2 makes the angle a 2 (both measured on the right side of the tangent), clearly a 2 - a1 is the "interior" angle between C1 and Cz (see Diagram 1.1); then it is furthermore true that a2 - a1 as so defined is also the angle between f (C1) and f (C2).

Diagram 1.1 This may be seen as follows: If zl E C1 and z2 E C2 are two variable points on the curves near zo such that lzl - zol = 122 - zol = T say, then

and, as r

+ 0, zl + zo,zz + ZO, 81 + al,02 + a 2 . . f(z1)-f(zo) = l i m fl(zo) = lim r+O z1 - 20 r-0

Hence, since f is analytic in

f ( ~ 2-) f (zo) zg - do

7

and since fl(zo) # 0, we can write fl(zo) = ~ e ~ where % a R ~ # 0 and 6 = arg fl(zo) is a specific fixed number. Now let 41 = arg(f (zl) - f ( a ) ) , so we may write f (zl) - f (20) = plei@l,say. = ~ e and " so limr+o(41 -81) = 6, whence limT,o = al + 6. Then limT,o Since the limit exists, f (C1) has a definite tangent at f (zo) which makes the angle a1 6 with the real axis. Similarly f (C2) makes the angle a 2 6 with the real axis, whence the result follows. We should further note that f preserves the sense of the angle. Also, clearly lim p1lr = Ifl(zo)l # 0 . 7-0

ml

+

+

In fact, the above shows that the angle between f (C1) and f (Cz) in question is obtained by a rotation by 6, and any small subregion of D containing zo goes into

3

1.l. Introduction

a "similar" subregion of f (D) determined by this rotation and a "stretching" by If1(.o)l. Because of this similarity, maps which preserve angles as above are called conformal. Note 1.1. Such a definition of conformal includes the possibility of a conformal map preserving the magnitude but not the sense of angles. An example of such a map (which is not analytic) is reflection in the real axis f (z) = 2 , or, more generally, the map obtained by taking the complex-conjugate of any analytic conformal map. Some authors call such maps "indirectly conformal". If we suppose f ' has a zero of order n at zo, then f (C1) and f (C2) still have definite tangents at zo, but the angle btween them is the angle between C1 and C2 multiplied by n 1. For if f ' has a zero of exact order n at zo, then in a neighborhood of zo,

+

f (2) = f (zo) + a(z - ~ 0 ) +~. . ~. where ' a#0, and so in the previous notation plei4' = f (zl) - f (zo) = a(zl - zo)n+l + . . . =

+ . . . (higher powers of r )

= Ialrn+lei(q+(n+l)el)

where q = arg a. So plei@l lim -- lalei(~+(n+l)al) rn+l

T+O

and lim $1 = 11

r-+O

+ (n + l ) a l , rlim $2 +O

= 11

+ (n + l ) a z

Before turning to a brief study of linear fractional transformations, we give some examples of non-linear transformations.

Example 1.l. f (2) = z2. If z = reie and w = f (z) = pei4, then p = r2 and $ = 28 ; hence the angular region a < argz < ,B is mapped onto 2a < argw < 2P, and if ,B - a > n the image region covers part of the plane twice. (The ambiguity can be removed by consideration of the Riemann surface for fi). If z = x i y and w = u iv, then u iv = x2 - y2 2ixy. Hence the pre-image of the straight line u = a is the hyperbola x2 - y2 = a and the pre-image of the straight line v = b is the hyperbola 2xy = b. Suppose a and b are not both 0 and let # 0 be a point where the hyperbolas meet, and 8 the angle at J. Since fl(z) = 22 and 2< # 0, 8 is also the angle between u = a and v = b; but this angle is n/2; hence 8 = n/2. If a = 0 and b = 0, then since f r ( z ) has a simple zero at the origin,

+

<

+

+

+

1. Conformal Mapping and the Riemann Mapping Theorem

4

if 8 is the angle between x2 - y2 = 0 and 2xy = 0, 28 is the angle between u = 0 and v = 0 which is n/2; hence 8 = n/4 (as can also be deduced directly). Similarly the lines a: = c, c # 0 and y = c, c # 0 map onto parabolas meeting at an angle of n/2, while x = 0 and y = 0 (the axes) map onto the halflines v = 0, u 0, and v = 0, u 2 0 each described twice. These half lines meet, as is expected, at an angle of n.

<

Example 1.2. f ( z ) = -. Writing z = x + iy, the strip between x = 0 and x = n/2 is mapped into the open unit disk with the interval (-1, O] deleted. the two bounding lines map on the boundary of the slit disk. The line {z : x = 0) maps onto the real interval [-I, 01 described twice: as y goes from co through positive real values to 0, f (iy) = ~ ; ~ ~ ~ [ : y goes from -1 to 0, and then back from 0 to -1 as y goes from 0 to co througk negative real values. The line {z : x = n/2} maps onto the unit circle described once (as y goes from co to 0 through negative real values, the lower semicircle is described, and as y goes from 0 to co through positive real values, the upper semicircle is described.) One may note in particular that co is mapped onto -1. The line x = a, 0 < a < n/2 is mapped onto a loop which cuts the real axis at -1 and at another point where Irn = 0 which may be easily computed. Each loop contains the slit (-1,O] in its Jordan interior, and is contained in B ( 0 , l ) . If 0 < a < b < n/2, the loop for x = b contains the loop for x = a in its Jordan interior.

(:;::[:;{

Example 1.3. f (z) = a + 16/z The circle {z : lz + 1 - il =

)

m)maps onto the curve of Diagram 1.2.

1.2. Linear fractional transformations

Diagram 1.2 Curves of this kind are known as aerofoils, and have had some importance in aerodynamic studies. The interior of the circle maps onto the exterior of the aerofoil. For further examples with diagrams of the mapping properties of a great variety of functions, the reader is referred to A Dictionary of Conformal Mapping by H . Kober. This was originally prepared for the British Admiralty in 1944-48, and was reissued by Dover Press, New York in 1952.

1.2

Linear fractional transformations

Definition 2.1. A function of the form

is called a non-singular linear fractional transformation (or, sometimes, Mobius transformation).

Theorem 2.1. I f f ( z ) is meromorphic i n the extended complex plane C, and maps it one-to-one onto itself, then f ( z ) is a non-singular linear fractional transformation, and conversely. , Proof. Since f ( z ) is meromorphic in ,@

it is rational. so f ( z ) = @ where p ( z )

and q ( z ) are polynomials with no common zeros. For a fixed wo:(& = wo for some z = zo if and only if woq(zo)- p(zo) = 0. if deg(p(z))> 1, w q ( z ) - p ( z ) is a polynomial of degree > 1 for all but a t most one value w and so f is not one-toc,d not both 0. If one. Similarly the degree of q ( z ) must be 1. So f ( z ) = ad - bc = 0, then if d # 0, f ( z ) = bld and if c # 0, f ( z ) = a l c ; and so f is not onto. ad - bc # 0 is meromorphic in C, ; t o prove it is Finally, clearly f (2) = onto:

s,

s,

If c # 0, f : -d/c

-+ m; if c = 0, then a # 0 and f

If a # a / c , let w = -,

then f : w + a .

: co

-+m.

1. Conformal Mapping and the Riemann Mapping Theorem

If a = a l c , c

# 0 , then f : co -+ alc.

That f is one-to-one is clear.

Theorem 2.2. The set of all non-singular linear fractional transformations forms a group under composition.

Proof. An obvious verification. Note that if az +b

m(3= a, the inverse 4 - I

-dz + b ( z )= cz - a .

Theorem 2.3. A non-singular linear fractional transformation is conformal at all z except z = - d / c (oo if c = 0 ) , where it has a pole.

Proof. If m(z) =

5, then # ( z ) = mf . ad-bc

We now briefly turn our attention to circles (and straight lines, which are degenerate circles).

Definition 2.2. Two distinct points of C, p and q are called inverse with respect to the circle C ( z o ,p), p > 0 , if they are collinear with zo, lie on the same side of zo, and I P - zol 1q - 201 = p2. Suppose p and q are distinct points inverse with respect to C ( z o ,p), and p = zo +meifi, m > 0 . Then collinearity implies arg(q - zO)= arg(p - zO)= p, and we 2 . also have Iq - zol = p 2 / m ; so q = zo $e2fi. If z E C ( z o , p ) ,then z = zo + peie for some 8, and

+

since pei6 - meifi = pe-ie - m e - i f i - -e-i(6+fi)(mei6 - peifi). Furthermore, since p does not equal q, m / p # 1. Thus we have proved

Theorem 2.4. The circle C ( z o , p ) is a subset of

and p and q are distinct points inverse with respect to C ( z 0 ,p). The converse of this theorem is also true:

Theorem 2.5. The equation = k , k # 1,# 0 always represents a circle with respect to which p and q are inverse points.

1.2. Linear fractional transformations

7

Proof. A somewhat wearisome computation which is left to the reader. The circle turns out to be 1 - IC2

Note 2.1. The geometric definition of circles by using inverse points goes back to Apollonius (c. 255-170 B.C.E.). Suppose p E B(z0, p),p # zo, then one can easily give a geometric construction of the point q which is inverse to p with respect to C(zo,p) as follows:

Diagram 1.3 If L is the line passing through zo and p, construct the perpendicular to L at p, and at the point where this line intersects C(zo,p) draw the tangent T to the circle. The intersection of L and T is q. This construction clearly can be reversed so that given a point in the complement of B(z0, p), the point inverse to it with respect to C(zo,p) can be found. Note 2.2. As p + zo along L in Diagram 1.3, q + oo (this may also be seen directly from Definition 2.2). We, therefore, make the definition: zo and oo are inverse with respect to C(zo, p ) . is a straight line, for z E S means z lies equidistant from p and q, hence lies on perpendicular bisector of the line joining them, and this is the set of all points equidistant from p and q. Because of this last note, it is conventional to consider straight lines as "degenerate" circles (they may also be viewed as circles passing through m.) Therefore, for the rest of this section, the word "circle" will mean "circle or straight line".

1. Conformal Mapping and the Riemann Mapping Theorem

8

Theorem 2.6. Under a non-singular linear fractional transformation, circles go into circles and inverse points with respect t o the original circle into inverse points with respect t o the image circle (inverse points with respect to a straight line meaning points symmetrical about the line and equidistant from it).

s,

1 ~ 1

= k , k # 01, where k is fixed. Let f(z) = ad-bc # Proof. Let K = {z : 0. Since f is one-to-one, w E f ( K ) if and only if f -'(w) E K , that is if and only if E K , that is if and only if -dw+b cw-a

+

-p

= k,

that is if and only if

+

assuming cq d # 0, cp d # 0, which proves the result under this assumption. If cq d = 0, f : q -+ oo, and the equation for f (K) becomes

+

k lad - bcl Icp+dl '

Iw-sI=H

8

and and m are inverse with respect to f ( K ) by Note 2.2. Similarly, if cp+d = 0, f : p -+ oo and the equation for f (K) becomes

We now give some examples of the use of linear fractional transformations.

Example 2.1. Problem: Find all non-singular linear fractional transformations f : z + w = f (z) of the closed upper half-plane {z : I m z 0) onto the closed unit disk {w : Iwl 5 1) with the real line mapping onto the unit circle.

>

is such a transformation. If a = 0, then f : oo -+ Solution: Suppose f (z) = 0 contradicting the real line mapping onto the unit circle, so a # 0. If c = 0, f : oo -+ oo which is impossible. So also c # 0, and f : -b/a + 0, f : - d / c + oo. Now since f is non-singular linear fractional, so is f -', and 0 and oo are inverse points with respect to C(0,l). Hence by Theorem 2.6, -b/a and - d / c must be inverse points with respect to the real line {z : I m z = 0). Hence we can write -b/a = a , - d / c = 6 where Im a # 0. So f (z) = =4 Now 1 f (0)I = 1 (since the

3

real axis maps onto the unit circle), and so

I

(s). . (s) I = 1, or la1 = lcl, whence ,

():

1.2. Linear fractional transformations

9

(,-,).

a = ceix, A real, and f (z) = eix Finally, since f (a) = 0 E B(O, I ) , we have I m a > 0. It remains to verify that all transformations of the form

f ( z ) = eix

(-)zz -- aa

,A real, I m or

>0

do, in fact, map the closed upper half-plane onto the closed unit disk in the desired fashion. But it is an easy computation that, if I m z = 0, then If (z)l = 1, while if I m z > O , then If(z)l < 1. It is worth noting that since there are three arbitrary constants: A, I m a , Re a in the solution, one can make three given points on the real axis correspond to three points on the circle. The reader, if he wishes, can invent any number of similar exercises for himself mapping specified circles or half-planes onto other circles or half-planes.

Example 2.2. Problem: Suppose f (z) is analytic in B ( 0 , l ) and I m f (z) Find an inequality on 1f '(0) 1.

Solution: If one could find a linear fractional transformation that q5 : f (z) -+ g(z) = q5(f (2)) with the properties: (i) 4 : { z : I m f ( z ) (ii) 4 : f(0)

> 0.

C#J of C onto C such

> 0) '3{ z : 1g(z)1 < 1)

-+0,

then 1g(z)1 < 1 for I m f(z) > 0, that is for z E B(O, I), and also g(0) = 0, whereupon Schwarz' Lemma leads to the desired inequality. Let f (0) = [. By Example 2.1, (i) will hold if g is of the form

and property (ii) requires a = J. So

(:[:; I:)

g(z) = eix - , whence cg(z) - Jeix f (z) = g(z) - ei" , and

Since g(0) = 0, 1 f '(0)l = 21m J1gl(0)l. By Schwarz' Lemma, lgl(0)l 5 1, and so we get the result If1(0)l I 2 I m f ( O ) .

10

1. Conformal Mapping and the Riemann Mapping Theorem

Linear fractional transformations are also closely related t o the concept of crossratio, perhaps familiar t o some readers from projective geometry. Many standard complex variable texts contain discussion of this connection. Thron's Introduction to the Theory of Complex Variables [227] also contains an elementary analysis of various subclasses of linear fractional transformations.

1.3 Univalent Functions Definition 3.1. A function f is univalent (simple, schlicht) in a region D if it is analytic (and single-valued) in D and does not take any value more than once in D. Note 3.1. The definition says that f : D + f (D) is one- to-one, but points in f (D) may have other points outside D as pre- images under f . The terminology "univalent" for the concept of Definition 3.1 seems to have won out in English. The original German word "schlicht" was used in English as a loan word for many years, and, for a while, "simple" had a vogue. Readers will find both terms used in other books and papers they may consult.

Theorem 3.1. If f is uriivalent in a region D , then fl(z) # 0 for all z E D .

Proof. If for some zo E D , f '(zo) = 0, then g(z) = f (z) - f (zo) has a zero of order n 2 a t zo. Since f is univalent, there is a closed neighborhood K of zo, K c D such that g(z) # 0 on the boundary of K ( f is non-constant) and the only zero of f 1 in the interior of K is a t t o (since the zeros of non-constant analytic functions are isolated). Suppose K = B(z0,6). Then inf,,c(zo,6) 1g(z)] = m is > 0. Suppose a is any complex number with 0 < la1 < m. Then by Rouchk's Theorem g(z) - a has as many zeros as g(z) does in B(zo,G), and hence at least two zeros there. But since -&(g(z) - a ) = gl(z) = fl(z) # 0 for z E B(zo,6) - {zo), g(z) - a does not have a multiple zero in B(zo,6). Hence g(z) takes the value a for two distinct values of z E B(zo,6) c D ; hence f (z) takes on f (zo) a for these two values, contradicting f univalent.

>

+

Theorem 3.2. A univalent function of a univalent function is univalent.

Proof. Clear. Theorem 3.3. The inverse of a univalent function f is a univalent function $ with domain $ = range f .

Proof. Clear. Theorem 3.4. If f is univalent and maps B ( 0 , l ) onto itself with f (0) = 0 and some point on the unit circle remaining fixed, then f (z) = z.

1.3. Univalent Functions

11

<

Proof. By Schwarz' Lemma since f (0) = 0 and If (z)l 5 1 for lzl 1, If (z)l 5 lzl for lzl 5 1. The same argument applied to the inverse function 4(w) defined by = 1 for 1x1 5 1, whence is constant by 4(f (z)) = z gives lzl 5 If (z)l. So the maximum modulus principle. So f (z) = eisz for some fixed 8,0 5 8 5 27r. But f fixes some point ei@, real; hence 8 = 0. $J

Theorem 3.5. I f f is univalent and maps the closed unit disk onto itself, then f is a non-singular fractional transformation. Proof. Clearly non-singular linear fractional transformations are univalent. Suppose f (0) = <. Then there exists a non-singular linear fractional transformation L such that L maps the closed unit disk onto itself and such that L(<) = 0. (In fact, since 0 and rn are inverse with respect to C(0, I ) , it is easy to see by Theorem 2.6 that e i " s )

5.2 - 1

is such a mapping.)

Let $(z) = L(f (z)); then $J maps the closed unit disk onto itself, is univalent, and $(0) = 0. Hence as in the preceding proof $(z) = az, where la1 = 1. So f (z) = L-I ($J(z))= L-l (az) is a non-singular linear fractional transformation.

Theorem 3.6. Suppose fn(z) is univalent in a region D for each n = 1 , 2 , . . . and suppose as n + oo, fn(z) + f (z) uniformly on compact subsets of D . Then f is either univalent or a constant. Proof. f (z) is analytic in D (by the "Weierstrass Double-series Theorem"). Suppose it were not univalent. Then there are zl, 2 2 E D; zl # z2 and a complex number such that f (21) = f (z2) = .

<

<

Let K1, K 2 be two closed disks with centers zl, z2 respectively such that K1 c D , K z C D, K1 n Kz = 4. Suppose f is non-constant. Then we can also choose K1 and K 2 so that f ( z ) # for z on the boundary of K1 U K 2 . Let infzEBd(KIUKz) If (2) - CI = m. Then m > 0, and we can find an no so large that for n no, Ifn(z) - f(z)I < m for z 6 Bd(K1 U K2). SO, since fn(z) - = (f n (z) - f (2)) + (f (z) - <), by Rouch6's Theorem fn(z) - has as many zeros in K1 UK2 as f (z) - <, namely, a t least two, contradicting the univalence of f n (z).

<

>

<

<

Note 3.2. The case of a constant limit in Theorem 3.6 can actually occur as the example fn(z) = z/n shows. Theorem 3.7. I f f is analytic at z = 0 and fl(0) # 0, then there is a p that f is univalent in B(0, p).

>0

such

12

1. Conformal Mapping and the Riemann Mapping Theorem

Proof. f (z) = CrZo anzn, a1 f 0, for all z E B(0, R) for some R. If f ( a ) = f (z2) for some zl, 22 E B(0, R ) , then CF=l an(zy - z;) = 0 and so

Now, as R -+ 0, Cr=2n l a , l ~ ~ - -+ l 0, since a power series is uniformly convergent in its disk of convergence. Hence, for all R sufficient small, nlanlRn-' < lal[, and so zl =z2.

A proof of Theorem 3.7 can also be given using Rouchk's Theorem in a by now familiar way. We close this section with a well-known result which provides yet another indication of the influence of boundary behavior on the values of an analytic function. Theorem 3.8 (Darboux). Let C be a simple closed rectifiable (oriented) curve which is the boundary of a region D. Suppose f is analytic on DUC and one-to-one on C , then f is univalent in D.

Proof. f (C) = I? is clearly a simple dosed rectifiable (oriented) curve. Let A = the Jordan interior of r. Let 20 E D be such that f (z) - f(z0) f 0 for z E C. Then letting wo = f (zo) the number of zeros of f (z) - f (zo) in D is

on making the substitution w = f (z). The left side of this equation is an integer 2 1 (since zo E D), the right side is f1 or 0 according as wo E A or wo @ A, and depending on the direction in which I? is described. Hence, the only possibility is 1,wo E A, and f takes the value wo just once (at zo) in D.

Note 3.3. This theorem of Darboux is susceptible to an extensive topological generalization. For example, among other related results, Meisters and Olech [156] (Duke Mathematical Journal (1963), 63-80) have proved the following.

>

Theorem. Let X be a compact subset of n-dimensional Euclidean space Rn(n 2) such that Bd X is an irreducible separating set of R,. Let E be a possible "exceptional set" with the properties: (1) E n Int X is discrete, (ii) B d X - E # 4. Suppose f is a continuous mapping of X into I%, which is locally one-to-one on X - E.

1..4. Normal Families

13

Then if f flBdX is one-to-one, f is a homeomorphism of X onto f (X). That an analytic function in the complex plane is a locally one-to-one mapping on its domain of definition, except for a discrete set E , can be seen by computing its Jacobian. We shall return to univalent functions in more detail in later chapters. Their study is one of the most active areas of present research in one complex variable.

1.4

Normal Families

Definition 4.1. A family 3 of complex-valued functions f defined on a region D of the complex plane is called normal if every sequence ifn)of functions in 7 either contains a subsequence {f,,) such that { f,,) converges uniformly, or a subsequence {fn,) which tends uniformly to oa (i.e. given 6 > 0, If,,(z)l >_ 6 for k 2 ko and all z), on every compact subset of D. Note 4.1. If F is a normal family of analytic functions, then the limit function F , say, of { f,,) is either an analytic function or identically co. Note 4.2. F need not belong to 7 . (This is the only difference between "normal" and "sequentially compact.") Note 4.3. If 7 is a normal family of analytic functions and 7' = { f ' : f 6 F},then 7' need not be normal. For, consider the family 7 of all functions fn(z) = nz2 - n2 defined on the whole plane. 7 is a normal family since f n + co uniformly on every compact subset of the plane. But fA(z) = 2nz, and 7' is not normal since fA(z) + c m for z # 0 but + O for z = 0. The existence of "good subsequences" is clearly a useful property for a family of functions to have, but no definition is much use unless it is satisfied in some way not immediately obvious, but nevertheless useful. In this connection there is a famous theorem due in various versions to Ascoli, Arzela (and others, though these are the names usually prefaced to it even in more generalized versions). The theorem has proved of use in several areas of analysis. We recall first

Definition 4.2. A family of complex-valued functions 7 is said to be equicontinuous on a subset E of the complex plane if and only if for each 6 > 0, there is a 6 > 0 such that whenever z,z0 E E and lz - zol < 6 , then If@) - f(zo)( < c simultaneously for all f E 7 . Theorem 4.1 (Arzelii-Ascoli). A family 7 of continuous complex-valued functions defined on a region D of the complex plane is normal if (i) 7 is equicontinuous on every compact subset of D; and (ii) For each z E D, { f (z) : f E F) lies in a compact subset of the plane.

1. Conformal Mapping and the Riemann Mapping Theorem

14

Proof. D contains an everywhere dense countable set of points (e.g. the points with rational coordinates contained in it). Let {Ck} denote these points in some fixed ordering. For a fixed value of k, say k = 1 the sequence {fn(C1)} lies in a compact subset by (ii), and so some subsequence of it converges. We now repeat the process starting with this subsequence and k = 2. Repeating this process for each successive fixed value of k, we thus find an array of subscripts

such that (a) Each row is a subsequence of the preceding row, and

(b) limi,,

fn,,i (Ck) exists for each k.

Consider the diagonal sequence ni,i; it is strictly increasing and ultimately a subsequence of each row of the array. Hence {fniji}is a subsequence of {f,} converging by (b) at all the points {Ck}. For convenience, write f n i for f,i,i. Now let K be a compact subset of D, then by (i) F is equicontinuous on K. Given 6 > 0, there is a S > 0 such that for zl, z2, E K and f E F whenever lzl -zzI < 6, then If (ZI)- f (z2)I < E. Since K is compact, the cover by neighborhoods of radius 612 has a finite subcover; take a point Ck from each of these. For all sufficiently large h and j , say h, j > io, I fnh (lk) - f n j (Ck)1 < E for all these Ci, (since f n i (Ck) converges as i -+ 00). Since we have taken a 612 sub-cover, for each z E K , there is a Ck such that ICk - ZI < 6, and so Ifn, (Ck) - f n j (211 < 6, Ifnh (z) - f n h (Ck)l < 6 by equicontinuity. So

Hence, since

6

was arbitrary

> 0, Ifni}is uniformly

convergent on K.

Families of analytic functions often can be shown to be normal by an application of the following consequence of the Arzel&Ascoli theorem: Theorem 4.2. A family 3 of analytic functions in a region D is normal if the functions in F are uniformly bounded on every compact subset of D. Proof. By Theorem 4.1, it is enough t o prove that uniform boundedness implies equicontinuity. Suppose the uniform bound is M.

1.4. Normal Families

15

For a fixed zo E D , consider the closed disk B(zo,r ) Then for all z E B(zo,r/2),

c D.

Let I? = C(zo,T).

Hence

So for z E B(zo,r/2),

Now let K be a compact subset of D. Clearly T can be chosen so that for every ,-z, E K ; the closed disks B(z0,r) C D. Cover K by disks centered at each point of K and of radius 7-14 and take a finite subcover. Let {&) denote the centers of the resulting disks, Mk = m a x Z E m (z)l, and S = maxk Mk (since there are ). if for only finitely many disks). Given E > 0, let 6 = 6(c) = min(r/4, ~ ~ 1 4 5 'Then, zl, zz E K , I Z I -zzI < 6, we have by the construction that for some Ck, Iz2--CkI,I< r / 4 and so lzl - GI < 6 7-14 < min(rI2, r/4(1+ €15')) 5 7-12. Since the finite set of disks centered at the points {&) covers K, we get by (1) that if lzl - z21 < d, then

If

+

Note 4.4. If we modify definitions 4.1 and 4.2 so that the functions f in the family F are allowed to take values in a metric space M , then Theorems 4.1 and 4.2 remain true. If, for example, we take M to be the Riemann sphere with the chordal metric (whereupon oo is like any other point), it can be shown their converses are also true. We have here only developed a small portion of the theory of normal families for the purpose at hand. Some more results on normal families appear in Chapter 2. The theory was developed extensively by Monte1 in particular. His book L e ~ o n s sur les Familles Normales des Fonctions Analytiques, first published in 1927, was reissued as a Chelsea reprint in 1974, and may be consulted by readers interested in pursuing these ideas further. It may be mentioned that there are proofs using normal families of the Picard Theorems (viz. Section 2 below), and extensions of some ideas in Chapter 3 can be given with this theory. Note 4.5. It is easy to see by using Cauchy's integral formula that if F is a family of functions that are uniformly bounded on every compact subset of a region D , then the family of derivatives is also uniformly bounded on every compact subset of D . It follows from Theorem 4.2 that if a family is to provide a counterexample of the sort illustrated in Note 4.3, then the family cannot be uniformly bounded on compact sets.

16

1. Conformal Mapping and the Riemann Mapping Theorem

Note 4.6. As a consequence of the results of this section, some authors call the property "uniformly convergent on compact subsets" by the term "normally convergent ." Note 4.7. Recalling that the chordal metric on the Riemann sphere is given by

it is fairly easy to see that a family 3 of analytic functions f is normal in a region D if and only if the expressions

are uniformly bounded on every compact subset of D. (See Theorem 2.2.4).

Note 4.8. A version of Theorem 4.1 for continuous functions on a Banach space (with values in a complete metric space) can be found in DieudonnB, Foundations of Modern Analysis[55]. Topological and uniform space versions of Theorem 4.1 appear in the last chapter of Kelley, General topology. Theorem 4.3 (Hurwitz). If I f n } is a sequence of analytic functions which is never 0 in a region D and if fn(z) converges uniformly to f (z) on every compact subset of D , then f (z) is either identically 0 or never 0 on D .

Proof. Suppose f (z) # 0. Then the zeros of f (z) are isolated. So, given zo E D , ) I = m > 0. there is an r with f (z) # 0 for z E B(z0, r ) - {zo). Then i n f z E ~ ( z oI ,f r(2) Hence -L converges uniformly to on C(zo,r). Also (by a well-known result fn ( z ) of Weierstrass), fA(z) converges uniformly to f '(2) on C(zo, r ) . Hence

But, by hypothesis, each integral on the left equals 0. Hence the integral on the right is 0 and so f (zo) # 0. Since zo E D was an arbitrary, the theorem follows.

Note 4.9. the sequence { fn(z)) in Theorem 4.3 forms a normal family of functions on D . The case of f r 0 can actually occur as the example fn(z) = +ez shows. Note 4.10. Theorem 4.3 can also be proved by the technique of using Rouchh's theorem with a constant function exhibited earlier. The reader may be interested in attempting t o construct this proof.

1.5. The Riemann Mapping Theorem

1.5

The Riemann Mapping Theorem

Let R be a simply-connected region in the plane with at least two boundary points. Then there exists a function g(z) which is univalent in R and maps R onto B ( 0 , l ) . If further it is required that g map a given zo E R onto 0 and satisfy gl(zo) > 0 , then g is unique. Theorem 5.1 (Riemann Mapping Theorem).

Before proving the Theorem, several remarks are in order, so that we may have a better understanding of just what it says. Note 5.1. It is impossible to map an open simply-connected region with just one boundary point onto the open unit disk, as it is clear from the fact that the boundary point (if it is not m) will be a t most a removable singularity of the function; application of Liouville's theorem then shows that (in all cases) the only possible such functions are constants. Two boundary points in fact implies continuum many boundary points, but only the existence of two is used in the proof. Note 5.2. The boundary of a simply-connected region in the plane need not look anything like a Jordan curve. For example, consider the region in

Diagram 1.4 Diagram 1.4, which represents a square with corners (0, 1,l + i , i ) ,with segments of height 112 perpendicular to the real axis at the points 2-"' deleted.

18

1. Conformal Mapping and the Riemann Mapping Theorem

This region is simply connected. On the other hand 0, which is clearly a boundary point, cannot be reached by a continuous curve from any point in the region's interior. Boundary points of this sort are called inaccessible. Because of boundaries of this sort, one cannot say anything in general about the boundary of R going onto C ( 0 , l ) in a univalent fashion under the Riemann mapping function g. For example, if F is the Riemann mapping function for this region and zo corresponds to 0, then any of arc of C ( 0 , l ) containing zo will necessarily contain a point whose pre-image is a t distance 2 112 from 0; thus on the boundary the map is not continuous. However, even accessible boundary points (those which an be reached by a continuous curve from an interior point) can cause trouble. Consider the simplyconnected region in Diagram 1.5:

Diagram 1.5 the upper unit semi-disk, together with for each rational plq (p, q relatively prime, p < q) the line segments of length l / q and argument xplq, with common endpoint 0, deleted. Suppose we have two sequences of points in the interior of the region converging t o 0 and lying along segments of argument ncr and TP respectively, where a and /3 are irrational. Because of the "rational boundary segments" lying "between" these two line segments, the images of the sequences under a Riemann mapping function F (which is conformal) must converge t o different points of C ( 0 , l ) . In fact, clearly there are continuum many points of C ( 0 , l ) all of which are "images" of 0 under F in the above sense. While the above shows that the notion of a simply-connected region permits some bizarre examples, nevertheless, the following two theorems, neither of which will be proved here, show that the "usual situation" is much better behaved.

Theorem (Osgood-Carathkodory). If F ( z ) is a Riemann mapping function for a simply-connected region R whose boundary is a Jordan curve C , then F ( z ) is continuous on C and maps it one-to-one onto C ( 0 , l ) . Theorem (on accessible boundary points). The images of accessible boundary points are everywhere dense i n C ( 0 , l ) .

1.5. The Riemann Mapping Theorem

19

A proof of the Osgood-Carathkodory Theorem can be found in Carathkodory's book, Conformal Representation[34]. A proof of the slightly less general, but usual, case of a simply-connected region whose boundary consists of a finite number of smooth Jordan arcs, can be found in Nehari [168]. The theorem was first conjectured by Osgood in 1900 and first proved by CarathQodoryaround 1911; this proof involved use of concepts from Lebesgue's theory of measure and integral. At the same time, Osgood and Taylor[l79] also provided a (quite different) proof. A further contribution to the theorem by Carathkodory appears in Gottinger Nachrichten, 14 (1913), pp.323-370. Carathkodory developed his theory of "prime ends" to handle the behavior of simply-connected regions with arbitrary boundaries. In particular, the introduction of "prime ends" allows a theorem involving one-to-one correspondences between C ( 0 , l ) and corresponding prime ends arising from convergence towards the boundary (from the interior) of the original simply-connected region. CarathQodory's three papers can be found in Volume IV of his Gesammelte Schrzften (Munich, 1916). The theorem on accessible boundary points can be found in Volume I1 of Bieberbach's Lehrbuch der Funktionentheorie (reprinted Chelsea, 1945, p.29). The first chapter of this book of Bieberbach and Chapter I11 of Volume I1 of Carathkodory's Theory of Functions contain further discussion of the mapping of the boundary. Note 5.3. The Riemann mapping theorem would appear to be significant even topologically, as a proof that all simply-connected planar regions with at least two boundary points are topologically equivalent. I know of no simpler proof than that below if one were to ask only for a continuous mapping. We now turn to the proof of Theorem 5.1; the one given below, which has become standard, is due to Fejkr and Riesz and depends on ideas introduced by Koebe and Carathkodory. Proof of Theorem 5.1. Let 3 be the family of all functions f which are univalent and bounded in R, map a given 2.0 E R onto 0 and satisfy fl(zo) = 1. If 3 is non-empty, let

and p = inf m ( f ) . JET

The existence part of the proof now proceeds in three steps: 1. 3 is non empty. 2. There is a function f * E 3 such that m(f*) = p.

3. f * (z)/p is the desired function.

20

1. Conformal Mapping and the Riemann Mapping Theorem

Proof of (1): Let a , b,a function

# b be two boundary points of R and consider the ~ ( 2 =)

z-a

\Iz

where it is understood we stay on the same branch for z E R (start with a C E R and analytically continue throughout R; this is possible since 5 # 0, # co for z E R , and since R is simply-connected, the monodromy theorem guarantees that the resulting function is analytic and single-valued throughout R). An easy computation shows that w(z) = so defined is also one-to-one in R. Let

\/= I

Then, by continuity, there exists a neighborhood B(w0,6) of wo such that each w E B(w,, 6) is taken on for some z in a neighborhood B(zo,6') C R. Hence, since w is single-valued in R , it cannot take on any value in B(-wo, S), provided 6 is sufficiently small. (The only candidates for pre-images lie in B(zo, 6') which maps into B(wo, 6)). It follows that for given constants C and d, C # 0, the functions

are bounded in R, and they are clearly still univalent in R. By choosing C and d appropriately we get a function in F. In fact, taking

(a computation shows that since a , b E BdR, and zo E R , wl(zo) # 0, co), we obtain the function

with a derivative a t zo

Hence F is non-empty. Proof of (2): If F has only finitely many elements, then the existence of f * is trivial. If F has infinitely many elements, then, by the definition of p, for every integer n , there is a function f n E F1such that m ( f n ) < p l l n 5 p 1. Hence, by Theorem 4.2, the set of functions { f n ) is a normal family, and so (since the f n are uniformly bounded), there is a subsequence { f,,) converging uniformly on compact subsets of R.

+

+

1.5. The Riemann Mapping Theorem

Let 4 ( z ) = limk+, f n k ( z ) . Then, by uniform convergence,

4'(zo)= lim fAk(zo)= 1 . k+w

The second result also shows that 4 is non-constant in R , and since the f n are univalent, from Theorem 3.6 we get that 4 is also univalent in R. Hence 4 E F. Finally, p 5 m ( 4 ) 5 lirnk,, p 2= p, and so 4 is the desired f * (there may nb be more than one such f * for all we know now, but the existence of one is all that is required). Proof of (3): Since f *'(zo)= 1, f * is not identically 0 on R ; hence p > 0. Let g(z) = f * ( z ) / p .Then g is univalent on R, g(zo)= 0, gl(zo)= lip, and Ig(z)l < 1, for z E R. It remains to show that g actually effects a mapping onto B ( 0 , l ) . Suppose not. Then there exists a: # 0, la1 < 1 such that g(z) # a: for all a E R.

+

Fix a value of

&iand consider one branch of h ( z ) =

dm

restricted to R with

h(zo) = &i. ( h ( z )is single-valued in R since > 1, and so by hypothesis, the radicand # 0 or oo for z E R.) Clearly h ( z ) is analytic and an easy computation shows it is univalent. Also, h 2 ( z ) = L ( g ( z ) )where L is a linear fractional transformation mapping B ( 0 , l ) onto itself; hence Ih(z)l < 1 for z E R. Let

The same argument shows that Ik(z)l < 1 for z E R. k is univalent (since h is) and since la1 < 1 h(zo) - 6 = 0 . k ( z O )= &h(zo) - 1 We wish to normalize k so that kl(zo)= 1. It turns out that

and

So, since gl(zo)= l/p, we have h1(z0)=

$fi(la:12

- I ) , and kl(zo)turns out to

= $. Let S ( z ) = @ k ( z ) . Then S ( z o ) = 0, S1(zo)= 1, S is univalent and bounded in R; so S E F,but since Ik(z)l < 1 for z E R, 2 4 4 m ( S ) = sup IS(z)l 5 z€R la:] • l P <

1. Conformal Mapping and the Riemann Mapping Theorem

22

contradicting the definition of p. Hence g is onto, and the first part of the theorem is proved. To prove uniqueness under the conditions stated, note that iff and 4 satisfy the conditions of the theorem and F ( z ) = f (4-'(z)), then F maps B ( 0 , l ) onto itself, is univalent, and F(0) = 0. Hence, by the same proof as for Theorem 3.4, F ( z ) = eiez,

I3 a real constant, 0

< I3 < 27r. But eie = F'(0) = -&f (4-'(2))

hypothesis. Hence 13 = 0 and F(z) r z.

Note 5.4. Riemann's Theorem is stated in his Gottingen dissertation of 1851; as Weierstrass pointed out the proof given there is insufficient. The first completely correct proof was given by H.A. Schwarz in 1896 (for convex regions whose boundary consisted of a finite number of smooth Jordan curves). Schwarz' method is set out in Hurwitz-Courant, Funktionentheorie [217]. A correct proof along the lines suggested by Riemann was first given by Hilbert around 1900. A variant by Herman Weyl of Hilbert's rehabilitation of the so-called "Dirichlet Principle", simpler than the original, appeared in his famous book Die Idee der Riemannschen Flache [245]. An English edition also appeared posthumously as The Concept of a Riemann Surface (Addison-Wesley 1955). The above proof of Fejkr and Riesz was first published in 1922 (by T. Rad6 [208]). Rad6's paper contains references to the most important earlier work of Koebe and Carathkodory in this connection. A rather unusual proof of the Riemann Mapping Theorem as a consequence of the solution of a particular extremal problem for univalent functions was given by Garabedian PI. Note 5.5. The Riemann mapping theorem is closely connected with the solution of the Dirichlet problem for simply-connected regions. Suppose $(x, y) is a solution of Laplace's equation in a region D and f (z) = u(x, y) iv(x, y), z = x iy, is a univalent mapping of D onto some other region f (D). Then $(x, y) goes into @(u,v)defined on F ( D ) . A straightforward compu4,, = 0 implies ,@, a, = 0. Thus the existence of tation shows that $,, a solution to the Dirichlet problem for a simply-connected region whose boundary is a simple (rectifiable) Jordan curve follows from the Riemann Mapping Theorem and the Poisson integral formula which provides a solution for the disk. In fact, if g is the Riemann mapping function for a simply-connected region D such that g(zo) = 0, g1(zo)> 0, then one can conclude from the maximum principle for harmonic functions that G(z,zo) = -1 logg(z)l is the Green's function for D with singularity at zo. Copson [50] consequently makes the remark that for anyone for whom the existence of a certain electrostatic potential is intuitively obvious, the Riemann mapping theorem is also intuitively physically obvious.

+

+ +

+

The most obviously unsatisfying thing about the proof of Theorem 5.1 is that it is purely an existence proof, but does not tell us how to construct a univalent mapping between a planar region and B(0,l). We address this question now.

1.5. The Riemann Mapping Theorem

23

Theorem 5.2. Given a simply-connected region R, there is a sequence of functions &(z) such that limn,, $,(z) = $(z), $(z) maps R univalently onto B(0, I), and the distance of Bd $,(R) from C ( 0 , l ) is 5 $, K a constant, as n + co.

Proof. In step (1) of the proof of Theorem 5.1 we showed that there exists a function univalent and bounded on R: multiplying, if necessary, by a suitable constant, we get the existence of a function F ( z ) such that F is univalent on R, and IF(z)l < 1 for z E R. Suppose F maps R onto Ro and Ro # B ( 0 , l ) (if it should, no further argument is necessary). Let be a point in BdRo nearest the origin, i.e. for all z E Bd Ro, 1x1 2 [
<

.

T(z) = continued throughout Ro (by the monodromy theorem as a singlevalued function, since Ro is simply-connected). T maps onto 0 and 0 onto fl. Hence we look at the map

<

Clearly, $0 is univalent, do(0) = 0 and I$o(~)l5 1 for all z E Ro. (The attentive reader will notice a certain resemblance to the arguments of step (3) of the proof of Theorem 5.1). Suppose maps Ro onto R1. Let $ be the (clearly existing) inverse of 40. $ is the composition of a linear fractional transformation mapping B ( 0 , l ) onto itself, z2, and another linear fractional transformation mapping B ( 0 , l ) onto itself (viz. proof of Theorem 3.5). Hence $ is analytic on B(0, l ) , and I$(w) 1 5 1 for w E B ( 0 , l ) . We already know that $(O) = 0, and so by Schwarz' Lemma, 1$(w)1 < lwl for IwI < 1, w # 0 (since clearly $(w) # eisw) and 1$'(0)1 < 1. Hence, writing w = $o(z), we get for z E R, z # 0, I$o(z)l > IzI and l$b(0)1 > 1. Thus the distance of R1, the image of Ro under $0, from C ( 0 , l ) is < the distance of Ro from C ( 0 , l ) . Now let <1 be a point in B d R l nearest 0. Say l n, @,,,(z) = $m($m-l (. . . $n+l(z) . . . ), @,, and maps it into B(0, I), and @,,,(0) = 0; hence by Rn+1 and hence in B(0, Schwarz' Lemma again,

24

1. Conformal Mapping and the Riemann Mapping Theorem

On the other hand, since 4k(O) = o for all k, manifold use of the chain rule gives

Combining this with the preceding equation, we get for a given n

> 1and all m > n ,

Now, an easy computation (cf. proof of step (3) for Theorem 5.1) shows that

A similar computation holds for each

$k

and hence, from (5.1) we get

-

Letting m -+ ca, (5.2) guarantees the convergence of the resulting infinite product 3 1 as k -+ ca. But (since ?$$ > 1). Hence,

:+&

and since pk is bounded as k + ca, it follows that limk+, pk = 1. Furthermore, since @,(z) = a(n)z + . . . , a(n) # 0, and 9,(z) = a(m)z . . . , a ( m ) # 0, is analytic and has no zeros in Ro exists and is not zero. Thus lim,+o (9, is univalent and cPn(0) = 0). Hence by the maximum and minimum modulus theorems,

+

Therefore, l i m ; ~ ~ can define

= 1 uniformly on compact subsets of Ro, and so we

(2;;)

w,,m(z) = log -

as an analytic function throughout Ro. It follows that lim Re w,,,(z)

n+w m+w

= 0,

z E Ro .

1.5. The Riemann Mapping Theorem

25

By differentiating in the Poisson integral formula (see Appendix), we find on putting z = reiB, ~ ~ , ~= U(T, ( z 8)) + iv(r, 8), that uniformly on compact subsets of Ro

whence by the Cauchy-Riemann equations (in polar-coordinate form) lim n-tm w,,,(z) m+w 0 uniformly on compact subsets of Ro. In other words lim n,m+m

=

cp,O = 1, whence @,(z)

lim @,(z) - @,(z) = 0 ,

n,m-tm

and so the functions @,(z) converge uniformly on compact subsets of Ro to a function @ (z). Since for z E B d R o , by the maximum modulus theorem, pn 5 I@n+l(z)l 1, letting n -+ oo, we get for z E Bd Ro, I@(z)l= 1. Since @(O)= 0, @ is non-constant, and so by Theorem 3.6, @ is univalent. Composing @ with F , the existence part of the theorem follows. To estimate the speed of convergence, take logarithms in (5.2), and m = 2n. Since is monotone decreasing for real z > -1, taking sight of (5.3) we have, since pk < 1,

<

2n

- logpn+l

2

C

log

k=n+l

>

5

k=n+l

log (1

+

(1- fi)2

) 2 k=n+l 5 ('

(5.5) -

&I2 2

2 1og3/2 .

and

Substituting (5.6) and (5.7) in (5.5) gives (setting a =

log(&)

(l-pl) log 3,2

)

26

1. Conformal Mapping and the Riemann Mapping Theorem

Hence, since the pn are monotone increasing in size,

Applying (5.8) v - 1 times to the expression

we obtain

and so,

Taking for n

> 2, v = [

I+

1 in this last inequality gives finally

which proves the result.

Note 5.6. The preceding argument as a proof of the Riemann mapping theorem goes back (including the use of Schwarz's Lemma) to Koebe [133]; the observation that this proof can be used to provide an estimate of how fast the functions an converge as n + oo to the Riemann mapping function is Ostrowski's[180]. The determination of the values of pn in the proof may be difficult; a large part of Ostrowski's paper is devoted to analogous proofs in which it is not necessary to know the distance from the origin to the boundary of an image domain. Of course, then we cannot necessarily expect in general convergence even at the rate of 0 In general, the functions @, converge so slowly as n + oo that to the best of my knowledge, this construction has never been explicitly carried out to approximate the mapping function for a given region. Consequently for simply-connected regions of various special shapes, other methods have been introduced for approximating the Riemann mapping function or its inverse. One of the stimuli to these investigations was aerodynamic problems. The advent of high speed computation caused a large increase in interest in such "constructive methods" in conformal mapping. However, mathematical concern with e.g, problems of integral equations has also contributed to interest in these methods. The interested reader should consult the book by Dieter Gaier[80].

(i).

1.5. The Riemann Mapping Theorem

27

As a simple example of another method for approximate construction of the mapping function, we have

Theorem 5.3. Let g be the Riemann mapping function for a simply-connected region R such that for a given zo E R, g(zo) = 0, gl(zo) > 0. Let u be any other function univalent on R such that u(z0) = 0 and ul(zo) = g l ( z O ) Suppose u maps R onto R*. Then the area of R* is 2 n (= area of unit disk), and = n only if u 5 g. Thus to determine g , one may attempt to find functions u such that J& I ~ ' ( z ) 1 ~ dy dx is minimal.

Proof. Let $ = u o g-l. Then $ maps B(0,l) onto R*, is univalent, $(0) = 0, and = = 1$'(O) = ~ ' ( g - ~ ( o ) ) By the area theorem (see Appendix), if J = area of R*, and if $(z) = 2 + C r = 2 anzn,

#

unless all the a, = 0.

Note 5.7. For some (nearly-circular) regions, the approximation indicated by Theorem 5.3 by truncating the power series for u1 and considering only polynomials of a fixed degree has actually been carried out in practice (viz. Hohndorf[ll5]). Although the explicit conformal mapping between two given simply- connected regions which is implied by the Riemann mapping theorem may be difficult to find in practice, there are cases of considerable importance where an explicit formula for conformal mapping can be given. One such is the case of a conformal mapping of B ( 0 , l ) or a half-plane onto the interior of a polygon. Since half-planes and disks can easily be transformed into one another by linear fractional transformations, these problems are equivalent, and we discuss the mapping of the upper half-plane {z : I m z > 0) onto the interior of a polygon. Theorem 5.4 (Schwarz-Christoffel Formula). Let P be a polygon bounding a simply-connected region with vertices at the points ak, Ic = 1,.. . , n, and interior angles akn, 0 5 akn < 2n, k = 1 , . . . ,n , a k # 1. Then

maps the half-plane {z : I m z > 0) conformally onto the Jordan interior of the polygon. Here the real numbers Ak are defined b y requiring F(Ak) = ak (if one of the Ak is oo, then that term is simply omitted in the above formula). K1 and K 2 are complex constants depending on the position and size of the polygon.

28

1. Conformal Mapping and the Riemann Mapping Theorem

Proof. Before beginning a proof, let us stress that the polygon P need not be convex (e.g. Diagram 1.6, where n = 7) and that the case of an unbounded polygonal region (one vertex at oo) is of considerable interest (e.g. Diagram 1.7). A triangle with one vertex at oo (a1= 112, a2 = 112, a3 = 0)

Diagram 1.6

Diagram 1.7 It is worth observing that all a k < 1corresponds to the case of a convex polygon. Suppose F(z) maps the upper half-plane univalently onto the Jordan interior of P. Let Ak,k = 1 , . . .n be the distinct points which are the pre-images of the ak under F (a simple argument shows that since we are dealing with polygonal regions, the Ak are well-defined by continuity). By the Schwarz Reflection Principle (see Appendix), F can, in fact, be continued analytically across each segment of the real axis determined by the points Ak, except possibly at those points themselves.

1.5. The Riemann Mapping Theorem

29

For now, assume no Ak is m . To determine the character of the singularity at Ab, note that for a sufficiently small neighborhood of Ak, F maps a small segment of the real axis onto two segments intersecting at an angle of xak. Since the map z --+ zs (0 > 0) takes rays from the origin into rays from the origin, but multiplies the angle between such rays by 0, it follows that

maps a small segment of the real axis containing Ak onto two straight line segments intersecting in an angle of x , i.e. onto a straight line segment (described once); hence once more by the Schwarz Reflection Principle, G is analytic at Ak. Furthermore, G1(Ak)# 0 (this follows from the univalence of F in the upper halfplane and its analytic continuation by Schwarz reflection). Hence, by Theorem 3.7 (translated), G is univalent in some neighborhood of Ak, and so in this neighborhood, G(z) = c ~ ( z - A ~+) c z ( z - A ~ ) ~ +... Since F ( z ) = F(Ak)

+ (G(z))*"

, where C1 # 0 .

(5.10)

we get on differentiating and dividing, for

in a neighborhood of Ak. Hence, since G1(Ak)# 0, by (5.10), in a deleted neighborhood of Ah

has a simple pole at Ak. where Gl (z) is analytic. Thus Treating each Ak in this way, we get the Mittag-Leffler expansion of

5,

where H ( z ) is analytic in the closed upper half-plane and in fact (by the Schwarz Reflection Principle) entire. Furthermore, F is analytic at oo (oo was not one of the Ak in this case), and ... ; hence in a neighborhood of m has an expansion of the form bo therefore, in a neighborhood of m ,

+%+3+

I . Conformal Mapping and the Riemann Mapping Theorem

30

z;=,%

$$#

and so is analytic at m . Since is analytic at m , it follows from Liouville's Theorem that H (z) is constant, and indeed by (5.l l ) , since vanishes at m , H(z) equals lim,,, Thus (5.11) gives

z;=l%

$$f = 0.

and integration and exponentiation produce the formula (5.9) of the Theorem. If one of the Ak, say Al, should be m , we first choose a linear fractional transformation

which maps oo onto A; (real positive) and y (real positive) is chosen so that all the other Ak map into finite points A; (where the map is conformal). Then, by what has just been proved

say, maps the upper half-plane onto the Jordan interior of the polygon with vertices at a k where F(A;) = ak. But ( = @ and so setting t = $, we get z+y '

However, since the xak are a11 the interior angles of a polygon, and so a1 1 = - C;=2(ak - I), and (5.12) becomes

+

F ( z ) = -K;(-ATy)"' - Kl

lz

C;=, xak = (n-2)x,

((AT - A;)w - rA;)"k-ldw

f i ( ~ - Ak)Ok-'dw

+ K,'

+ K Z , for some K l , K2

k=2

(Observe that $$ = C 1 ( O . ) Hence the effect of Al = oo is simply to eliminate that term from the product in the integrand.

Theorem 5.5. (Schwarz-Christoffelformula for B(0,l)): If F maps B ( 0 , l ) conformally onto the Jordan interior of a polygon p, and notation is as in the preceding

1.5. The Riemann Mapping Theorem

Theorem, then

where Bk =

s,

and C1 and C2 are cornplez constants.

-is s.

maps B ( 0 , l ) onto the upper Proof. The linear fractional transformation C = A computation shows that the half-plane. The inverse of this map is z = Schwarz-Christoffel formula for B ( 0 , l ) has the same form as the formula for the upper half-plane with different constants.

Note 5.8. Given an n-gon with vertices a t the Ak, the crk are known and it is necessary to find the Ak or Bk in order for the Schwarz-Christoffel formula to be explicit. Using linear fractional transformations, it is clear that three of the Ak say, may be chosen a t will. And so for n = 3, we see that the mapping function depends only on the angles (triangles with the same angles are similar). For n > 3, there remain n-3 values of Ak corresponding to the remaining ak. In general, no solution which gives these constants is known, and unless the polygon has some special form (such as a great deal of symmetry), a t present they can only be evaluated numerically. The triangle is discussed in Example 5.3, the rectangle in Chapter 8 below, particularly section 4 through Definitions 4.1. Note 5.9. There are other explicit formulas for dealing with mappings of a halfplane or a disk onto the Jordan interior or exterior of a polygon-like region whose bounding curves are arcs of circles. This latter case leads (for triangles) to Gauss' hypergeometric function F ( a , b, c; z). For these formulas, see Nehari[l68] and Sansone and Gerretsen[215]. Physical applications of the Schwarz-Christoffel formula can be found in Henrici[ll2]. Henrici also considers the problem, important for applications, of mappings with "rounded corners". All of these references have a number of examples of the Schwarz-Christoffel formula. Example 5.1. Consider Theorem 5.5 and the mapping

This is of the Schwarz-Christoffel form with Bk = e2nikln,k = 1 , 2 , . . . , n, and = 1 - 2/n, k = 1,. . . ,n. Thus F maps B ( 0 , l ) onto the interior of a regular n-gon (with interior angles (1 - 2171)~).The points qk = e2nikln,k = 1, 2 , . . . , n , map onto the vertices of the polygon, and

32

1. Conformal Mapping and the Riemann Mapping Theorem

is the radius R of the circumscribed circle. Thus

by Euler's First Integral. Since the length C of a side of the polygon is 2Rsinr/n, we have that

(since r ( x ) r ( l - x ) = &) Example 5.2. The function

maps B ( 0 , l ) onto the interior of an &gon with the angles at the vertices corresponding to e2k"ils, k = 2,4,6,8 being 0 and at those corresponding to e2k"i/8, k = 1,3,5,7, being r / 2 . Thus the region looks like, say,

Diagram 1.8 It is easy to check (since i4 = 1) that the widths of each infinite strip are the same, and so the dotted lines indicate a square. In fact, this width C is

1.5. The Riemann Mapping Theorem

33

However, a slightly easier integral to evaluate is the length of the diagonal of the dotted square, and so we get

Making the substitution v = t a , expanding using Euler's First Integral again, we get that

& in a geometric progression

and

+

r(k+l) = k-lI2 O ( ~ C - ~as/ k~ + where the series converges since ---4) co. (This is, in r(k+ ,) fact, a hypergeometric series evaluated at -1. Using results on the hypergeometric r2(') function, one can obtain for l the exact evaluation l = .)

++ 9

Example 5.3. F ( z ) = J : wa-I (1 - w)P-ldw ,maps the upper half plane onto the , ~ y where , y = 1 - a - 8 , and Jordan interior of the triangle with angles ~ aTP, with O,1, oo going into respective vertices of the triangle. If a is the side opposite to angle ~ athen , the length of a is

&

on making the substitution w = and by Euler's First Integral again. Since a /3 = 1- y, this last expression = sin .rrar(a)r(p)r(y). By the law of sines (or . by . similar computation), the other two sides have lengths sinipI'(a)I'(p)r(y) and sin TyI'(a)l?(p)r(y): The vertices of the triangle-are at 0, !l$M, .-ria r(a+O)

+

a

4 m.

Note 5.10. It can be shown that the function inverse to the one discussed in Example 5.3, which maps the interior of the triangle onto the upper half-plane, can be continued analytically throughout the plane as a single-valued function only if a,/3,y are reciprocals of integers. Assuming (as above) that all the vertices of i, , , the triangle are finite, there are three cases only: ( a , P, y) = (h, Each of these leads to a doubly periodic function (see Chapter 8).

5 , i).

(i, 9) (h, i, i)

Note 5.11. The reader will, no doubt, wonder why we have not discussed mapping of multiply-connected regions. For simply-connected regions with more than

1. Conformal Mapping and the Riemann Mapping Theorem

34

one boundary point, the Riemann Mapping Theorem shows that B(0,l) provides a canonical domain. For doubly-connected regions (e.9. open annuli), there is an infinite one-parameter family of canonical domains. For regions of connectivity n , n 2 3, there are 3n - 6 parameters in which two domains of connectivity n must agree in order for them to be mapped into one another. The mapping may be constructed as the limit of a convergent sequence of maps by a version of Koebe's construction of Theorem 5.2. Explicit consideration of multiply-connected regions may be found in the cited book of Nehari [168], (Chapter VII), and in Golusin [85], (Chapter V and VI), where the existence and uniqueness of conformal mapping functions onto canonical domains (e.g. "Parallel slit domains") for regions of con2 is proved. Constructive methods are discussed in Gaier's [80] book nectivity already cited. A survey article by Gaier [81] appears in Jahresbericht der D.M.V. 81, 1978, 25-44. Infinitely-connected regions are also discussed there.

>

As an example of what may happen in regions of connectivity the following simple

> 2, we prove

Theorem 5.6. There does not exist a univalent function f mapping the annulus {z : 0 < rl < IZ - zol < r2) onto the annulus {z : 0 < rs < lz - zol < re) and continuous on the closed annulus unless rzlrl = 7 - 4 1 ~ 3 . Proof. Consider first, the special case of the two annuli A1 = {z : 0 < r < lzol < 1) and A2 = {z : 0 < R < lzol < 1). Suppose r # R and a function f mapping Al univalently onto A2 and continuous on A1 existed. By the Schwarz Reflection Principle (see Appendix), it can be continued analytically for all z # 0, co. Call the resulting function again f which is univalent for all z # 0, then f (z) has a removable singularity at 0, and in fact lim,,~ f(z) = 0. It follows from the Casorati-Weierstrass Theorem that the isolated singularity at co must be a pole (i.e. f (co) = co). Hence f is univalent, maps C onto C, and has a pole at oo, so f must be rational, and in fact, must be a linear fractional transformation; Removing the singularity a t 0, f (0) = 0 and f (co)= co, give so f (z) = b = 0, c = 0. Furthermore, (z)l = 1 for lzl = 1 (by continuity), so f (z) = eiez, 0 real, and so if lzl = r , If (z) 1 = r , a contradiction. By translation and dilation, the theorem follows.

s.

If

Several of the books cited in this chapter contain discussions of the properties of special conformal maps.

Chapter 2

Picard's Theorems 2.1

Introduction

In 1879 Picard proved that an entire function takes on every value with at most one exception, (Picard's "Little Theorem"), and that in any neighborhood of an isolated essential singularity, an analytic function takes on every value except a t most one, (Picard's "Big Theorem"). Hadamard (1892) and Bore1 (1896) began to incorporate Picard's results into entire function theory (see Chapter 3 for some of these results), while Landau, Schottky, and Carath6odory (1904-5) found deepenings of the theorem itself. As a consequence of two important papers which were published in 1924, the theory branched in two directions. First, Andr6 Bloch discovered a new "elementary" proof of the Picard, Landau, Schottky results. "Elementary" here means the following: Picard's proofs involved the use of a particular transcendental function, the "elliptic modular function", which in fact, is related to a certain map of Schwarz-Christoffel type (See Chapter 1, Section 5); Bloch found a way to eliminate all use of such functions. Furthermore, his approach presents new ideas and problems, some of which are still unsolved. Bloch's original presentation contained a condition of univalence; this was removed by Landau. Second, in the same year, Nevanlinna gave his proof of Picard's Theorem which led to the contemporary theory of meromorphic functions, or "Nevanlinna theory", an introduction to which can be found in Chapter 4. Finally, the circle of problems discussed here is also connected with Montel's theory of Normal Families (See Chapter 1, Section 4). The presentation of the "elementary" proofs of the Bloch, Landau, Picard, and Schottky Theorems follows Landau's 1927 volume, Darstellung Einige Neuere Ergebnisse der Funktiontheorie.

2. Picard7s Theorems

2.2

The Bloch-Landau Approach

Theorem 2.1 (The Bloch-Landau Theorem). Let f ( z ) be analytic i n B ( 0 , I ) , and suppose 1 fl(0)l 2 1, then the image of B ( 0 , l ) under f contains an open disk of radius R > 0, and in fact, we can take R = 1/16.

< <

r 1, and consider the numbers Proof. Let M l ( r ) = maxl,llr 1 f l ( z ) I , 0 ( 1 - &), k an integer 2 0. Then for k = 0 , M l ( 0 ) 2 1 by hypothesis, and since by the maximum modulus theorem, M I ( 1 - &) M1(1), we have that

Hence, there is a largest Ic 2 0 , say ko, such that nience, let r = 2-". Then

( 1 - &) 2 1. For conve-

By the maximum modulus theorem and (2.2),there is a complex number 5 with [[I = 1 - r such that 1 f1(5)1 = M l ( 1 - r ) f . Let g ( z ) = f ( z + 5) - f (C). Then g ( z ) is analytic on B(O,r), g(0) = 0 , and

>

On the other hand, for z E B ( O , r / 2 ) ,we have

Hence,

+ < Ml(1-

max lg (z)l = max I f l ( z [)I I~I~TI~

Izll~l2

7-12) < 2 / r ; ,

on using (2.2). So for z E B ( O , r / 2 ) ,

Let y be a complex number such that g ( z ) # y for z E B ( 0 , r / 2 ) . We will show that Iyl 2 Since g(0) = 0 , it follows that the image of B ( 0 ,r / 2 ) under g contains an open disk of radius 1/16; hence the image of B ( c , r / 2 ) under f contains a disk of radius 1/16 (centered at Since if z E B ( C , r / 2 ) , then IzI 151 7-12 = 1 - r / 2 < 1, this proves the theorem. Knowing y # 0 is not enough to prove the theorem for even some R, since we need to know that y is bounded away from 0 for all f .

&.

<

+

f(c)).

37

2.2. The Bloch-Landau Approach However, since y # O(g(0) = O), 1 and takes the value 1 a t 0. Let

is analytic and non-zero in B(O,r/2)

(where we can pick the principal branch and continue throughout B ( 0 , l ) analytically since g(z) # y and is analytic there). Then, since h is analytic in B(0, I ) , it has a power series

By Parseval's Theorem (which has a well-known almost trivial proof for analytic functions),

since lg(z)l

< 1, for

on using (2.3). Hence, )yl1

I

E B(O,r/2). So, since co = 1 and cl =

e,

we get from

&.

Note 2.1. Bloch originally required that the disk of radius R in the above be taken on univalently. Landau removed this condition. Although it is immaterial for the development of Picard's Theorem, there is a certain interest in knowing the largest disk covered in either case. Let L (Landau's constant) be the radius of the largest open disk necessarily contained in the image of B ( 0 , l ) under a function analytic on B ( 0 , l ) with 1 fl(0)l = 1. Let B (Bloch's constant) be the radius of the largest such disk where, in addition, it is required that the disk be taken on univalently. Clearly B L. The exact values of B and L are still not known. The best known bounds seem to be r(5/6)r(l/3) 112 < L 5 r'(1/6) and

<

2. Picard's Theorems

38

(see Theorems 3.3-3.6 below.) It is conjectured that the upper bounds are, in fact, the correct values of B and L. Although we have defined our functions f , to be analytic in the closed disk B(0, I), it perhaps worth noting that the value of B is the same if we use the open disk as domain instead. Indeed, if B' is defined similarly to B except with requiring f to be analytic only in B(0, I), then clearly B' 5 B. Given E > 0, by definition there is a function f analytic in B(0, I), such that the radius R of the largest disk it takes on univalently is < B' E . Then, if 0 < a < 1, the function g(z) = f (az) is analytic in B (0, and a disk it takes on in the image of B ( 0 , l ) has radius 5 R / a ; also g'(0) = fl(0);

4

+

Letting a

+ 1 and E -+

0, we get that also B

6)

< B', and so B = B'.

N o t e 2.2. The attentive reader of Chapter One and the above, will be convinced of the utility of the monodromy theorem in studying the assumption or omission of values by a function, since if g(z) is univalent and omits a , k(g(z) - a ) may be analytically continued (as a single-valued function) throughout a simply-connected domain where k is some function with a branch point at 0. This idea will continue to find repeated application in various ways. T h e o r e m 2.2 (Picard's "Little Theorem"). : A non-constant entire function takes on every value except at most one. Proof. By contradiction. Suppose F ( z ) is entire, non- constant, and omits a and /3 as values, a # /3. Then (and only then)

omits the values 0 and 1,is entire, and non-constant. So there is no loss of generality in assuming F omits 0 and 1. We now construct a function g, depending upon F , with gl(0) = 1 which omits a disk of radius 1/16, thus contradicting the Bloch-Landau Theorem. Since F ( z ) # 0 for all z and F(z) is entire, we can write

(1) F(z) = e2"ih(z)where h(z) is entire, and since F never takes the value 1, h never takes integer values. In particular (2) h(z) # 0, so h(z) = u2(z) where u(z) is well-defined and entire, and

(3) h(z) Then

#

-

1, so h(z) - 1 = v2(z) where v(z) is well-defined and entire.

(4) u2(2) - v2 (2) = h(z) - (h(z) - 1) Hence

1 and so u(z) - v(z) is never 0.

2.2. The Bloch-Landau Approach

39

(5) U(Z)- v(z) = ef("), say, where f is entire. To express F in terms of f , by (4) and (5), u(z) So

+ v(z) = A = e-f(") 4.)-4.1

.

and by (2) and (I),

(6) F ( z ) = exp ($(e2f("1

+ e-'f(")))

.

We now claim that the image of the plane under f as defined by (1)- (5) contains no disk of radius 1. A renormalization off will then provide a function g contradicting the Bloch-Landau theorem. To prove this claim, we show

(A) Every disk of radius 1contains some point y(m, n) of the form y(m, n ) = flog(+ + J s ) where m and n are integers, m 2 1, and

+ 9,

(B) The image of the plane under f contains none of the points y(m, n). The reader who follows through the proofs of (A) and (B) which follow, will readily see the motivation for the y(m, n). (A) follows since the difference of successive ordinates of the points y(m, n ) is 7~/2and the difference of successive abscissas = log(d3

Hence, for every complex

<,there is a y = y(m, n)

lReC - Reyl and so

I(

- y(m, n )1 <

< 112, and

and e-f'"~) = if(

for m = 1

(the nearest one) such that .rr

[ImC - I m y ( 5 4

fi , <2

= 1, which proves (A).

(B) follows since if for some m, n, f (zo) = f log(+ zo, then for this zo ef("") = if(

+ I),

+ d s )+ 9, for some

+ d K X ) * l i n = (fifJ a ) i n rJKX)i-n .

,

2. Picard's Theorems Hence by (6)

contradicting the assumption that F never takes the value 1. It remains to renormalize properly to obtain a contradiction to the BlochLandau Theorem. By construction f is entire and non-constant, so there is a J with fl(J) # 0. Consider

Since the image of the plane under f contains no disk of radius 1, the image of the plane under g contains no disk of radius &. But expanding g in a power series around J, we have

which satisfies the hypotheses of Theorem 2.1, and this is a contradiction.

Note 2.3. Lawrence Zalcman[254] has given an interesting proof of Picard's Little Theorem for all "practically occurring entire functions". Zalcman's proof is based on the Borel-Carathhodory inequality, which will be needed in the next chapter and is proved in the Appendix. Note 2.4. Instead of the chain of arguments leading to the construction of f , the log(-2)); then proof could be shortened by saying that "Let b(z) = kArccos under the hypothesis that F never takes the values 0 and 1, b(F(2)) = f ( z ) is analytically continuable as a single-valued function in B(0, I), etc." This has the advantage of brevity and the disadvantage of providing no indication whence the seemingly bizarre function b. It is clear from the proof that the value of R in Theorem 2.1 is immaterial to the proof of Theorem 2.2; what is material is that it be independent of the function

(5

f. Theorem 2.3 (Landau). If a is any complex number, then there is a number R(a) (depending only on a) such that if F(0) = a , F1(0) = 1 and F is analytic in B(0, R ( a ) ) , then for some C E B(0, R(a)), F ( < ) is either 0 or 1. Proof. Suppose for a given R, F is analytic in B(0, R), and # 0, # 1 there. As in the proof of Theorem 2.2, construct the same function f corresponding to F. Then f (0) depends only on F(0) = a , and since

2.2. The Bloch-Landau Approach taking the logarithmic derivative,

whence 1 = F'(0) = ria(e2f(0)- e-2f(0))f'(0)

.

Hence ft(0) # 0, and its value depends only on a. We now need an appropriate normalization: for z E B(0, l ) , f(RZ) is analytic and = co 9(z) = R f '(0)

+ z + c2z2+ . . . .

Since F ( z ) # 0, # 1, as before, the image of B(0, R) under f contains no disk of radius 1, and so the image of B ( 0 , l ) under g contains no disk of radius Rlff(0)l' Hence, by the Bloch-Landau Theorem (Theorem 2.1)

which is finite and depends only on a . Note 2.5. By using a slightly different normalization, Theorem 2.3 can be phrased in terms of R(a, p) where F(0) = a, F' (0) = P. That is, the following surprising result is true: given any function analytic at 0, there is a number R depending only on the first two coeficients of its power series such that somewhere in B(0, R) f takes on either the value 0 or the value 1. As we shall see later (Theorem 4.3 and Note 4.5), although the precise values of the constants B and L discussed in Note 2.1 are not known, nevertheless, the best possible value of the "Landau radius" R can be determined. In 1906 Landau considered the problem similar to the statement of Theorem 2.3 in which, in addition, the coefficients a l , . . . ,a, of the power series are prescribed. In 1911, Carathkodory and Fejkr considered such problems in the context of harmonic functions and answered, in particular, this question definitely (Rendiconti Palermo). For the case n = 1, treated by Carathkodory, see Theorem 4.3. Note 2.6. Picard's Little Theorem (Theorem 2.2) is a special case of Landau's Theorem 2.3. For if G is entire, non-constant, and never takes 0 or 1, then by choosing an 11 with G1(q) # 0 and putting

F is also entire and must take on 0 or 1 for some z with 121 5 R(G(o)); hence so must G for some z which is a contradiction. Landau discovered Theorem 2.3 and its relation to Picard's Little Theorem in 1904 some twenty years before Bloch's Theorem by a proof necessarily quite different from the above. Konrad Knopp [I291 has written how he remembered the class lecture in analytic function theory when Landau first presented the result.

2. Picard's Theorems

42

Theorem 2.4 (Schottky's Theorem-weak form). For each complex a and each 8 , 0 5 8 < 1, there is a positive finite k(a,8) such that if F(z) is analytic in B(0, I), F(0) = a , and F ( z ) # 0, # 1 in B(0, I), then IF(z)l

< k(a,8) f o r z

E B(O,8).

Proof. Given F satisfying the hypotheses of the theorem, once more let f be the corresponding function as constructed in the proof of Theorem 2.2. Suppose ]
is analytic in B ( 0 , l ) and gl(0) = 1, whence as in the proof of Theorem 2.2 (since F ( z ) # 0, # I ) , the image of B ( 0 , l ) under g contains no open disk of radius 1 Hence by Theorem 2.1, (l-e);f,(C)I> or (1-Q)If1(C)l.

&,

and if f'(<) = 0, this last is trivial. So for all C, with

]
5 8 < 1, integrating along

Since f (0) depends only on F(0) = a ,

Hence by (6) in the proof of Theorem 2.2

Note 2.7. For fixed 8, as a + co, k(a, 8) may + co. In fact, in 1932, Ostrowski [I811 showed that if ko(a,8) is the best bound in Theorem 2.4; then for fixed 8, as a + co, with 1 . ~ 1 = r, k!z ko(a,8) al-. ( 2 5 6 ) e .

-

Raphael Robinson [211] obtained explicit upper and lower bounds for ko(a,8). (Robinson's constants in the paper cited are different from the above as he uses 1 and -1 for the two omitted values.) Various other results for kO(a,8),some best possible, for small 8 or small a were obtained by Hayman [99] and Jenkins [124]. Functions like e c ( e ) show the need for the factor a W in Ostrowski's result.

2.2. The Bloch-Landau Approach

43

Note 2.8. Landau's Theorem 2.3 is a consequence of Theorem 2.4. If R > 0 and G is analytic on B ( O , R ) , G ( 0 ) = a , G'(0) = 1 and G ( z ) # 0,# 1 for z E B ( O , R ) , let F ( z ) = G ( R z ) . Then F is analytic and # 0 , # 1 on B ( 0 , l ) . By Theorem 2.4, for z E B(O,1/2), say, IF(.)I = IG(Rz)l < k ( a , 1/21

Theorem 2.5 (Schottky's Theorem-Strong Form). For every w and every 8, 0 5 8 5 1, there is a function $(w, 8 ) such that if F is analytic and # 0 , # 1 on B ( 0 , I ) , then if IF(O)I 5 w, IF(z)l 5 $(w, 8 ) for z E B(O,8).

Proof. With no loss of generality we can suppose w 2 2. There are then two cases; 0 5 arg F ( z ) < 2n, according as la1 = IF(0)l is 2 or < 1/w. Supposing first la1 2 and using the notation developed in the proof of Theorem 2.2,

i,

1 h ( z ) = -log F ( z ) , whence 2ni

I 1% lal l

1 2n

log w 2n + I ,

5 -dlog laI2 + 4n2 5 -+ I S 2n

say. Working backward through u ( 0 ) and v ( 0 ) as defined in proof of Theorem 2.2 (noting that I u ( o ) ~ v ( o = ) l Iu(0) v(O)(),we arrive at a bound depending only on w

<

+

I f (011, say If (011 p(w)Hence, by the proof of Theorem 1.4, for lzl 5 8, If (z)l < If (011 5 $ ~ ~ (0w) , and so IF(z)l < $ Z ( W , ~ ) . If, on the other hand, la1 < 5 112, then repeating the argument with 1 - F ( z ) instead of F ( a ) and 2 instead of w (since 11 - a ] < 2 ) , we get

for

+ fi

5

and the theorem in this case as well.

CI

Theorem 2.6 (Picard's "Big Theorem"). Suppose F ( z ) is analytic in a region K , = { z : 0 < lz - zol 5 p} and zo is an isolated essential singularity of F ( z ) . Then in K,, F ( z ) takes on every value except at most one.

Proof. By contradiction. If F does not take on the values a and b in K p , then B ( z ) = (Z O f p t ) - a is analytic in the "punctured unit disk": { z : 0 < lzl 1 ) and leaves out 0 and 1; so with no loss of generality, we may assume that a = 0, b = 1, 20 = 0 , p = 1.

,-,

<

2. Picard's Theorems

44

Since zo = 0 is by hypothesis an isolated essential singularity, by the CasoratiWeierstrass Theorem, there is a sequence {z,) such that lznl

< e-4n

for all n , Izn+ll

< I ~ n ln-+m ,lim zn = 0, and IF(zn) - 21 < 112

(the reason for the bound e-4" will become clear momentarily). Let w, = log z, then

along the negative real axis. By Theorem 2.2 (Picard's "Little" Theorem) since eZ is entire and never 0, it must take on every other value and so we can write for z in the punctured unit disk F(z) = F ( e w )= G(w) and F(z,) = G(w,) . Since the map z = ew takes the punctured unit disk onto the open left half plane (IzI < 1,z # 0 + eRe" < 1 @ Re w < 0, w # m ) , G is analytic in {w : Re w < O), G(w 27ri) = G(w), and G(w) # 0, # 1 for Re w < 0. The vertical segment in the strip 0 I m w < 21r which passes through w, (and so has abscissa Re w,) is the image of C (0, I zn 1). Let H,(w) = G(w, 47rw). Then H,(O) = G(w,) = F(z,) for all n, and so

+

<

+

IHn(0) -21

< 112, and

IHn(O)I < 5 / 2 and H,(w) #O,# 1

+

for Re(w, 47rw) < 0, that is, in particular, (by (2.5)) for 47r(Rew - 1) < 0, 2.e. for Re w < 1, and so certainly for I w ~ < 1. Hence by Theorem 2.5, for w E B(O,1/2) say, IH,(w)l 5 P, a constant, for all n. 27r, and so if It - wnl 5 27r, Now lwl 5 112 if and only if I(wn 47rw) - w,I then IG(t)l P . Replacing the dummy variable t by w, and using the definition of w and w,, we get that for all z with log 27r, IF(z)l 5 P . In particular, then, if

+

<

<

I

5)I < ( Hence, by the Maximum Modulus Theorem, the same

IzI = 1 . ~ ~ 1 IF(z)I , 5 P. inequality holds in the annuli between the circles C(O,lzn1)' and so for all z with 0 < 121 < lzll, IF(z)I P . But this means 0 is a removable singularity contradicting the assumption that it was an isolated essential singularity.

<

Note 2.9. Theorem 2.6 can also be stated in terms of meromorphic functions (at poles the value m is taken on) and this is sometimes known as "Picard's Big Theorem". In this form the theorem says: Suppose f (z) is meromorphic in a region R except for an isolated essential singularity, and the image of R under f omits three values. Then f is constant. To prove this, let a, b, c be the omitted values and g(z) =

(s) (w).

3),

Then g(z) # 0, # 1,# m in R (at poles of f , g(z) = and so g is analytic in R except for an isolated singularity and omits 0 and 1 in R and so by Theorem 2.6 is constant. So f is also.

2.3. The Elliptic Modular Function

45

Note 2.10. An immediate consequence of Theorem 2.6 (sometimes also called Picard's Little Theorem) is: An entire function, not a polynomial, takes every value except at most one infinitely often.

2.3

The Elliptic Modular Function

2.4

Introduction

The proofs of Section 2 make the "Big" and "Little" Picard theorems and their relatives seem like computational consequences of Bloch's Theorem. Before 1924, however, an entirely different approach was used for this circle of ideas based on the so- called elliptic modular function. Interest in this approach did not cease with either Bloch's or Nevanlinna's discoveries (for the latter, see Chapter 4), for proofs via the elliptic modular function could be made to yield sharp results. To construct the elliptic modular function, consider the region

Diagram 2.1 By the Riemann Mapping Theorem applied twice, there is a function which maps R univalently onto the upper half-plane. By a special case of the OsgoodCarathkodory Theorem (see Appendix), under this map the boundary of R will go onto the real axis in a continuous one-to-one fashion. Furthermore, using a linear fractional transformation if necessary, we can ensure that the half-line {z : Re z = 0, Im z 0) maps onto {w : w 5 0); the half-line {z : Re z = 1, Im z 2 0) maps onto {w : w 2 1); the semi-circle {z : lz - 1/21 = 112, Im z > 0) maps onto {w : 0 < w < 1). Call this function t(z) (note that there may be more than one function with these properties: uniqueness is not a concern here). Continue t(z) by Schwarz reflection (see Appendix) over the three arcs. The boundary of the resulting domain consists of straight lines and circular arcs; so we can continue again

>

2. Picard's Theorems

46

by Schwarz reflection. By continual repetition of this process t(z) is analytically continued into the largest domain possible. What does reflection over the initial circular arc look like? The semi- circle {z : z = 112 1/2eis,0 0 5 7 ~ ) is mapped onto the real interval [0,1] by g(z) = log(2z-1) (whose inverse is gV1(w) = 1/2+1/2eTiW). So, if = peie E R, reflection over {z : lz - 1/21 = 112, I m z > 0} is given by

2

<

+

-z

-i

I+log(2C - 1) = - log 12c - 11 7T 7T

+ -1 arg(2C - 1) conjugation 7T

i 1 9-I - log 12C - 11 + - arg(2C - 1) + 7T

7T

Alternatively, reflection over the semi-circle amounts to find the point inverse to with respect to C(1/2,1/2); that is (see Definition 1.2.2) solving I< - 1/21 Iq 1/21 = 114 subject to the condition that q and are collinear with 112 and lie on the same side of it. This leads again to the last expression in (4.1). For example, the line { $ i$ : 112 < II, < m) C R is mapped onto the segment { $ i x : 0 < x < $1, under reflection over the semi-circle, and the line

<

<

+

+

($+ rn + i$ :

+ rn +

< $ < m)

i m

C R gets mapped onto a circular arc with

and 112. The straight line on the boundary of endpoints $ R, {z : Re z = 1, I m z > 0) gets mapped onto the semi-circle K1 = (314 ieie : 0 0 .rr) when reflected over the semi-circle (112 1/2eie : 0 6 T } . Similarly, the line {z : Re z = 0, I m z > 0) gets mapped onto K2 = ieie : 0 5 9 5 T ) . Similarly, on reflection over K1, the points 0,1/2,1 go respectively onto 2/3,1/2,1, while m maps onto 314, and so the straight lines orthogonal to the real axis with these as endpoints get mapped onto semi-circles with endpoints (2/3,3/4), (1/2,3/4), (3/4,1) respectively. Under reflection over K l , the points 0,1/2,1 map respectively onto 0,1/2,1/3 and the lines map onto corresponding semi-circles. K2 reflected over K1 maps into a semi-circle with endpoints 112,213, and K1 reflected over K2 into a semi-circle with endpoints 1/3,1/2 (see Diagram 4.2). Repeated reflection gives rise to umbrella-shaped regions which seem down" on the real axis. to LLclose

< <

{i +

+

+ < <

2.4. Introduction

Diagram 4.2: The domain of t ( z ) after the second stage of reflection over circular arcs. Boundary arcs are indicated by solid lines.

Definition 4.1. The function which results from continuing t ( z ) by reflection into the largest possible domain will be called p ( z ) . If a point is added to the domain of p ( z ) at the first state of continuation, its image lies in the lower half-plane, if at the second stage the image is again in the upper half-plane, etc., that is, p maps a point in the upper half-plane onto a point in the lower or upper half-plane depending on whether C is obtained as an element of the domain of p after an odd or even number of reflections. Now the "boundaries" of the domain after each stage of reflection consist either of straight lines Re z = n, n an integer, or arcs of circles centered at the real axis and cutting it in certain points. To find out what those points look like, write 0 as and 1 as and note that at the end of the first stage of reflection, we have acquired as potential boundary points in the interval [O,1] (images of 0, 1 or oo) the points

c

i,

and at the end of the second stage, the points

(see above Diagram 4.2). These two sequences have the following property: (*) They are irreducible rational numbers ordered in the increasing order of magnitude, and if mln, plq, r l s are any three successive ones, then after reducing to lowest = n+s . terms, 2 q

Definition 4.2. The sequence of irreducible rational a l b i n [O,1] such that

ordered i n order of increasing magnitude is called the Farey series of order N, written FN.

48

2. Picard's Theorems

The sequence (4.2) above is the Farey series of order 2 and (4.3) the Farey series of order 4. It turns out that the property (*) is characteristic of Farey series. In fact, FN+l may be obtained from FN by inserting between two successive terms a l b and cld of FN the fraction $ reduced to lowest terms provided the resulting denominator is 5 N 1. Furthermore, it is easily shown that (*) is equivalent to the fact that if a l b and c l d are any two successive elements of a Farey series of some order, then (**) bc - ad = 1. (See Appendix for proofs.) What we further need t o observe for our use here is the following:

+

Claim: If C is a circle orthogonal to the real axis and passing through the points which are successive elements of a Farey series of order N, then on reflecting . over C, the point 5 = 9-n maps into the point If we prove this claim, then we will have shown that successive reflections eventually generate the Farey series of any order and thus, eventually any rational in [O,11 appears as a "boundary point".

,: :,

3

E + :) ( hence, on reflecting over it, the real point 5 goes into Proof of Claim: The circle in question has center

and radius

$ (: - E),

But since m / n , p / q are successive elements of a Farey series, pn - m q = 1 by (**) and so (4.4) equals

m 9 + pn 2nq

S +

Sqn(2qnr - s m q - spn) .

But T = c ( p - m) and s = c(q - n ) for some integer c, hence on substituting, the second fraction in (4.5) becomes

q-n 2qn(qnp - q n m - mq2

-

9-n

+ pn2) - 2qn(n + q)

since pn - m q = 1. And so (4.5) equals

mq+Pn 2nq

+

q-n - qn(m+p)+pn2+mq2+q-n 29n(n + 9) 2qn(n + 9 )

+

+

+ +

But, since pn - m q = 1, pn2 mq2 q - n = q n ( m p) n ( p n - m q ) p n ) + q - n = q n ( m + p) and substituting this in (4.6) proves the claim.

(4.6)

+ q(mq -

2.4. Introduction

49

Thus, one cannot continue by reflection over any rational point in [O, 11 and so (by reflection over straight lines) over any real rational point. Hence, since the rationals are dense in the real numbers, the real axis forms a boundary to continuation of t(z) by reflection. In fact, every point of the real axis is a singular point of p(z) (i.e. the real axis forms a natural boundary for p(z)). For suppose p(z) could be continued somehow over some real point zo; then in every neighborhood of zo, p takes on all values in both the upper and lower half-planes, and so can hardly even be continuous a t zo. p is single-valued by the monodromy theorem (the upper half-plane is simplyconnected). Furthermore, p takes on all values in the upper half-plane and all values in the lower half-plane (infinitely often) by construction. Since in the original domain R , t mapped the boundary of R onto the real axis, we see that every point on the real axis, except possibly O,1, oo is taken on infinitely often by p a t points in the interior of the upper half-plane (on "boundary arcs which become superseded during reflection"). There remains the question whether the values O,1, oo are taken on by p. But t was univalent and mapped O , 1 , oo onto O,1, co a t points of the real axis, which are excluded from its domain of definition as an analytic function. Hence, we have proved

Theorem 4.1. p is an analytic function mapping the upper half-plane {z : I m z > 0) onto @ , - {0,1, oo) (the whole plane except for 0,1, co). p is called an elliptic modular function (nothing has been said about uniqueness). p is a well-known function and can be used to prove

Theorem 4.2. Picard's "Little" Theorem

.

(Theorem 2.2).

Proof. Let v be an arbitrary, but fixed, branch of the inverse of p, and suppose g(z) is an entire function omitting 0 and 1 (this involves no loss of generality as in the earlier proof). Then v(g(z)) is entire (since p takes on all values except O,1, oo, single valuedness is guaranteed by the monodromy theorem) and I m v(g(z)) > 0 for all z. Hence h(z) = ei"(g("))is entire and

By Liouville's Theorem, h(z) is constant; hence v(g(z)) is, hence g is ( p is singlevalued). C]

Note 4.1. The reader is no doubt mildly puzzled by the naming of the sequences FN,"Farey series"; the name has unfortunately stuck for historical reasons. Farey, in fact, was a geologist and the first proofs of the crucial properties of these sequences were given by Haros and Cauchy (independently). for further historical references, see the notes to Chapter I11 of Hardy and Wright[97].

2. Picard's Theorems

50

Note 4.2. J. E. Littlewood has remarked that Picard's Little Theorem is an example of an important result whose statement and proof can be stated in one line each (the function called p in the above being well-known in the theory of elliptic functions). See Theorem 8.6.10 and A Mathematician's Miscellany (1953), p.19. Note 4.3. p is an example of an "automorphic function". An analytic function f is automorphic if there is a group G of non- singular linear fractional transformations such that for all A E G, f (A(z)) = f(z). The extensive theory of automorphic functions begins with the classical work of Henri Poincar6 and Felix Klein and is still actively pursued currently. Note 4.4. Picard's "Big" Theorem (Theorem 1.6) can also be obtained directly by use of p (or more properly speaking, a fixed branch of its inverse mapping v). However, here there is the difficulty that since v(g(z)) is to be defined in a deleted neighborhood of zo, there is no monodromy theorem available since the domain is not simply-connected; consequently it is not clear a prior2 that v(g(z)) can be defined uniquely everywhere in the deleted neighborhood. This difficulty can be gotten around by using the automorphic properties of p. For a proof along these lines, see Thron 12271. The monodromy theorem is thus an essential part of the above simple proof (which is Picard's) of Picard's "Little" Theorem. An improved version of Landau's Theorem can be given using p. Theorem 4.3. Landau's Theorem (Theorem 2.3).

Proof. (Carath6odory) : Let f(z) = Cr=,anzn be analytic in B(0, R). Suppose f omits 0 and 1 in B(0, R). As in the proof of Theorem 2.3, consider a fixed branch v of the inverse of p and v(f (z)) in a neighborhood of z = 0. By the monodromy theorem v(f (z)) can be continued as an analytic (single-valued) function throughout B(0, R) to give a function g(z) = v(f (2)) analytic in B(0, R) and satisfying Im g(z) > 0. Now note (verification is trivial) that if Im a > 0, Im z > 0, then

we have lh(z) 1

< 1 for z E B(0, R), h(0) = 0, and h is analytic in B(0, R) (since Img(z) > 0, Img(0) > 0, g(z) # g(0)for z E B(0, R)). Hence, setting h(z) =

9(z)-9(0)

'

Therefore, by Schwarz' Lemma, lh(z)I and

5

1 ,121

for z E B(0, R)

1

2.4. Introduction Now, by definition of h, hl(0) =

!Fo,m1o; g(z) - -d o-) -

&I - d o ) vl(ao)al v(f (z)) - v(f (0)) 1 -= lim z--to z .(f (z)) - v(f (0)) 2i Im(v(a0))

'

Hence

That is, if R

> R(ao,a l ) for some z E B(0, R), f

takes either the value 0 or 1.

Note 4.5. This is the third proof presented of Landau's Theorem 2.3. The bound for R(ao,al) obtained here is, in fact, sharp. Suppose we take f (I) =

;?@).

9

Then, maps B ( 0 , l ) into the upper half-plane and consequently f is analytic in B ( 0 , l ) and does not take on 0 or 1in B ( 0 , l ) . But a0 = f(0) = ,u(i); SO v(ao) = i, and also then,

So, for this function 2 I m v(a0) 2 lvl(ao)l la1 I lvl(ao)l&J

=1

but neither 0 nor 1 is taken on in B ( 0 , l ) . It turns out somewhat surprisingly that analytic functions omitting two fixed values have a close relationship to "normal families". Before presenting this, it will be convenient t o have a (rather interesting) criterion for a family of functions t o be normal. With this in mind, recall the

Definition 4.3. Given two (finite) complex numbers zl and z2, the chordal distance between zl and 2 2 is defined by

and d(z1, CQ) =

2

J-

(zl, finite)

.

A straightforward computation shows that the chordal distance is the metric induced by stereographic projection, that is, it is the Euclidean distance between points on the (unit) Riemann sphere corresponding to zl and z2 under stereographic projection.

2. Picard's Theorems

We also need the

1

Definition 4.4. I f f is continuously differentiable at the point z , the chordal derivative

x(f )

of f at z is defined b y X ( f ) = limh+o

dw.

Z

Hence,

x ( f ) = lim

2lf ( 2 + h) - f (z)I - 21f1(z)l l h l J ( l + I f ( z + h)I2)(1+ If(z)I2) l + l f ( z ) I 2

It is worth noting the immediate facts that (1) ~ ( f =) continuous at z.

x (j),

(2) ~ ( f is)

We now have the following interesting and useful normality criterion:

Theorem 4.4. A family F of functions analytic in a region R is normal if and only if on every compact subset of R, the chordal derivatives of the functions in .F are uniformly bounded.

Proof. Necessity. Suppose .F is normal and there is a compact subset C of R and a sequence { f , ) of functions in F such that suponc ~ ( f , ) + ca as n + 00. Then, either (a) There is a subsequence {f,,) of {f,) which is uniformly convergent on any compact subset of R; whence its limit would be analytic on C ; whence ~ ( f , , ) would be bounded on C ; contradicting the assumption that suponc x ( f n )ca ias n + m . Or goes (b) A subsequence {f,,) of {f,) goes uniformly to co;whence

{k)

x

2 would be bounded on C (in fact -+ 0 on C as lc + oo) and so by (I),we again have the contradiction that ~ ( f , , ) is bounded on C. Sufficiency. Assume for all f E F , ~ ( f is) uniformly bounded on compact subsets of R. Clearly f followed by stereographic projection maps any arc r onto an image with length uniformly to 0; whence

In particular, if

(fnr

)

is the straight line joining zl and z2, then

where the integral is along a straight line. Hence, if ~ ( f 5) M on compact subsets of R d(f(XI), f ( ~ 2 )5 ) Mlz2 - zll

2.4. Introduction there and so on compact subsets of R

for all f E 3 . Since a compact subset C is closed and bounded, the equicontinuity of 3 on each C follows, and hence by Theorem 1.4.1 (the ArzelB-Ascoli Theorem), 3 is a normal family.

Theorem 4.5. Let 3 be a family of functions analytic in a region R and such that for every f E 3, on every compact subset of R, Im f (z) > 0. Then 3 is a normal family. Proof. It is easy to compute that with a , b, c, d complex constants,

Hence, in particular

x

(%)

= ~ ( f ( z ) as ) an easy computation shows.

On the other hand, w(z) = - . - maps the open upper half- plane onto B ( 0 , l ) . Hence, if I m f (2) > 0, < 1. It follows that the family G = : f E 3) is uniformly bounded on compact subsets of R; hence by (4.8) { ~ ((z)) f : f E 3) is uniformly bounded there, and so, by Theorem 4.5 again, 3 is normal.

l e i

{e

Note 4.6. Clearly, Theorem 4.5 (and (4.7) above) can be exploited t o provide many similar results. We now come t o a result going back to Paul Monte1 (in 1912) which connects normal familities and the omitted values of functions by way of p(z) and which can be used to provide another different proof of Picard's "Big" Theorem (Theorem 2.6). Theorem 4.6. Suppose 3 is a family of functions f analytic in a region R, each of which omits 0 and 1 as values. Then 3 is normal in R. Proof. Let p be given by Definition 4.1. Let U C R be an open neighborhood of zo. Given f E 3,for a sufficiently small neighborhood of zo, since f omits 0 and 1, there is a branch v of the inverse of p such that v(f (z)) lies in one of two adjacent regions of the form used in constructing p(z) (depending on whether f (z) lies in the upper or lower half-plane). Clearly with no loss of generality, we can pick the branch v so that these regions are given by the region D of Diagram 4.3,

2. Picard's Theorems

Diagram 4.3 and then continue v(f (z)) analytically throughout U to obtain a function g analytic in U. Since I m g ( z ) = I m v ( f (z)) > 0 for z E U, by Theorem 4.5, the family G of functions g corresponding to the functions f E 3 is normal in U. Let {f,) be a sequence of functions in F and {g,) = {v(fn)) the corresponding sequence in G. Then {g,) has a subsequence, say {gnk) which either converges uniformly to oo or converges uniformly t o a function G analytic in U. In the second eventuality, by the open mapping theorem, if G is non-constant, it is never real (an open neighborhood of a real point contains points in the lower half-plane, but since for z E U, I m gnk(z) > 0, for z E U, I m G(z) 2 0). Furthermore, if G were a real constant, since values G takes on lie in the closure of 27,G(z) 1,-1, or 0. We thus have four cases.

-

Case I: gnk(z) -+ oo uniformly as k -+ co for z E U. Then for any closed disk U' C U, and z E U1,p(gnk(z))= fnk(z) -+ oo uniformly as Ic -+ oo, whence f,, + oo uniformly on compact subsets of U. Case 11: G is non-constant and analytic in U. Then by the argument above, G(U) is open and G(U) n {z : I m z = 0) = 0. So, if U' is a closed disk c U, g(U1) is bounded and has a positive distance from the real axis. For z E U, gnk( z ) -+ G(z) uniformly as Ic -+ oo; hence

-

uniformly on the closed disk U'.

Case 111: G(z) 1 (or -1) i.e. gnk(z) -+ 1 (or -1) uniformly for z E a closed disk U' C U. So v(fnk(2)) -+ 1 (or -1) uniformly as k -+ oo for z E U'. Hence, fnk(z) -+ 1 uniformly for z E U'. ( p is analytic in the open upper half-plane and continuous a t 1 and -1 from above, but not analytic at either point. Defining p by continuity (from above) for real values, p(1) = p(-1) = 1.) Case IV: G(z) disk U' c U.

0, then similarly t o Case 111, fnk(z)-+ 0 uniformly in a closed

2.4. Introduction

55

Hence, taking the four possible cases together, we have that 3 satisfies the definition of normality in some closed neighborhood of zo E R. Since zo was an arbitrary point of R, and since compact subsets of R can be covered by finitely many disks having the property that 3 is normal in them by the above argument, it follows that 3 is normal in R.

Note 4.7. If 3 is a family of functions meromorphic in a region R , clearly the same proof shows that 3 is normal in R if every f E 3 omits (the same) three values. We shall need this version of the theorem in Chapter 5 (Theorem 5.1.3). As a consequence of Theorem 4.6, we have

Theorem 4.7. Picard's "Big" Theorem (Theorem 2.6).

Proof. With no loss of generality, we can assume f has an isolated essential singularity at 0. Suppose there is a p such that in the punctured disk B(0, p) - {0), f omits two values, which with no loss of generality, we may assume as before to be 0 and 1. Let fn(z) = f (zln). Then each fn is analytic on B(0, p) - (0). Consider the compact subset r = {z : lzl = p/2) of the punctured disk. Then either (a) There is a subsequence {f,,) of {f,) converging uniformly to a function F analytic on I?, or (b) There is a subsequence {f,,) of {f,) converging to oo on r . In Case (a), let M = m a x , ~ rIF(z)l. Then

>

and all nk sufficiently large, say nk N. But If, (z) 1 = If (z/nk)1, and so letting w = zlnk, we get that for Iwl = and nk 2 N , If (w)I 2M. Thus, f is uniformly bounded on a sequence of circles whose radii decrease monotonically to 0; hence, by the maximum modulus theorem, f is bounded in a deleted neighborhood of 0, and so 0 must be a removable singularity contradicting the hypothesis that it is an isolated essential singularity. = + 0 uniformly on a sequence of circles, In Case (b), similarly 1 fn , (z)

<

&

&

h

and is uniformly bounded in a deleted neighborhood of 0. So, 0 is a removable singularity of ,; and $ can be extended to an analytic function g which is 0 a t 0 and otherwise g(z) = It follows that f has a pole at 0, which is again a contradiction.

h.

Note 4.8. Although the statement of Theorem 4.6 originates with Monte1 [162], it is implicit in a paper of Caratheodory and Landau [38]. CarathQodoryand Landau apply Schottky's theorem (Theorem 2.5), and the Vitali Convergence Theorem, thus this approach can also be used to prove Theorem 4.6.

2. Picard's Theorems

56

Note 4.9. Actually the argument of Theorem 4.7 can be made to yield a stronger theorem: If z = 0 is an isolated essential singularity of f (analytic in a deleted neighborhood of 0), then there exists a t least one ray argz = 6 such that for every 6 > 0, f assumes every value except a t most one in the sector 18- arg zl < E . This is one version of Julia's Theorem. There is also a version for meromorphic functions. We will return to these questions in Chapter 5. Note 4.10. Still another approach to the theorems of this chapter is via Ahlfors' Theory of covering surfaces for functions meromorphic in the unit disk. A way to accomplish this was shown by Dufresnoy in 1941, and may be found (with much other material) in Chapter VI of Hayman7s[100](Chapter V of that book is devoted t o Ahlfors' Theory). Ahlfors' theory will not be discussed in this book, though it has an affinity t o Nevanlinna's Theory of meromorphic functions, to which an introduction is provided in Chapter 4 (viz. Note 4.3.1.).

2.5

The Constants of Bloch and Landau

In Theorem 1.1, the lower bound 1/16 is obtained for the constant L defined in Note 1.1. Much better results are known. The best known lower bounds for B and L result from an argument of Ahlfors. The best known upper bounds are provided by examples discovered by Ahlfors and Grunsky. it may be conjectured that the upper bounds are the correct values of the constants but this is still an open question. Ahlfors' proof is motivated by using differential-geometric notions t o extend Schwarz' Lemma[6]; "An Extension of Schwarz' Lemma" is, in fact, its title. However, the details of the proof can be given without reference to differential geometry. To help in understanding it, it is useful first to recast Schwarz' Lemma in the "invariant form" which is attached to the name of Pick[l91]:

Theorem 5.1. Suppose f is analytic on B ( 0 , l ) and satisfies a E B ( 0 , l ) . Then, for z E B ( 0 , l )

If

(z)l

< 1 there.

Let

(i) (ii)

Proof. We may note that Theorem 5.1 with a = 0 and f(0) = 0 is just Schwarz' Lemma. Now the linear fractional transformation $ ( z ) = maps B ( 0 , l ) onto itself with 0 going to a. Similarly, $(z) = maps B ( 0 , l ) onto itself with f ( a ) 1-f ( a ) z going to 0. consider the function

*

2.5. The Constants of Bloch and Landau

57

on B(O, 1). Then g(0) = 0, g is analytic on B ( 0 , l ) and Ig(z)l 5 1. Hence, by Schwarz' Lemma, Ig(z)l 5 lzl on B(0,l). Hence I+ o f (z) I 5 14-I (2) 1, which is (i). Dividing both sides of (i) by Iz - a1 and letting z + a , gives (ii).

Note 5.1. In Schwarz' Lemma (by the maximum modulus theorem), Jg(zl)l = J z l ) for some zl E B ( 0 , l ) only if g(z) = eisz, 6' a real constant. Hence, equality holds for some z E B ( 0 , l ) in Theorem 5.1 (i) only if $ o f (z) = eie$-I (z) and thus is a linear fractional transformation mapping B ( 0 , l ) onto itself.

Note 5.2. Theorem 5.1 (i) can also be stated in terms of the hyperbolic metric for B(0,l). It is this formulation which is the beginning of Ahlfors' considerations. For discussion of the hyperbolic metric, see for example, Carathkodory [34], particularly Chapter 11, Nevanlinna [169], and Sansone and Gerretsen [215]. The last cited reference contains a discussion of other metrics as well. Nevanlinna gives versions of Landau's and Schottky's Theorems (Theorems 2.3 and 2.4). Before proceeding with Ahlfors' method we need

Definition 5.1. Let X be a continuous real-valued non-negative function on B(0, I ) , and suppose X(a)

# 0. Let A, be a function such that in some neighborhood of a ,

(i) Xa has continuous second partial derivatives. (ii) Xa(a) = X(a) (iii) X,(z) 5 X(z)

where A as usual is the Laplacian operator

Then, Xu is said to support X a t a .

Note 5.3. As the next theorem shows, the requirement (iv) is motivated by the (see the right side of Theorem 5.1 (ii)) then, writing the fact that if X(z) = Laplacian in its polar coordinate form

&

58

2. Picard's Theorems

For the left side of Theorem 5.1 (ii) a similar computation holds except a t the set of points C where fl(C) = 0. For, a straightforward computation shows that if k is analytic a t z and h = h(x, y) is real-valued with continuous second partial derivatives, then if $(z) = h(k(z)), in a small neighborhood of z,

Taking h(2) = log

(*

1

and k(z) = f (z), we get by (5.1)

Since (assuming f'(z) # 0) log 1 fl(z)l is harmonic in a neighborhood of z, A log I f l ( z ) J= 0 there, and so also

,w

In differential geometry, the quantity is called the Gaussian curvature K(X) of the metric induced by X(z); it is invariant under conformal mappings; that is if f is conformal, K(X(f)) = K(X). The proof of this is essentially given in the preceding paragraph, depending only on the change of variable formula for A given by (5.2).

Theorem 5.2. Suppose X(z) is a non-negative continuous function on B(0, 1)' and that for each a E B ( 0 , l ) for which X(a) # 0, there is a function A, supporting X at a in sense of Definition 5.1. Then X(Z) for all z E B ( 0 , l ) .

<&

Proof. For 121 < R < 1, let u(z) = Logh(z) and v(z) = Log ( R . Then u(z) is continuous a t all points of B ( 0 , l ) where X # 0. We will now show that

+)

<

U(Z) v(z) for all z E B(0, R) . For, suppose not, then for some zo 6 B(0, R) u(ro) - v(zo) > 0. Furthermore, as z approaches a point of C(0, R ) , u(z) - v(z) + -m. Similarly, as z approaches a zero of A, u(z) - v(z) + -GO. Hence, u(z) - v(z) has a positive maximum in B(0, R) . Suppose this maximum is taken on at a. Then clearly X(a) # 0, and so by hypothesis, there is a function A, supporting X a t a. Then (Log A, (a)) - v(a) = (Log X(a)) - v(a) = u(a) - v(a) by Definition 5.1 (ii), and so by continuity Log(X,(z)) - v(z)

> 0 in some neighborhood

of a .

>0

2.5. The Constants of Bloch and Landau By Definition 5.1 (iii), then, in a sufficiently small neighborhood of a,

0

< (Log A,(z))

- v(z)

5 u(z) - v(z) .

(5.3)

Let F ( z ) = Log(A,(z)) - v(z). Then (5.3) means that in some sufficiently small neighborhood of a, say B(a, po), F has a positive maximum at a (if in every neighborhood of a there are points z such that F(z) 2 Log(A,(a)) - v(a), the chain of inequalities u(a) - v(a) = log A,(a) - v(a) = F ( a ) I: F ( z ) 5 u(z) - v(z) 5 u(a) - v(a) shows that F is constant in a neighborhood of a). Let p < pol then using the polar coordinate form of A, we have

where, on the right, Fa=F ( a period 2n, hence

+ reie).

On C(a, p), F ( a

+ peie) is periodic with

and we have

On the other hand, A F ( z ) = A log A, ( z ) - Av(z) 2 4 ( ~ , ( z ) )-~Av(z)

(5.5)

R21z,2

by Definition 5.1 (iii). Since u(z) by definition = Log , a simple computation, the same as that done in Note 5.3 (which is the case R = I ) , shows that also Av(z) = 4e2"(") for z E B(a,po) .

(5.6)

Since by (5.3), log A, (z) - v(z) > 0 in B(a, PO),we get from (5.6) that A(v(z)) < ~ ( A , ( z ) )in~ B(a,po) and hence by (5.5), AF(z) > 0 in B(a,po), and so from (5.4), since p < po, we conclude

Integrating both sides of (5.7) from 0 to po we get

2. Picard's Theorems or, since F has a local maximum at a in B(a, po),

a contradiction.

<

It follows that logX(z) = u(z) v(z) = log ( n 2 ~ , z , T )for all z E B(0, R),O R < 1. Exponentiating, and letting R + 1, the theorem follows.

<

Note 5.4. Clearly Theorem 5.2 involves proving a maximum principle. In fact (as Ahlfors says), the function of the proof is subharmonic in B ( 0 , l ) and it is also true that the maximum modulus principle holds for subharmonic functions, whence Theorem 5.2 can be deduced. The reason for the lengthy proof which has been given above is t o avoid the introduction of the topic of subharmonic functions. Note 5.5. Maurice Heins [log] proves a slightly generalized version of Theorem 5.2 involving exceptional sets, but this has no effect on the application below. Theorem 5.3. (Ahlfors): Let B be Bloch's Constant. That is if 3 is the family of all functions f analytic on B ( 0 , l ) and such that 1 fl(0)l = 1, then B = inffGFP(f) where P ( f ) is the supremum of the radii of disks taken on univalently in the image of B ( 0 , l ) under f . (Compare Note 2.1.) Then B 4 1 4 .

>

We will call a disk taken on univalently a "univalent disk". Proof. Suppose f E 3. For each z E B(0, I), define p(z) as follows: If f '(z) = 0, p(z) = 0; otherwise p(z) is to be the radius of the largest univalent circle in the image of B ( 0 , l ) under f which has center f (z) (since f is not necessarily univalent on B(0, I), this image disk lies on the Riemann surface for f ) . Note that if we define D(z) as the largest such disk, then f-l(D(z)) is an open subset of B ( 0 , l ) whose boundary either contains a point where f ' is 0 or else a point of C ( 0 , l ) (for otherwise by the usual sort of compactness argument, we can cover Bd(D(z)) by a finite number of disks where, by Theorem 1.3.7, f takes on values univalently and thus extend D(z) to a larger disk D*(z) centered at f ( z ) such that f is univalent on f (D*(2)). Clearly p(z) P ( f ) for a E B ( 0 , l ) and since we need only consider f for which P ( f ) is finite, we may assume p is bounded. We now show that p is continuous. If f l ( a ) # 0, then for all z in the neighborhood f -'(D(a)) = Bat, say, of the length of the shortest line segment from f ( z ) to a, fi(z) # 0 and p(z) C (f (a),p(a)) and 5 the length of the longest line segment from f (z) to C (f (a),p(a)); thus I p(a) + If (z) - f (a)l 7 p(a) - If (z) - f (a)l 5

<

>

or,

< If (a) - f (2)I , for f '(a) # 0 and 2 E Ba

I P ( ~) ~ ( aI )

2.5. The Constants of Bloch and Landau

61

But, if f l ( a ) = 0, then in some deleted neighborhood Na of a, f l ( z ) # 0. If z E Na and is sufficiently near a, f-l(D(z)) n C ( 0 , l ) = 0, and so a zero of f must lie on Bd(f-l(D(z)), but this can only be at a, and so p(z) = If(z) - f(a)l for fl(a) = 0, and z E Na .

(5.9)

Equations (5.8) and (5.9) show that p is continuous. We now use p to construct a function X(z) to which we may apply Theorem 5.2. Suppose a E B ( 0 , l ) and fl(a) # 0. Let b be a point on C(f (a),p(a)) which is the image of a point where f 1 = 0. The function f (z) never takes the value b in B( f (a), p(a)) and so a single-valued branch of can be defined throughout this open disk. Now, as is readily computed, if g is analytic at z, gl(z) # 0, and if p(z) = then Alogp(z) = 4 ( p ( ~ ) )in~ a, neighborhood of z. (Compare Note 5.3.)

d

w2',

m

id-,

f as above, and A a constant to be determined Hence, taking g(z) = later, we get a real-valued function def

Xa(z) =

Alfl(z)l 21f (z) - bI1j2(A2- If (z) - bl)

which is defined provided If (z) - bl

< A2, and satisfies

. A log Xa (z) = 4(Xa( z ) ) ~

(5.11)

As well as (5.10), we consider the function (defined whenever A2

> p(z)

and

A f' z

fl(.) i o, for then P(Z)# o), ('1' = 2(p(1))1!2:A?1p(z))' ~f lim,,, X(Z)exists for a point a where f l ( a ) = 0, then we can define X throughout B ( 0 , l ) as a continuous function. But if f l ( a ) = 0, then by (5.9), p(z) = If (z) - f (a)l and lim X(z) = lim

z+a

"a

Hence, if A

>d

2

(J-)"~

z-a

I=-al

( A -~ I ~ ( z-) f ( i ) l )

m ' say, the function defined by

is continuous and non-negative in B ( 0 , l ) . Furthermore, from the definition of p, If (a) - bl = p(a), if f l ( a ) # 0; while if f l ( a ) = 0, p(a) = 0, and we take b = f (a). Thus, comparing (5.10) and (5.12) and repeating the preceding calculation when fl(a) = 0, we have

2. Picard's Theorems

62

Also, t1I2(A2 - t ) is an increasing function on the real interval [O, A2/3] and taking A > (3P(f))ll2, 0 5 p(z) 5 P(f) < $A2. Thus, for z in a sufficiently small neighborhood of a ,

If

(z) - bl

< If

(a) - bl

1 + E = p(a) + E < -A2 3

for

E

>0

(5.14)

sufficiently small. On the other hand, for z in a sufficiently small neighborhood of a , b g' B (f (z), p(z)) and so

Hence, from (5.14) and (5.15), for z in a sufficiently small neighborhood of a , 0 p(z) If ( z ) - bl < 1/3A2 and so by (5.10) and (5.12)

<

<

for z in a sufficiently small neighborhood of a. But (5.13), (5.16) and (5.11) are just (ii), (iii), (iv) of Definition 5.1, and so Xa (z) supports X(z) a t a. From Theorem 5.2 now follows A(.) ~ e n c e if, A > d by hypothesis,

m

,

L- for z 1-1212 A f' z

2(p(z))l!~((A?!p(z))

E B(O, 1)

.

< . Taking z = 0, since 1 f'(0) 1 = 1 - i=ip

(again since t1I2(A2 - t ) is increasing in [0, A2/3]). But A can be arbitrary provided and so letting A + (3/3(f))'l2, (5.17) becomes only if it is >

dm,

and so B = inf P(f) 2 f E 7

&@.

Note 5.6. Ahlfors' original paper also applied Theorem 5.2 to the problem of explicit bounds in Schottky's Theorem. In addition to the original paper, an exposition by Ahlfors of his Theorems 5.2 and 5.3 can be found in his book, Conformal

2.5. The Constants of Bloch and Landau

63

Invariants. Here there is a brief discussion also of what Ahlfors calls ultrahyperbolic metrics (i.e. X satisfies Definition 5.1 (iii)). Maurice Heins [log] has studied in detail how to define a more general class of metrics (which he calls S - K metrics) for which the inequality of Theorem 5.2 holds, and how its theory can be developed parallel to the theory of subharmonic functions. In particular, he obtains a sharpThis is the ening of Theorem 5.2 from which he deduces that, in fact B > best-known result to date. Raphael Robinson [211] has shown that there exist functions f analytic in B ( 0 , l ) such that B is the largest radius of a univalent disk for f and that these all have C ( 0 , l ) as a natural boundary. The existence of such "Bloch functions" also plays a role in Heins' proof that B > Ahlfors (loc. cit.) also applied his ideas to obtain a lower bound for Landau's constant.

m.

9.

Theorem 5.4. (Ahlfors): Let L be Landau's constant. That is L is defined as B is in the statement of Theorem 5.3 with the word univalently deleted. Then L 2 112. (Compare Note 2.1.) Proof. Again, start with a function f analytic in B ( 0 , l ) such that I fl(0)l = 1. Let l ( f ) be the supremem of the radii of disks taken on in the image of B ( 0 , l ) under f. As before, let p(z) be the radius of the largest disk centered a t f (z) and contained in the image of B(0,l) under f . Clearly p(z) l ( f ) . Also, as in the previous proof, Ip(z) - p(a)l 5 If (z) - f (a)[ and so p is continuous in B ( 0 , l ) . Let a E B ( 0 , l ) and let b be a point on C(f (a),p(a)). Then f (z) - b is never 0 for z E B ( 0 , l ) . Let A be a positive constant to be determined later. Then a can be defined in B ( 0 , l ) . single-valued branch of log ( f ( z ) - b

<

A)

Again, we use the fact (compare Note 5.3) that if g is analytic at z, gt(z) # 0, and

then

A 1%

=4 ( ~ ( z ) ) ~

in a neighborhood of z. Taking g(z) = l-'og(7l+) f as above, we get (using the l+'og( *) fact that for complex w, 11 wI2 - 11- wI2 = 4 Re w) a real-valued function

+

def

Ka(z) =

which is defined and positive provided

Ifl(z)I

If

(z) - bl

< A, and satisfies

2. Picard's Theorems

on the set of points where fl(z) # 0. Again as before, we consider as well the function

Clearly, p(z) # 0 for z E B ( 0 , l ) . The function t log($) is increasing in the real interval [0, Ale]; so taking A > el(f), and again arguing as in the proof of Theorem 5.3, we get that for z in a sufficiently small neighborhood of a, 0 < p(z) (z) - bl < Ale. It follows that

< If

for all z in a sufficiently small neighborhood of a. Finally, since p(a) = If (a) - bl by the definition of b,

But (5.20), (5.19), (5.18) are just (ii), (iii), (iv) of Definition 5.1, and so by . Hence, if A > eC(f), Theorem 5.2, K(z) 5

&

and so, taking z = 0, since 1 fl(0))l = 1 by hypothesis,

again, since tlog(A/t) is increasing in [O,A/e]. But A can be arbitrary provided only if it is > e l (f ) , and so letting A + eC(f ) gives

and consequently

L

> 112.

Note 5.7. If (notation as in Theorem 5.4) A

Re Log

> el(f), then

A

A

f (2) - b

If (2) - bl

------ - Log

>0.

Furthermore, ~ ( z = ) maps the right half-plane {z : R e r > 0) onto B(0,l). This motivates the choice of the function g in the proof of Theorem 5.4. Pommerenke [202] has shown that L > 112. He also provides a proof different from Heins' that B >

9.

2.5. The Constants of Bloch and Landau Note 5.8. It has been shown by Ahlfors and Grunsky [8] that

and a similar method as remarked by Sansone and Gerretsen (op. cit. 11, p.670) shows that

The last reference also contains a proof of (*). These proofs involve the construction of explicit conformal maps of the sort referred to in 1. Note 5.8, and their explicit representation by hypergeometric functions. For example, for B , the conformal map in question maps the (Jordan interior of the) "curvilinear triangle" whose sides are circular arcs with vertices 1, einl3,e2in/3,a11 angles n/6, and center 0 onto the (Jordan interior of the) equilateral triangle with the same vertices and center. It is also worth noting that before Ahlfors and Grunsky established (*) and Ahlfors, Theorem 5.4, it was not known that B < L.

Note 5.9. Ralphael Robinson 12131 applied the ideas of Ahlfors' generalization of Schwarz' Lemma directly to Picard's "Big" and "Little" Theorems, (Theorems 2.2 and 2.6), Landau's Theorem 2.3, and Theorem 4.5, as well as certain generalizations (see also Chapter 4). Note 5.10. In addition to B and L, one might also consider the constant A defined as is L except that now we insist the functions be univalent. Clearly L 5 A. Bounds for A are more obscure than those for B and L. That A > 0 was known to Hurwitz in 1904, long before Bloch's Theorem. The best-known lower bound seems to be A > .5705, a result of James Jenkins [125]. Earlier Landau [137], had obtained A > 9/16 = .5625. Landau also shows L < A in this paper. The best known upper bound would appear to be A < .658 obtained by Ralphael Robinson [214]. This involves a conformal map of B(0,l) onto the unit disk slit along six radii partway toward the center. Perhaps worth mentioning is the easier upper bound A 5 n/4 (noticed by Bloch for B) which arises from consideration of the map w = f (t)= $ log which maps B(O.1) onto the strip -a/4 < Irn w < n/4. Finally, as indication of some of the many other problems of the sort considered in this section, we mention the problem of Hurwitz alluded to in the above paragraph, whose definitive solution was given by CarathBodory (see Theorem 8.6.11).

(e)

Theorem 5.5. If f is analytic in B(0, I), f (0) = 0, f'(0) = 1, and f # 0 in B ( 0 , l ) - {0), then the image of B ( 0 , l ) under f contains a disk of radius 1/16, and this is sharp. The interested reader who consults the ample literature will find many similar problems with full or partial solutions.

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Chapter 3

An Introduction t o Entire Functions Picard's Theorems are initially somewhat surprising facts about entire functions. Basic results about such functions are part of the working equipment of anyone interested in complex analysis. In some ways, this chapter is different from the two preceding. The focus here, rather than being on a central theorem (the Riemann Mapping Theorem, the Picard Theorems) is on basic facts and an introduction to the extensive theory of functions with no singularities in C. However, earlier ideas, particularly involving valuedistribution, will appear once more here. Chapter 4 attempts to be the same sort of introduction to Nevalinna's theory of meromorphic functions. The stimulus of Picard's theorems to both these theories will be apparent. Chapter 5 deals with particular problems leading off from some of the material in this chapter. The reader of other literature should be warned that entire functions are called integral in British terminology (entidre and ganz unfortunately may be translated either way).

3.1

Growth, Order, and Zeros

One of the most basic results about entire functions is an explicit connection between the moduli of the zeros of a function analytic in the open disk and the modulus of the function itself, known as Jensen's Formula. Theorem 1.1. Suppose f is analytic on B(0,R) and f (0)# 0. Let T I , ra, . . . be the moduli of the zeros o f f i n B(0,R) arranged in a non-decreasing sequence. T h e n if

3. Entire Functions

r log -- -

log1f(reie)ld8 - logjf(0)l .

The formula has an alternative statement which is often even more useful: Let n ( x ) be the number of zeros of f in B(O,x), 0 < x < r (multiple zeros counted multiply). Then log I f (reie)ld8- log 1 f (0)l . For the rest o f this chapter, n ( x ) will be counting function defined in the preceding paragraph. Proof that (1.1) and (1.2) are equivalent: For rn tion we have rn log -= n log r T1

. . .T n

But m = n ( x ) for rm

x n

< r 5 rn+l, by partial summan-1

logr, =

m=l

m(log T,+~ - log rm) m=l

5 x < rm+l and n = n ( x ) for rn 5 x < r , and so n-1

rn /'m" log r l . . . T n = m=l 'm

qdx 1: +

$dx

=

1' %

dx

Proof of Theorem 1.1. Suppose first rn # rn+l and T , < T < rn+l, then since in this range n ( r ) is constant, both sides o f (1.2) have a continuous derivative with respect t o r. Furthermore,

Hence, for rn

< r < rn+l,

69

3.1. Growth, Order, and Zeros

where C is a constant. As r + r,, the modulus of a zero, the left side of (1.3) is clearly continuous. Hence, we need only show that the right side of (1.3) is continuous as r + r, in order to prove that (1.3) holds for all r . Without loss of generality, we can assume that r, is the modulus of a simple zero and (by rotation, if necessary) that its argument is 0. Then

LO^^ f (reis)1 =

~ o g f (reis)

+

l-tei"

1

r Log I - -cis rn

1

,

Because we assume that r, is the modulus of a simple zero, the first term is a continuous function of r in a neighborhood of r, (and indeed + log(r, 1 f '(r, 1) as r + r, when 6' = 0). Hence, it remains to show that if = a, then

5

Jd

2"

Log 11 - aeis Id19

is continuous as a + 1. But for 0 < a < 1, setting z = eis, Log (1 - aeis)d6' =

Log 11- aeieIdO = Re

Log (1 - a z ) dz=O, iz as may be seen, for example, by expanding Log(1- a z ) in a power series (Iazl is < 1) and integrating termwise. If a > 1, a similar argument shows that

12"

Log11 - aeis1dO =

Jd2" I

I

e dO Log 1 -s:-

+ 27r Log a = 27r Log a .

(1.5)

Finally, the improper integral

This last integral can be evaluated by standard methods of contour integration (and appears as an exercise in several books); however, it can also be done in a completely elementary and nearly trivial fashion as follows:

Jd"

log sin 6'dO = 2 "12

=nlog2+21

logsinOdO+21

But log cos(O - 7r/2)dO =

~12 logcos6'd6'.

3. Entire Functions

70 Hence

S,"

logsin0d0 = -T log2, and substitution of this in (1.6) gives

Equations (1.4), (1.5), and (1.7) provide the desired proof of continuity as a -+ 1. Hence (1.3) holds for all r > 0 and letting r -+ 0, we see that C = - log 1 f (0) 1, which proves the formula.

Note 1.1. There are several other proofs of Jensen's formula; another will appear in Chapter 4. Formally in the expression S %dz which counts the zeros, let z = reis, divide by r, integrate with respect to r , and take real parts to produce (1.2). Unfortunately, this is not clearly valid since a t zeros the integrand is infinite. A particularly satisfying proof for readers who know about harmonic functions can be obtained from the mean-value property for harmonic functions. One multiplies f by a suitable product to eliminate the zeros to obtain a related function F such that IF(z)l = If (z)l on C(0, r ) and log lF(z)l is harmonic in B(0, r ) . This proof results in the form (1.1) of the formula. A formula for the case f(0) = 0 can be obtained by considering a suitable is analytic and F(0) # 0. function F ( z ) = f (z)z-"which

k

Note 1.2. Jensen's formula may be interpreted as saying roughly that the more zeros an entire function f (2) has, the faster it must grow as 121 -+m (the converse of this idea is obviously false as iterated exponentials show). Note 1.3. There are somewhat more recondite formulas similar t o Jensen's which apply t o a half-plane or a rectangle instead of a disk. These are results of Carleman and Littlewood, respectively. Proofs of both may be found in Titchmarsh's Theory of Functions [229]. Carlemann's formula has considerable application t o certain problems in entire function theory which are discussed in Chapters 6 and 7 of Boas, Entire Functions [27]; Littlewood's finds application in analytic number theory. One recurring theme of the theory of entire functions, already evidenced by Jensen's Formula, is the connection between the zeros of an entire function and its growth as lzl -+ m. In order to discuss this, we need some definitions to make our concepts of growth more precise.

Definition 1.1. Iff is an entire function, then the order (of an entire function) p of log log M T f is defined by p = LT+, l o g T ( ) , 0 5 p 5 m , where M ( r ) = maxl,l=, 1 f (z)1, and by convention constants have order 0. Note 1.4. Thus a function f is of finite order p if and only if its associated max) every E > 0 and for no 6 < 0, as imum modulus function M(r) is o ( ~ T ' + ~for r + m. Roughly speaking, then a function of finite order grows no faster than some function of the form eP("), P a polynomial, z = reis, as r -+ m .

71

3.1. Growth, Order, and Zeros

Definition 1.2. Iff is an entire function of finite order, and f (0) # 0, let r l , 7-2,.. . denote the moduli of zeros off (if any) arranged in non-decreasing order and let pl = inf{a

>0:

1

7 converges) . n=l rn

pl is called the exponent of convergence of the zeros o f f . I f f has no zeros pl = 0. Similarly, the exponent of convergence of the a-points of f is defined as the exponent of convergence of the zeros of f ( z ) - a.

Theorem 1.2. If f is an entire function of finite order p and the exponent of convergence of its zeros is pl, then pl 5 p. Proof. By Jensen's Formula (Theorem 1. l ) (assuming f (0) # 0)

JdT

T d x=-

:r

Jd'"

LO^^ f (re")

Id0 - Loglf (0)1

,

and since f is entire, this holds for all r . Since n is non- negative and non-decreasing, we can estimate the left side of (1.8) (with r replaced by 2r) as follows

Hence from (1.8),

iTJd2=

n(r) log 2 5 -

log If (2rei6)Id0 - log If (0) 1

But, by Definition 1.2, since f has finite order p, the right side of (1.9) is 5 KrP+' for every E > 0, where K is a constant depending only on E . It follows that if P is any number > p, then n(r) < K I ~ O where K1 is a constant which does not depend on r . , so, for every E > 0, and K2 a constant indepenHence n = n(r,) < ~ l r iand dent of n ,

By Definition 1.2, C r = l verges; so

(5)

P1-'

diverges, hence by (1.10), Cr='=,n

-(P

- 6 )

1

di-

5 1; letting 6 -+ 0, and then /3 -+ p we get pl 5 p.

Note 1.5. f (2) = eZ shows that pl may be < p. Nevertheless, we now have for entire functions f of finite order, a rather explicit connection between the zeros of f and its growth as Izl -+ oo.

72

3. Entire Functions

One of the useful things about entire functions of finite order is that a more precise version of the Weierstrass Product Theorem, known as the Hadamard Product Theorem, holds for them.

Theorem 1.3. (Hadamard) If f is an entire function of finite order p with zeros z l , z z , . . . and f(0) # 0, then

where P ( z ) is the Weierstrass canonical product formed from the zeros o f f ( z ) and Q ( z ) is a polynomial of degree 5 p. Once this result is stated, one way t o attempt a proof is to consider the Weierstrass product f ( z ) = e9(")zkn,=, (1 eRn('lzn),where zn are the zeros (other than possibly 0 ) of f , R,(z/z,) is a polynomial which is a truncation of the formal power series for - Log 1 - f- , chosen of smallest degree so as t o guarantee convergence of the product and g is entire; take logarithms, and use the finite order condition to deduce that a sufficiently high derivative of g vanishes identically. This method can actually be carried out and was by Landau. However, it depends on a useful inequality, which may be considered a version of the maximum modulus theorem applied to the real part of an analytic function, and known as the

f-)

(

1

Lemma (Borel-CarathBodory Lemma). Let f be analytic in B(0,R ) and let M ( r ) = rnaxl,l=, If ( z )I and A(r) = maxlZl,, Re f ( z ) . Then for 0 < r < R ,

and, furthermore, if A ( R )

> 0, then

For a proof and discussion, see the Appendix.

Proof of Theorem 1.3 (Landau). There is no loss of generality in assuming f (0) # 0. By the Weierstrass Product Theorem, then f ( z ) = eQ(") ~ ( z, )

where P(n) =

nr=l(1 - 2) eRn(zlzn)and R , (2) is a polynomial which is a f-)

chosen of smallest degree truncation of the formal power series for -log (1 so as t o guarantee convergence of the product. By hypothesis f is of finite order p. Let v = [ p ] . Then, since by Theorem 1.2, the exponent of convergence of the zeros of f , pl, is 5 p, we have deg Rn 5 [ p l ] 5

73

3.1. Growth, Order, and Zeros [p] = v. Taking logarithms in the Weierstrass Product and differentiating v times, thus gives

= Q("+')(~)+ "" dzU+ln=l Log (1 --

M

= Q"')

(z) - v!

+1

$) + Rn (t) (1.11)

1

n= 1 (z, - z)"fl '

and to prove the theorem we need to show that Q("+l) is identically zero. This will be done by using the Borel-Caratheodory Lemma on expressions of the form t

1 (z, - z)"+l n=l

Let gR(z) =

# nlrm15R (1 - $) max lgR(z)l lzl=2R

<

-1

. Then, since f has order p, for r > 0,

max

as R + m , sincefor lzl = 2 R a n d (znl modulus theorem, for lzl 2R,

<

'

< R, 11-$1

2 1. So, by themaximum

Now gR(0) = 1; let hR(z) = Log gR(z) where we take that branch for which hR(0) = 0. Then hR(z) is analytic for lzl 5 R, and

Furthermore, by the maximum modulus theorem, lgR(z)l 2 lgR(0)l = 1 and so Re hR(z) 0, and so by part (b) of the Borel-Carathkodory Lemma for 0 < r < R,

>

rnax ( h t + l ) ( z )1 5 121=1.

+

2"+3(v I)!R (R - T)"" L"glgn(z)l

where K is a constant depending only on p and Taking r = R/2 gives max ~h:+')(z)l lzl=R/2

5 K ~ R P - ~ - ~, + '

where K1 does not depend on R.

E,

<

+

K ~ " + ~ ( vI ) ! R P + ~ + ~ 7 (R - r)u+2

on using (1.11).

3. Entire Functions But, by the definition of hR and (1.10),

Now, if lzl = R/2 and lznl > R, then lz, - zl and (1.13) yield

> ;lz,l,

and so for lzl = R/2, (1.12)

where K1 and K 2 do not depend on R. By the maximum modulus Theorem, (1.15) also holds if lzl 5 R/2. But v = [p]; so v 1 > p 2 pl ; hence both terms on the right in (1.15), + 0 as R + oo (if E is chosen small enough) whence the theorem follows.

+

Note 1.6. A different proof of Theorem 1.3 (Hadamard's) is based on the result that for functions f of finite order, if m(r) denotes minlzl,, If (z)I, then except for small neighborhoods of zeros o f f , m(r) does not grow much faster than &. The growth of m(r) has been the subject of considerable research; the basic results can be found in Boas, Entire Functions [27], Chapter 3. Although pl may be

< p, in many cases actually pl

= p as is shown by

Theorem 1.4. If P ( z ) is a Weierstrass Canonical Product of finite order p and pl = exponent of convergence of the zeros of P , then pl = p. Proof. By Theorem 1.2, pl 5 p. Let {z,) be the zeros of P ( z ) , and let rn = lznl where the r, are, as usual, arranged in non-decreasing order. Then

where, by the definition of R,,

3.1. Growth, Order, and Zeros for some integers Tn and so for lzl

Hence, for lzl = r and a

ILOgP(z)I<

< lznl,

> 1,

I L O ~ ( ~ - ~ ) + R ~ ( L ) I +

zn

Tn

zn

x

OC1

T,>ark=Tn

I r --=X1+C2, k rn

say.

Now, since P is of finite order, by Theorem 1.2, if m is an integer > p, CrZl$ converges. If we define g as the smallest integer such that Eml converges, then Tn - 1 5 g. Furthermore, if pl is an integer either g = pl - 1 or g = pl, (depending on whether or not c-&- converges) while if p1 is not an integer, g = [pl]. In any case, either

-&

(a)

g = pl - 1

or (b) In case (a),

C2

For all 6 sufficiently small pl

can be estimated as follows (since a

+ 6 < g + 1.

> 1):

as r + w. In case (b), we have for C2

(

= 0 r9+1

C Tn>aT

= O(rht6) as r

1

)

ypl-p, -6

+ m, since

=

o ( r g + l (a~)-g-~+p1+6 )

03

C n=l

Hence, for all sufficiently small positive 6, from (1.16),

1 Tn

converges.

3. Entire Functions

76 But, for E l , we have in the first place that for lzl = r and r,

where C is some positive constant. But, given

6

< ar,

> 0, since g 5 pl, and Tn - 1 < g,

Substituting in (1.17), we get that for every E > 0, lLog P(z)l = O(rpl+'), as r + oo, and hence p 5 pl, which, with Theorem 1.2, proves the Theorem.

Note 1.7. For a general entire function T, - 1 = deg(Rn) in the canonical product may + oo as n + oo. However, for functions of finite order, this cannot happen and this is the essential fact which allows us to prove Theorems 1.3 and 1.4. Theorem 1.4 has some immediate corollaries, important enough to be dignified as theorems: Theorem 1.5. Let f be an entire function of finite order p, with pl the exponent of convergence of the zeros o f f . Then if p is not an integer, pl = p. Proof. By Theorem 1.3, F ( z ) = e Q ( " ) ~ ( zwhere ) Q(z) is a polynomial and (Q(z)) 5 p. Since deg(Q(z)) [p] < p. By Theorem 1.2, pl 5 p. By Theorem the order of the entire function P ( z ) is pl. If pl were < p, then f would be product of two functions of order < p, and so, as an easy computation from definition of order shows, would have order < p, which is a contradiction.

<

deg 1.4, the the

Theorem 1.6. Iff is an entire function of finite order p, and p is not an integer, then f has infinitely many zeros. Proof. By Theorem 1.5, pl = p, hence since p is not an integer, pl

> 0.

Note 1.8. The functions ezk,k a positive integer, and eez show that the functions of finite positive integral order, or of infinite order may have no zeros. In general, in the Hadamard factorization f (z) = e Q ( " ) ~ ( zof) a function of finite order p if p is not an integer, the growth of f is determined by the growth of P, whereas if p is an integer the growth of eQ may dominate (but not necessarily). In any case, p = max(degQ(z),pl). If g is the smallest positive integer a such that

9.1. Growth, Order, and Zeros

77

Figure 3.1: Alternating regions. The family of entire maps ezk for k = 1,2,4. Black regions denote the existence of zeros and of very close values, whereas white regions include values growing to co. These regions are distributed along alternating regions with equal angles rlk.

Cr=, -&converges (where r,

are the moduli of the zeros of f arranged in nondecreasing order), then g is called the genus of the canonical product P and G = max(deg Q(z), g) the genus o f f . If an entire function of finite order is given in some form other than its Hadamard Canonical Product, it may be difficult to determine its genus. The distinction between functions of finite integral order and those of finite non-integral order appears throughout the theory. For entire functions of finite order, mile Bore1 found a remarkable deepening of Picard's Little Theorem (Theorem 2.1.2) with an elegant and straightforward proof. In order to prove this, however, we need to know the connection between the order of an entire function and the order of its derivative. This is an immediate consequence of Theorem 2.1 to be proved in the next section. However, a straightforward if somewhat computational, proof can also be given: Theorem 1.7. I f f (z) is an entire function of finite order p, then f l ( z ) has order P. Proof. Let z = reie and K ( r ) = rnaxl,l,, Ifl(z)l. Then, since f (2) = J: fl(t)dt f (01,

+

On the other hand, by Cauchy's formula,

3. Entire Functions Hence, by the maximum modulus theorem, 1 1 K(r)=maxlff(z)l<-(2nr)max If(w)l<-. IzI=T 2~ r2 Iw-zl=r

?-

Combining this with (1.18) we have,

Hence log M ( r )

+0

(i) M(r)

5 log(rK(r)) < 1% M(2r)

and consequently, 1

log log M(r) + O (M(T) 10, log r

M(T))

Taking the limit superior as r p = lim T

5

log log(rK(r)) < log log M(2r) .-log(2r) log~ log2r l o g ~'

+ co,we get logr

Theorem 1.8. (Borel) Let f (z) have finite order p where p is a positive integer. Then, given a complex number a, the exponent of convergence of the a-points off is p except possibly for one value of a. Proof. Suppose for two values a and b, the exponents of convergence for the apoints and b-points of f were both < p (the exponents of convergence are p by Theorem 1.2 applied to f (z) - a and f ( z ) - b). Then by Theorem 1.3,

<

f (z) - b = zk2eQ2(")~2(z) ,

(1.20)

where Ql(z) and Q2(z) are polynomials. So, by Theorem 1.4, the orders of Pl(z) and Pz(z) are each < p. Since f (z) has order p so do f (z) - a and f (z) - b have order p as an easy computation shows. It follows since PI and P2 each have order < p, that degQ1 = degQ2 = p. Subtracting (1.20) from (1.19) gives

b - a = z k l e Q l ( z )(~zl) - Z ' " ~ ~ Q ~ ( ~ ) P ~ ( Z )

(1.21)

3.1. Growth, Order, and Zeros or, rearranging,

Since P2 has order < p and Q2 is of degree p, the right side of (1.22) is of order p, and hence so is the left. Since Pl has order < p, it follows that Ql (z) - Q2(z) has degree p. Differentiating (1.21) gives

+ zklP,'(z) + Q', (z)) eQl(") = (k2zk2-lp2(z) + zkzP ~ ( z )+ Q ~ ( z )eQz(') ) . (klzk1-lp1(z)

Now, by Theorem 1.7, the order of Pl = order of Pl which is < p. Hence, the coefficient of eQl(") in (1.18) is an entire function of order < p, and so by TheoP ~ ( zP3(z) ) is a canonical product and rem 1.3 can be written as z ~ ~ ~ Q ~ ( " )where deg Q3(2) 5 p - 1 (since p is an integer). Similarly, the coefficient of can be written as ,zk4eQ4(") p4(z) where P4(z) is a canonical product and deg Q4 p - 1. Hence (1.23) can be rewritten as

<

Both sides of (1.23) must have the same zeros to the same multiplicities; hence k3 = k4, P3= P4, and consequently

But, as observed above, the left side of (1.25) has degree p and the right side degree p - 1, a contradiction.

<

Note 1.9. Define a "Borel exceptional value" of an entire function f of finite order p as a value of the complex number a such that the exponent of convergence of the a-points of f is not p. Theorem 1.8 shows (since the order of f (z) - a = the order of f (2)) that if p is a positive integer, there is at most one Borel exceptional value. Clearly, a Picard exceptional value (i.e. a value never taken on) is Borel exceptional for functions of positive order. Hence Borel's Theorem 1.8 and Theorem 1.6 present a substantial deepening of Picard's Little Theorem (Theorem 2.1.2) for functions of finite positive order. Note 1.10. The attentive reader will note that functions of zero order have been excluded from some of the above considerations. Non-polynomial functions of zero order do exist; one merely needs to construct a canonical product which converges sufficiently rapidly (or a sufficiently rapidly converging power series:

3. Entire Functions

80

F ( z ) = Cr=, n-n2zn is an example of an entire function of zero order by Theorem 2.1 below). It is easy to see that non-polynomial functions of zero order have no Picard exceptional value. For, if f takes the value a only finitely often and has order zero, then by Theorem 1.3 f (z) - a = czkP ( z ) , where P is a finite canonical product and c a constant. Further information on functions of zero order can be found, in particular, in M. L. Cartwright [42], Integral Functions.

Example 1.1. f (z) = ez cos fi is entire cos fi = x r = o

{q:

(

%)

1 kzk

, of order 1,

n a positive integer). and has zeros a t the points Hence, the convergence exponent of the zeros is 112. Hence, by Theorem 1.8, the convergence exponent of every other set of a-points is 1; that is, if [,(a) indicate converges for points a t which ez cos fi takes the value a # 0, then Cr==, every E > 0 and no E < 0.

6

3.2

Growth, Coefficients, and Type

We have already used several times, the fact that the order of a product or sum of two entire functions is a t most the larger of the respective orders, referring t o that ubiquitous aid "the reader" for the proof "by an easy computation". The computation really is easy, nevertheless, the result, as well as Theorem 1.7 and several other elementary facts, follows immediately from a connection between the order of an entire function and the size of its coefficients which is

Theorem 2.1. The entire function f(z) = ~ ~ = , a n is z nof finite order p if and = p, say, is finite. (Note that since f is entire, a, + 0 only if limn,, log as n + m. If an = 0, the value of the quotient is taken as 0.) If p is finite, then L' = P . Proof. We first show p 2 p , then that p

5 p.

(a) p 2 p. If p = 0, there is nothing to show. Suppose 0 < p 5 m and let 0 < E < p. Then, by definition of limit superior for infinitely many n ,

where K = p - E if p

<

and K = E if p = m. NOWfor every r

From (2.1) and (2.2), Log M ( r ) - n l o g r

> log lan] > -n Klog n

> 0,

3.2. Growth, Coefficients, and Type

Log M (r)

> n (log r

1

- - log n)

(2.3)

K

for infinitely many n and every r > 0. Considering the right hand side as a function of n , it is easy to see that it is maximal when r is such that n = Motivated by this (or simply because of the simplification obtained), by considering r of this K form we get Log M ( r ) > for an unbounded sequence of values of r . Hence Log Log M ( r ) > - (1 log K) log r log r

f.

+

for an unbounded sequence of values of r. Taking the limit superior of both sides, we get p infinite

and so letting E we get p p.

>

+ 0 in the case where p is finite and E + cm if p should be infinite,

(b) p 5 p. If p = cm, there is nothing to prove. Suppose p < oo, then given E

> 0, for all sufficiently large n, say, n

> no, 0 <

< nr+.7,

lanl

for n

n'o"

< p + E.

log(,&) -

That is,

2 no.

(Note that by the Cauchy-Hadamard formula, this implies f is entire.) Hence,

Now, as straightforward computation shows, the maximum of t-tlp+frt considered as a function of t occurs when t = $r"+'. This suggests breaking the second series somewhere in this vicinity. We thus get that

where C is a positive constant. For E l , we have

3. Entire Functions

82

since the series converges. For E2, if n 2 (2r)pf', then r n s 112 and so E2 5 $ = 2. Substituting these results in (2.4) and taking logarithms, we have

xr=O

<

Log M ( r ) = O(rp+' log r ) , as r and so

+

+ oo

+

+

Log Log r C Log M ( r ) < ( p E) Log T log r Log r for all sufficiently large r . Taking limit superior of both sides as r -+ co and then letting E + 0, we get p p. Log

<

Note 2.1. As already remarked, the fact that an entire function and its derivative have the same order, or that order of the sum or product of two entire functions, is a t most the maximum of the orders of the individual functions follow immediately from Theorem 2.1, as do other deductions about order easily expressed by power series. Nevertheless, similar results do not hold for the concept of genus mentioned in Note 1.8 even though it is closely related to order. In fact, as shown by Lindelof, the function

where 1 < cu < 2 has order 1 and genus 0, but F ( z ) - 1 has genus 1. Also, F ( z ) + F(-z) has genus 1, F'(z) has genus 0. Results of this sort show that although genus seems a quite natural notion, there is no simple way of obtaining it from knowledge of either the maximum modulus or the power series coefficients of an entire function. On the other hand, in the spirit of Borel's Theorem 1.8, Valiron has shown that given an entire function f of positive integral order, all the functions f (z) -a, except possibly one, are of the same genus. For the above function F ( z ) - a has genus 1 for all a # 0. Proofs of these results may be found in Georges Valiron, Integral Functions [233] (Lindelof's example is discussed in Chapter 111, section 14 and Valiron's Theorem in Appendix B of this book). If f is entire and of finite positive order, a natural refinement of the concept of order is

Definition 2.1. I f f is entire and of finite positive order p, then the type r o f f (z) is defined by r = G Log M(r) , (O< r m ) . r+OJ rp

<

(Hence, if f is of finite positive order p and finite type r , then M ( r ) = 0(e('+')") for every E > 0 and no E < 0).

Definition 2.2. A n entire function is said to be of growth ( a ,b) if it is order or of order a and type b.

<


3.2. Growth, Coeficients, and Type

83

As order, the type of an entire function can also be obtained from the coefficients as follows.

Ed

Theorem 2.2. Let f (z) = 03 anzn be an entire function. Let p be any finite nlanlpln. (Note that the choice of limit is positive number, and let v = limn,, motivated by Theorem 2.1 with p the order o f f .) Then

( a ) I f O < v < co, f is of orderp and type^=

$.

( b ) If v = 0, f is of growth (p,O).

(c)

Proof. If v

If v = oo, f is of growth 2 (p, co) (possibly of infinite order).

< co, then for all n 2 no, say, lanl < 1, and Log n

+ p/n Log lan[ 5

Log (v

+ E);

+

p ~ o g ( h 2) 1 - 10g(v E) n log n logn ' Taking reciprocals and then limits superior as n + co on both sides, we get that the order of f (z) 5 p. If v > 0, one can show similarly, using the definition of limit superior, that the order of f (z) 2 p. Hence, if 0 < v < co, the order of f (z) = p. Now suppose 0 5 v < co; then for E > 0 and n 2 no,

and so

where C is a positive constant. Considering the summand as a function of n, and noting that for constant a > 0 t and t 2 1, ):( has a maximum t = a / e , we see that the summand is largest for

q,

n near and that the maximum value of a summand is 5 e w . On the other hand, just as in the proof of Theorem 2.1, we wish to estimate a geometric progression and this requires that n be > (v + c)rP. Hence, writing the series in , have (2.5) as C1 + Cg where Cl is the part for which n 5 (v + 2 ~ ) r Pwe

3. Entire Functions and 1 1-

= 0 ( 1 ) as r -+ c o ,

Lk.L

(v+2.)

-

substituting (2.6) and (2.7) in (2.5) gives M(r)

< (v + 2c)rPexp ( ( v

LOq~(r) +

:y)+

0(1) .

(2.8)

+

So 5 p. O(r-P) as r -+ m. Since p > 0, it follows on taking taking limits superior as r -+ m and then letting 6 tend to 0, that T 5 -&. Similarly, if 0

<

v

<

m , for infinitely many n , lan/ 2 ( Y ) ~ ' ~ . Hence, by

Cauchy's inequality, M ( r ) 2 lanlrn 2

(7r

sequence of n's. Motivated as in the preceding paragraph, we take r = and get M ( r )

>e

(E)',

~ ) for ~ a11 ' ~r and an unbounded

.P(~-c)

' P

for an unbounded sequence of r's, and it follows that

T > $ .

If v = co,the above argument with v - E replaced by LO M ( T ) that lim,,, = co.

E

(and then

E

-+ co) shows

qP

Note 2.2. f and f ' not only have the same order (Theorem 1.7), they have the same type. This follows either from the same proof as Theorem 1.7 or directly from Theorem 2.2. The proofs of Theorems 2.1 and 2.2 indicate that to some extent, the order of an entire function f may be in fact, determined by a term in the power series for f of largest absolute value (since f is entire, there is a t least one such term). This is, in fact, so and the "maximum term" is actually a reasonable approximation t o the maximum modulus. While theorems involving the maximum term are not as important for later developments as Theorems 2.1 and 2.2, they clearly have a certain intrinsic interest. Definition 2.3. Let f ( z ) = Cr., anzn be an entire function. The maximum term o f f , p(r), is defined by p(r) = max, lanlrn. Theorem 2.3. I f f is an entire transcendental function of finite order, then lim 1% M ( r ) = 1. ~o~P(T)

T - i a

Proof. Since f is of finite order, and by Cauchy's inequality, p(r) 5 M(r), we have that lirn,,, = /?, say, is finite. Hence, given E > 0, for all n and all sufficiently large r , lanlrn 5 p(r) < e ~ ~ .+ '

3.2. Growth, Coeficients, and Type An easy computation shows that e so we get that for all n 2 no say,

85

,P+'

r

-n

-

has a minimum at r =

(&)'", and

> no, and all r ,

and consequently, for n

But M ( r ) 5 C F = o lanlrn, and by using (2.8), we may split this sum into two parts where the "tail" is estimated trivially by a geometric progression. In fact, for all r sufficiently large, say r ro, if n 2e(P e)ro+' = n l (r), say, then n no and so by (2.91,

>

>

+

>

So Log M (r) 5 ( 8

+ E)log r + log p(r) + 0(1), as r -+ m .

Hence, from (2.10) we get that 1 5 lirn,,, again, p(r)

Lqfg~(')+ m as r

a. F-

But by Cauchy's inequality, since f is not a polynomial,

(2.10) 7-

m.

But by Cauchy's inequality

< M ( r ) , and so, in fact lirn,,,

og M ( T )

= 1.

An immediate corollary of Theorem 2.3 is

Theorem 2.4. I f f is entire of finite order p, then

Lo9 Lo9 ,J(r) lim Log r

T+CC

If p

> 0,

and f has type

T,

=P

then

log,J(r) = T . lim T-OO

rP

Definition 2.4. I f f is analytic in the angle W = {reie : a and

(

)

5 0 5 P, 0 5 r < CQ),

e i e E ~ , ~If< (T ~ ~ ~ " 1 ) . ; log log ( m a x Rlog r =p7

?-+a

then p is called the order o f f in the angle W

86

3. Entire Functions As an application of Theorems 2.3 and 2.4, we have the following result of P6lya.

Theorem 2.5. Let F be a transcendental entire function of finite order p. Suppose F ( z ) = C r = o anzn. Then there is a sequence { e n ) where each 6 , = 1 or -1 such that the entire function 00

has order p in every angle W. Proof. If p = 0, clear (for any sequence {en)). Suppose p > 0. Let {p, : n = 0 , l . . . ) be a strictly increasing sequence of positive numbers such that limn,, p, = p. Let {$, : m = 1 , 2 , . . . } be a countable set of real numbers dense in [0,2,rr],and define the sequence (4, : n = 0,1,. . . ) by

etc., i.e. 4 l n + m - l ) ( n + m ) +,-I

= $rn

for n = 0 , 1 , 2 , . . . ; m = 1 , 2 , . . .; what is essential in the definition of the sequence (4,) is that each $, appears infinitely often in it. We now construct inductively two sequences of integers {A,), {K,), and a sequence of real numbers {r,) such that p(r,) = la,,, Ir;.; {K,) and {A,) are interlaced; and an appropriate lower bound is obtained by choosing the sequence {€,) properly in each interval [A,, A,+I). Set Xo = 0, and suppose A, has been constructed. Since by Cauchy's inequality, ogM r + m as r + m , Theorem 2.3 implies that + m as r -+ m ; hence given b, 0 < b < 1, for all sufficiently large values of r ,

w

Furthermore, by definition of the sequence {p,), for arbitrarily large values of r,

Thus we can find a sequence {r,) tending t o infinity with n such that (2.11) and (2.12) hold simultaneously for r = r,, in fact, again since -+ co as r -+ m , we can further require that r, satisfy (

)=a

r

say, where

K,

> A, .

(2.13)

3.2. Growth, Coefficients, and Type This defines r, and K,, given A,. On the other hand, since lav)rL= 0 ,

lim

t-+w

v=t

there is an integer An+1 such that

Plainly, An+1 > K,, and thus we have defined the sequences {A,), {K,), {r,). Furthermore, 0 = A0 < KO < A1 < 6 1 < . . . . We now define rm for integers m E [A,, An+1) by r& = 1 and

Clearly r, is uniquely determined by (2.15) as 1 or -1. Let

Clearly G cannot be of order > p in any angle (since F by hypothesis has order p and Theorem 2.2). We break the sum in (2.16) into three parts for z = rnei@-. By (2.11) and the definition of r,,

By (2.13) and (2.54) (since r,, Xn+1-1

= 1 and A,

1 rmam(rne'@n)m= p ( r n )

m=Xn

Xn+1-1

x

m=X,

< K, < A,+l),

~ ~ a ~ ( r , e ~ @ n ) ~ a,, ( ~ , e ~ @ n ) ~ n (2.19)

2~(rn)

3. Entire Functions

88

Taking (2.17), (2.18) and (2.19) together in (2.16), with b = 114 say, we get by (2.12)

, infinitely often, and since limn,, since the sequence (4,) contains each $ it follows from (2.20) that for each ray {rei*- : 0 5 r < w ) , -

lim

,--too

pn = p,

log log G(rei*- ) >P, log r

and since G is of order p (by Theorem 2.2, since F is of order p by hypothesis), equality must hold in (2.19). Hence, along each of a dense set of rays, {rei*- : 0 5 r < w), -

lim

T+W

log log G(rei*- ) =D log r

and so G is of order p in every angle. Note 2.3. Actually, we can achieve even more. Analogously to Definition 2.4, we can define the type in an angle of a function of finite order p in that angle. A similar proof then shows that if F is of finite order p and type T , then the sequence {em), em = 1 or -1, can be chosen so that the entire function G not only has order p, but also type T in every angle. In fact, by the continuity of the Phragm6nLindelof indicator function h(0) discussed in section 3 of this chapter (see Theorem 3.7 below), we can say that in this case T is the "type of f along every ray" (and not just along a dense set of rays). Note 2.4. Theorem 2.5 also holds for functions of infinite order. For, by Cauchy's inequality, p ( r ) 5 M ( r ) , while if lim,,, is finite, then by (2.10),

-

log log M ( r )

1imr-m log, is also finite. Hence, if F is of infinite order E,-.,;;,,, = w , and the argument of Theorem 2.5 can be repeated to show that, in fact, G is of infinite order in every angle.

Note 2.5. If f is an entire transcendental function, f (z) = Cr=3=, anzn, and p(r) its maximum term, then the central index v(r) may be defined for r > 0 as the ) defined as the index of the largest value of n for which lanlrn = p(r). ( ~ ( 0 is first non-zero a,.) It is not difficult to see that v(r) is monotone non-decreasing, v(r) = oo. Supposing piecewise constant, continuous from the right, and lim,,, f (0) # 0 (so u(0) = O), one can then show (just from the definition) that Log p(r) =

lF

dt

+ Log /f(0)1 .

3.3. The Phragme'n-Lindelof Indicator From Jensen's Theorem (Theorem 1.1), Log M(r) 2

lT

Tdt + Log lf(O)I

And this, together with Theorem 2.3, suggests not only do p(r) and M ( r ) behave somewhat similarly; but, perhaps more surprisingly (seeing the definitions), that v(r) and n ( r ) do. In fact, it is not hard to prove that if f is entire of order p, Lo '(') = p. On the other hand, E,,, then lirnT,, 5 P [by

w-

,GT

Jensen's Theorem n(r) Log 2 5 Log M(2r) - Log 1 f (0)1 .) The parallels between p and M , v and n , are even closer, and for a setting out of these, the interested reader is referred to Chapter 1 of Section IV, of the well-known book by Pcilya and Szego [200], Aufgaben und Lehrsatze aus der Analysis (Volume 11) (this has been translated into English also). For example, both Log p(r) and Log M ( r ) are convex functions of Log r (the latter fact, known as the Hadamard Three Circles Theorem, is proved in the Appendix). The study of the behavior of an entire function near a point where its modulus is large in terms of the coefficients of the power-series expansion of f is known as Wiman-Valiron theory. Wiman's [250] original papers appeared in Acta Math 37 and 41. A comprehensive survey of the theory was given by Hayman [loll. A somewhat different approach than that usually taken to the theory, is indicated by Fuchs [77] in Complex Analysis. One of the notable results of the theory is that given 6 > 0, there is a sequence of arbitrarily large r , say {r,) such that M(r,) < , u ( r , ) ( l ~ ~ , u ( r ~ ) ) ' Iand ~ + ~if, f has finite order p such a sequence with M(rn) < (P + ~ ) ( 2 ~ ) ' ~ ( r n ) ( l o g ~ ( ~ n ) ) ~ These results are due to Wiman and are capable of still further refinements, (Valiron [234]; Hayman, op.cit. Chap. 11, Section 4). A result of this type holding for all r is M ( r ) 5 p(r) (2v (r 1) (Valiron, op. cit. Chapter 11, Section

+ &)+

4).

3.3

The Phragmh-Lindelof Indicator

Suppose an entire function f is of finite positive order and finite type. Then there is a p, 0 < p < oo, such that

lim Log M ( r , f ) = T < C O . rP

T+W

This immediately suggests that a more refined study of the behavior of f can be undertaken through the study of the function

- Log lim

T+OO

if

(reis)l rP

3. Entire Functions

90

as a function of 8(-T < 8 5 T). Such a study, in fact, proves extremely informative. As a further motivation for such an undertaking, we first prove an important generalization of the maximum modulus theorem, published by Phragmbn and Lindelof in 1908 (Acta Mathematica, Vol. 31) [190].

Theorem 3.1 (PhragmBn and Lindelof). . Let R be a simply connected region i n C, bounded by a simple closed contour r. Let P E r, and suppose f is analytic on (RUJ?) - P . Suppose also, for all z E I?- P , If(z)l 5 M . Suppose further, that there is an auxiliary function a ( z ) such that la(z)l 5 1 for z E R, and such that, given E > 0, there is a system of curves C, with C, c R U J? and C, n J? consisting of two points, where, for every E R ; for some C,, E Jordan Interior of C, U r, with the property that for all z E C,, for all n, and for all E > 0, Ia(z)ltlf (z)I M . Then If (z)l 5 M for all z E R. (Intuitively, the C, are a system of curves connecting the two sides of r around P and arbitrarily close to P.)

<

<

<

<

Proof. Let F ( z ) = (a(z))'f (2). Then F is analytic in R and given E R, choose a C, such that E Jordan Interior of C, U J?. Then for z E C, U I?, IF(z)I 5 M , and so by the maximum modulus theorem, IF(<)I M , so If (<)I 5 Mla(<)I-', and so letting E -+ 0, If (<)I 5 M .

<

<

Note 3.1. The proof of Theorem 3.1 is so easy, that the consequences which result from this idea are even more interesting-as Landau remarks when proving a slightly different version: "Please read this proof once more! It is so deceptive, that otherwise one does not believe it." The point P may, of course, be (and usually is) taken to be oo- in fact since a preliminary linear fractional transformation can always be made, this involves no loss of generality. In practice, Theorem 3.1 is not used directly; it proves simpler in a particular case t o pick a particular auxiliary function suited to the region considered-thus one speaks of the "Phragm6n-Lindelof Principle". As an example, we have

Theorem 3.2. Suppose W is a wedge-shaped region bounded by two straight halflines L1, L z , making an angle n l a at 0 ( a > 112). Let f be analytic i n Suppose If (z)l 5 M for z E L1 U L:!. Suppose further that as r + oo

v.

P

~f(reiel = O(er ) where

P < a, uniformly i n W .

(*I

T h e n If(z)l 5 M in W . Proof. By a rotation, if necessary, there is no loss of generality in assuming that L1 and Lz are the lines 8 = f n / 2 a . Let E > 0, 0 < 6 < a - /3, and F ( z ) = e-czP+s(f (z)), z = reie. Then

3.3. The Phragme'n-Lindelof Indicator

+ <

91

+

If 8 = f7r/2a, then since P 6 < a , cos((p S)8) > 0, and so on the lines 8 = f7r/2a, IF(z)l 5 If (z)l 5 M , by hypothesis. Also, if z = ~e~~ where 181 7r/2a, then I F ( R ~ ~5 ~K) ~~ ~ ~ - ' ~ ~ + ~ ~ ~ ~ ( by hypothesis and the right-hand side approaches 0 as R + oo. Hence for R 2 1-0 say IF(z)l 5 M on the arc ~ e " , 0 101 7r/2 as well, and so by the maximum modulus theorem IF(z)l 5 M in the sector {reis : 0 5 r 5 R, 101 7r/2a) for all R 2 ro. Hence, IF(z)l M throughout W ; whence If (z)l ewP+'c04(fl+6)e) M throughout W. Letting + 0 , the result follows.

< <

<

<

<

Note 3.2. The wider the angle in Theorem 3.2, the slower f must grow in W for the theorem to hold. Some growth condition is needed if one is to generalize the maximum modulus theorem to unbounded regions, as may be seen by the following example: Let S = {z : -812 < I m z < 7r/2), and suppose g(z) = eez. Then g is continuous on 3 and analytic in S. Furthermore, on Bd S- {oo) writing z = x+iy, we have, 1g(z)1 = lee"C0S(fn/2)1 = 1. However, clearly, as z + oo through the positive real numbers g(z) + oo. Theorem 3.1 also indicates why a function of infinite order is necessary for such an example. Note 3.3. Clearly, it is not necessary for f to be analytic in a whole wedge, but only in the unbounded part resulting from removal of a sector, provided only that on the resulting piece of boundary we still have If (z)1 M . Using this, the theorem can be extended to the case cu < 112, where the angular region covers (part of) the plane more than once.

<

One can, in fact, improve Theorem 3.2 slightly:

Theorem 3.3. Suppose the hypotheses of Theorem 3.2 are satisfied except that (*) is replaced by the condition

If

(reie)[ = O(esr") for every S > 0 .

Then again

If

(z)l

(**I

5 M for all z E W.

< <

Proof. As before, the angle can be taken as -7r/2a 5 8 7r/2a. Given 6 > 0, let F ( z ) = e-'"= f (2). Then, for r real and positive I F ( r ) 1 Ke("-')'", and so taking 6 = €12 say, F ( r ) -+ 0 as r -+ co through positive real values. Hence F is bounded above, say by M i on the positive real axis, and IF(z)l can be supposed to assume the value M' on the positive real axis. Let M * = max(M, M'). Then by Theorem 3.2, applied separately to the angular regions formed by 8 = -n/2a, 8 = 0 and 8 = 0, 8 = 7r/2a, we get that IF(z)l 5 M* throughout W .But M * 5 M, for if M * > M , then M * = M' > M ; hence IF(z)l assumes its maximum at a point (on the real axis) interior to W, and so by using the maximum modulus theorem, one can conclude that F is constant, whence M * = M i = M . ~ letting " ~ , 6 -+ 0, the result Hence IF(z)I 5 M , and If(z)I 5 ~ e " " ~ ~and follows.

3. Entire Functions

92

Note 3.4. It follows from the argument of Theorem 3.2, that a non- constant entire function of positive order p < $ cannot be bounded on a ray through the origin, (take p = p, a = the rotated lines L1 and LZ are given by 0 = f n ) . Actually an entire function of growth (1/2,0) cannot be bounded on a ray through the origin. For suppose f (z) is such a function. Let g(z) = f (z2). Since f is entire and bounded on a ray, g is entire and bounded on a line, which, by rotation if necessary, can be assumed to be the imaginary axis. Furthermore, g is of growth (1,O). Hence, by Theorem 3.3, g is bounded in the right half-plane {z : Re z 2 0) and also in the left-hand plane {z : Re z 5 0), hence constant by Liouville's Theorem. So f is constant. Let m(r) denote the minimum modulus of f . It follows easily from the above result that if f is entire and of growth (112, O), then KT,, m(r) = oo. In fact, much more can be said about the minimum modulus of entire functions of "slow growth"; in particular, there is the well-known "cosnp Theorem" for functions of positive order < 1 conjectured (independently) by Lindelijf and Littlewood and proved (independently) by Wiman and Valiron. For this and related results, see Boas (op. cit.) Chapter 3, and Cartwright (op. cit.), Sections 4.41-4.45. Hayman [I021 has considered functions of order > 1 (including those of infinite orders). The present state of knowledge for functions of order p, 0 < p 5 1, seems to be as in P.D. Barry [14] where the context is general subharmonic functions and papers by Barry and Hayman in Mathematical Essays dedicated to A. J. MacIntyre [15]. If f (reis) actually has a limit as r -+ co (for a fixed value of 0) even more can be said; such questions will be considered in Chapter 5, however, the following theorem seems to belong here.

i,

Theorem 3.4. Suppose W is a wedge-shaped region bounded by two straight halflines L1, L2 making an angle n / a at 0. Suppose f is analytic and bounded in W and lim T+CC f (reis) = a and lim l-.+m f (reiQ)= b. Then a = b and f (reiQ)+ a reae

EL^

T~"EL~

uniformly in W. Proof. Let z = reis. As before, we can assume without loss of generality that the lines are given by 0 = f n/2a. Furthermore, we may assume a > 1 (since if a < 1, a substitution of the form z = Ck will reduce to the case a > 1). Now suppose f (z) -+ a along L1 = {reiQ: 0 = -n/2a) and f (z) -+ b along L2 = {reiQ: 0 = 7r/2a). Then, since (f (z) - a)(f (z) - b) = (f (z) - ;(a+ b))' - ;(a - b)2, the function g(z) = (f (z) - $(a b))' - :(a - b)' -+0 on L1 and on L2 as r + oo, and is analytic and bounded in W. Let G(z) = &g(z) where X > 0 is to be chosen later. Then ' Ig(z)], and so, given r > 0, IG(z)l < IG(')I = d T 2 + ~ x ~ ~ ~ ~ ~ + < x~lg(z)I " < r, for r > r1 = r ~ ( r and ) reis E LI U L2, and also IG(z)I wsay, for z E If we pick X = X(r) = then for r 5 rl also IG(z)I 5 M M Me < r. Then, by Theorem 3.2, IG(z)l 5 6 in W .But d lX+ 2( ~ ) 1+(kI2 =

+

r.

< d-

T,

<&

3.3. The Phragme'n-Lindelof Indicator

+

93

+

w.

,(z) = ~ G ( z and ) so 1g(z)1 5 (1 $ - ) I G ( Z 5 ) ~ (1 $16 in Letting r + m and then E -+ 0, we get that g(reis) + 0 uniformly in Hence, given E > 0, there is a circular arc C = {Reis : -7r/2a 5 8 5 7r/2a) on which Ig(z)l = 1 f (z) -a[ If (z) - bl 5 E, and so a t every point of C, a t least one of the holds (R, of course, depends possibly inequalities If ( z ) - a1 < E'/', If (2) - bl < on 6). Clearly we may assume that If (z) - a1 < & for z E L1 and If (z) - bl < & for z E L2 (for otherwise a = b already). Let 80 = sup(8 : Reie E C, If (Reis) - a1 < &). Then, since f is continuous on C, If (ReisO) - a1 &. On the other hand (if a # b ) , Reis0 is also a limit point of points on C where If (Reie) - bl < &. Hence, letting zo = Reieo, 2& 2 If (zo) - a1 If (20) - bl 2 la - bl, and so we must have a = b. Finally, applying to f and F ( z ) = f (z), the argument used for g and G previously we have that f (z) + a uniformly in W.

m.

<

+

Note 3.5. Theorems 3.2,3.3, and 3.4 are stated for wedge-shaped regions, however, by a suitable conformal mapping, one obtains other regions for which the PhragmknLindelijf principle is true. For example, the case of a vertical strip is especially important in the theory of ordinary Dirichlet Series (series of the form Cr==l The map w = i Log z takes a wedge in the z-plane onto a strip in the w-plane. It is in the context of a direct proof of a theorem for the strip that Landau makes the remark quoted earlier (Note 3.1).

G).

Suppose F is entire and of finite positive order. Then Theorems 3.2, 3.3, 3.4 already indicate that the growth of f along one half-line strongly influences its growth along other half-lines. For a closer examination of such questions, we make

Definition 3.1. I f f is entire and of finite positive order p, then the Phragme'nLindeliif indicator function h(8) is defined by

Note that h(0) may be +ca or -ca. Note also that by definition, h is periodic with period 27r.

Note 3.6. As is already apparent, the indicator function h(0) can be defined and results involving it hold (possibly with minor variations) for functions which are only of "finite order in an angle". However, we shall generally assume f is entire. In theorems such as Theorem 3.1, 3.2, 3.3, the growth of the function f ( z ) is compared with the growth of eaZPas lzl + m. If z = reis, and a = a + iP, this suggests comparison of h(8) with functions of the form B cos p8 + C sin p8. Definition 3.2. A real-valued function H(8) of the form H(8) = B cos p8+C sin p8; B, C constants, is called a sinusoid with multiplier p.

94

3. Entire Functions

Theorem 3.5. Suppose f is entire of finite positive order p, and that 0 5 < 82 < 27r, where 82 - 81 < nip. Suppose firther h(B1) 5 hl < ca, h(B2) 5 h2 < ca and let H ( 8 ) be the sinusoid with multiplier p such that H(O1) = hl and H ( 0 2 ) = h 2 . Then h ( 8 ) 5 H ( 8 ) for all 8 E [81,821.

+

Proof. Suppose we are given 6 > 0 and let Ha ( 8 ) = as cos p8 bs sin p8 be the sinusoid with multiplier p such that Ha(&) = hl 6 and Ha(B2)= h2 + 6. Let

+

Then,

1 re^') I = ~f (reie)le-Hs(e)TP ,

and so

~ ~ ( ~ ~5 ie ~e p~( h>~ +l " e - ( h ~ +6 )1~for p

r >rl =rl(6),

Similarly, I ~ ( r e ~ ~is2bounded )1 as r + m. Hence, by Theorem 3.2, F is bounded in the wedge-shaped region determined by the half-lines 6' = O1 and 8 = 82. But,

and hence

h(8) 5 H6(8) for 8 E

[ e l ,821

.

As 6 + 0 , H6(8) + H ( 8 ) and so the theorem follows. Note that hl or h2 may be -ca. In this case the proof shows that h ( 8 ) = -ca for 8 E [el,821 as well.

Note 3.7. The sinusoid with multiplier p of the theorem is easily seen to be (for hl- .,hz- finite) h2 sin p(8 - 81) - hl sin p(8 - 82) H(8) = sin p(82 - 81) The basic properties of h(8) depend upon the following fundamental inequality:

Theorem 3.6. I f f is entire of finite order p; h(8) is its Phragmkn-Lindelof indicator and 0 < O2 - 01 < 7r/p, 0 < 83 - 82 < n / ~then , if h(81) and h(82) are finite, h(81)sin p(83 - 82)

+ h(82)sin p(81 - 83) + h(83)sin p(82 - 81) > 0 .

Proof. As an elementary computation shows that for any sinusoid H ( 8 ) with multiplier p,

Suppose h(O1),h(O2)are finite, and let H be the (unique) sinusoid such that H(B1) = h(O1), H(82) = h(O2). We will now show that h(03) H ( 0 3 ) . Choose 8* E [81,82] such that O3 - 7r/p < 8* < 82. Then h(8*) 5 H ( 8 * ) by Theorem 3.5, and so

>

3.3. The Phragme'n-Lindelof Indicator

95

by the remark at the end of the proof of that theorem, h(03) # -00. Suppose contrary to what we wish to prove, h(03) < H(O3); then there is a 6 > 0 such that h(O3) 5 H(O3) - 6. Let sin p(O - O*) Hs(O) = H(O) - 6 sinp(03 - 0*) . Then h(O*)

< H(O*) = Ha((?*)

and h(O3) I H(O3) - 6 = Hs(O3) . Since 02 E (O*,O3) and 0 < 83 - O* < r / p , Theorem 3.5 gives h(02) Ha(B2) < H(02) contradicting the hypothesis that H(02) = h(02). Hence h(03) H(03), which, taking sight of (3.2) proves the theorem.

<

>

Note 3.8. The inequality of Theorem 3.6 can be written in the interesting form h(O1) cospOl sin pel h(O2) cos pO2 sin p02 h(O3) cosp03 sinp03

20.

Theorem 3.6 shows that h(O) is "sub-sinusoid"; that is that it has the same property with respect to sinusoids that convex functions have with respect to linear functions. Theorem 3.7. Suppose f is an entire function of finite positive order p, and h(0) its Phragme'n-Lindelof indicator function. If h(O) is finite for O E [01,03], then it is continuous in [01,03], and has a right and left-hand derivative at every point of (01, 03). If h is not differentiable at some point O* E (81, 03), then the left-hand derivative at O* is 5 the right-hand derivative at O*.

Proof. Let O2 E (61, 03), and suppose (with no loss of generality) that 83 - O1 < n l p , let H1,2(0) be the unique sinusoid with multiplier p such that H1,2(01) = h(el), Hl,2(e2) = h(02). Define Hz,3(0) similarly. Then by Theorem 3.5, for 8' E [42,031,

while by the proof of Theorem 3.6, for 0' E [O2,O3]

Similarly for 8' E

[el,1321, by Theorem 3.5

and by Theorem 3.6 for 19' E [dl, 821,

96

3. Entire Functions

(For suppose not, that is suppose for some fixed 8' E by Theorem 3.6, since all the sines are positive, h(82)

L

<

[el,821, h(8') < H2,3(Of);then

+

h(8') sin p(83 - 02) h(83) sin p(B2 - 8') sin p(03 - 8') (8') sin p(83 - 82) h(83) sin p(82 - 8') sin p(83 - 8')

+

Substituting the value of H2,3(8') gives the contradiction h(B2) < h(02).) Using (3.3)-(3.6), we see that for all 8 E [O1,fl3],8 # 82,

Since sinusoids are differentiable, it follows that lime+e2h(8) = h(B2) and so h is continuous. To show h differentiable from the right and the left, consider first the case in which h(8) = 0. Then Theorem 3.6 applied to 8,8 a,8 P, where 0 < a < n/p and 0 < p - a < nip, gives

+

h(8 Hence, for 0

< X < 'TIP,

+ a ) sin pp 5 h(8 + P ) sin pa .

sinpx

is a monotone decreasing function of x. Further-

more, by (3.7), (with 8 for 82 and 8 below as x

+

+ 0, hence lim,,o+

+ x for 8), ho s i n p x - ho x

bounded h(O+x)-h(B) exists, and so limx+o+ x '

2 s i n p x is

sin px li~n,,~+ = limx,o+ h(O+r) sin . If h(8) # 0, the above arguments may be applied to F ( z ) = e-h(e)zpf (2). Similar arguments show that h is differentiable from the left at every point of (el,e3). Similar considerations of points on either side of 0 shows that the left-hand derivative is always the right-hand derivative.

<

Note 3.9. h(0) is not necessarily differentiable in (el, 03). Consider f (2) = ez+e-". f has order 1. Since If (reie)l = lezreie+ lle-rcose, one easily sees (considering the cases cos8 > 0 and cos8 5 0 separately) that h(8) = I cos8l and at 8 = fn/2, this is not differentiable. Theorem 3.8. Suppose f is entire of finite positive order p and h(0) is its PhragminLindelof indicator function. If h hs finite in the interval ( a , P) than given E > 0, for r > ro = TO(€), Log (reie)[ < rP(h(8) e) uniformly in any closed subinterval of (a, PI.

If

+

Proof. Let [a,b] be a closed subinterval of a,P. Since by Theorem 3.7, h is continuous in [a, b] it is uniformly continuous there, and so given E > 0, we can find a sequence of points {dk),k = 0,1,2,. . . ,n, such that a = 80 < < . - . < ok = b with the following properties:

3.3. The Phragme'n-Lindelof Indicator

+

Figure 3.2: Critical values. Let the entire map f (z) : e" e-": the superimposition of the domain (background) and of equipotential curves (foreground) can help to figure out some features. First, the roots of f (z), located at the black discs along the imaginary axis and the intersection of such curves denoting the existence of critical points, located at ixn for n = f1, f2,. . . . On left and right, the lighter background areas show that f (z) takes on values growing to foo.

(i) For 8 E [8k-l,8k], k = 1,2,. . . , n , lh(8) - h(8k)l

<

E;

(ii) If Hk is the (unique) sinusoid with multiplier p such that Hk(8k-l) = h(8k-1), and Hk(8k) = h(Ok), then IHk(8) - Hk(8k)l E for all 8 E [&-I, 8k], k = 1,2, . . . n - 1 (since Hk is continuous).

<

But Hk(Bk) = h(Ok) and so combining (i) and (ii), we get Ih(8) - Hk(611 5 2~ Bk], k = 1 , . . . ,n. But an examination of the proof of Theorem 3.5, for 6 E reveals that actually what is shown there (equation (3.1)) is that in the wedgeshaped region determined by 8 = 81 and 6 = 02, Log If (reie)[ M H(8)rP where M is a constant. Hence, in our present context, for 8 E [ek-l, 8k], k = 1 , 2 , . . . , n , and r > ro =

<

(€1,

TO

Log

If

(reie)/ < (Hk(8)

+

+ f)rP < (h(8) + 3e)rP .

Replacing 6 by €13 throughout, and taking the union of the intervals [8k-1, Ok], the result follows. We now turn to some consequences of further investigation of h(B), some of which may be somewhat surprising.

Theorem 3.9. Let f be a function of finite positive order p and h(8) its Phragme'nLindeliif indicator function. If h(8) is finite for 8 E [a,P], then every subinterval in which h(6) > 0 has length 2 xlp, and every subinterval in which h(8) < 0 has

98

3. Entire Functions

length 5 nlp. Furthermore, each open subinterval in which h(8) < 0 is followed by a point where h(8) = 0 and an open interval where h(8) > 0 . Consequently, an interval in which h(8) 0 is preceded and followed by interval in which h(8) > 0. It follows immediately if h ( y l ) = 0 and h ( y 2 )= 0, then either 71 - 7 2 Yz T I P or h ( y l ) I 0 for some y' E ( 7 1 , ~ ~ ) . Proof. If h(8) > 0 in an interval (81,& ) ( e l < 193) of length < n l p where (81,83) c [ a , P ] ,and not in any interval of larger length, then since by Theorem 3.7 h is continuous, h ( & ) = h(83) = 0. Thus Theorem 3.6 gives h(82)sinp(O1 - 83) 2 0 for any O2 E (el,0 3 ) , and so (since h(82) > 0 by hypothesis) sinp(03 - el) 0, contradicting 0 < 83 - 81 < TIP. Argument for the length of the interval in which h(8) < 0 is analogous. Similarly, suppose h(8) < 0 in [81,82),where 0 < 82 - 81 < n l p and h(82) = 0. Then Theorem 3.6 gives for any 83 such that 0 < 83 -82 < nlp, and 83 E [ a ,P] that h(81)sin p(83 - 82) h(83)sin p(82 - 81) 2 0, whence h(83)sin p(82 - 81) > 0 and so h(83) > 0. Thus h(8) must be positive in an interval of length n l p following 02. Arguing similarly, if h(8) < 0 in (82,031 and h(82) = 0, we find in this case h(8,) > 0.

<

+

>

Theorem 3.10. Suppose f is entire of finite positive order p and h(8) is its Phragme'n-Lindelof indicator function. If h(8) is finite in an interval [ a ,y] where y - a = n / p and h ( a ) = h ( y ) = 0, then h equals a sinusoid in [ a ,y ] .

Proof. Suppose P E ( a ,y ) , and let

Then (since y-a = n l p ) , H ( a ) = h ( a ) = 0, H ( y ) = h ( y ) = 0, H ( P ) = h(P). Hence by Theorem 3.5 (used twice), h(8) I H ( 8 ) for all 8 E [ a ,y]. So, if h(8') < H(8') for some 8', then by Theorem 3.6, if 8' E ( a ,P), since h ( y ) = 0 and y - a = n l p ,

'

h(0') sin ~ ( -7P ) < H(0') sin P(-/ - P ) sin p(y - 8') sin p(y - 8') sin p(y - P ) cos p(8' = h(P), = h(P) sin p(y - 8') cos p(P -

h(P)

9)

y)

a contradiction. A similar argument holds for 8' E [ a ,71.

(P, y).

Hence h(8) = H ( 8 ) for all 8 E

Theorem 3.11. If f is an entire function of finite order p 2 112; h(8) is its Phragmdn-Lindelof indicator function, and h(8) is finite for all 8, then

rnp h(8) 2 - max h(8) . 0

3.3. The Phragme'n-Lindelof Indicator

99

Proof. Suppose h(8) assumes it minimum at 8 = O2 (h is continuous and bounded on [O,2x1 by Theorem 3.7, and so assumes its minimum somewhere in [O,2n]). By rotating the argument of f if necessary, we can assume without loss of generality that 82 = 0. Let a E (n/2p,n/p), and let a = maxe h(8). Then, in particular, h(8) 5 a, for 181 5 a and so taking 81 = -a, = a , 82 = 0 in Theorem 3.6, we have, h(-a) sin p a - h(0) sin 2pa h(a) sin p a 4 0, or since pa E (n/2, n), 2h(0) cos p a h(a) h(-a) 2a; but cospa is negative and so we have h(0) asecpa = -a1 secpal for all a E (n/2p,n/p). Letting a -+ we get the theorem.

+

<

+

<

>

y,

In a similar vein to the preceding theorems, but seemingly somewhat more specialized, we also have Theorem 3.12. Suppose in the right half-plane {z : R e z 4 0), f is analytic, If (reie)/ = O(ekT)there as r + m, where k < n. Suppose further f (n) = 0 for all non-negative integers n. Then f (z) 0. Then F is analytic in {z : Re z 4 0). Furthermore, Proof Let F ( z ) = &. is bounded on the semicircles C(0, n 112) n {z : Re z 0}, n = 0,1,2, . . . . Hence F ( z ) = O(eklZI)on these semicircles. This is also true on the imaginary axis and so, by the maximum modulus theorem if n - 112 = (n - 1) + 112 < r < n + 112 and -n/2 8 n/2, then IF(reie)l = ~ ( e ~ ( ~ +=l O(eh). / ~ ) ) Hence,

>

+

< <

IF(reie)l = O(eh) as r

+ m in { z : R e z 4 0 ) .

(3.8)

In fact, on the imaginary axis we have, for all sufficiently large r,

We now show that (3.8) and (3.9) (together with the condition k that F z 0. Let

< n)

imply

and consider the quadrant formed by the lines 8 = 0 and 8 = n/2. By (3.8), with z = reie, Ig(r)I = O(e-"ekT) = O(1) as r -+ m and by (3.9),

Hence, by (3.8) and Theorem 3.2 (since here a = 2 and ,Ll = I ) , there is a positive M such that 1g(z)1 M throughout the quadrant. But then by (3.10), IF(reie)l 5 erH(0)M, where H(8) = k cos 8 (k - n) sin 8. A similar argument can be applied to the quadrant formed by 8 = -n/2 and 8 = 0, except that here the function in the exponent is k cos 8 + (n - k) sin 8. Taking the two quadrants together, we find that there is a constant M > 0 such that

<

IF(reie)I

<

+

for - n/2

< 8 < n/2 ,

(3.11)

3. Entire Functions where H*(6') = kcos6' - (n - k)lsin81. Now let G(z) = et"F(z) where t > 0; then by (3.11),

The function of 6' in the exponent is 5 0 at four places in [-n/2, n/2], namely at f n/2 and at f arctan = f y , say. Hence, we can apply Theorem 3.2 once more, this time to the wedges [-in, y], [-y, y], [y, n/2] (since -n/2 < y < n/2), and get that in fact, ~ ~ ( r e ~M~ for ) l - 7r/2 6' 5 7r/2 .

<

<

But then by (3.12), for -n/2 < 6' < n/2, )F(reie)l 5 Me-tTC0Seand so, letting t + co, F G 0, and hence f (z) = F(z) sinnz G 0. Note 3.10. Theorem 3.12 is usually known as "Carlson's Theorem". The function f (z) = sinnz shows that n is best possible as a bound for k. The proof given has obvious affinity with the proof of Theorem 3.5. Since theorems on the PhragmknLindelof indicator remain true when adapted to functions analytic in an angle, an alternate proof can be given based on an appropriate version of Theorem 3.9. In fact, despite its specialized appearance, Carlson's Theorem is a result which is fundamental in the sense that it is a corollary of several different results in the theory of entire functions, and was the first example of such theorems. For theorems of this sort, see Chapters 9 and 10 of Boas, Entire Functions, where uniqueness and growth properties of an entire function f of growth (1, T),T < co, as determined by the behavior of f at specified sequences of points, are discussed. Some of the results obtainable can be quite surprising; for example, a famous theorem of Polya and Hardy which stimulated a great deal of further work is that if f is an entire function of growth (1, T),T < log 2, and f (n) is integral for n a non-negative integer, then f is a polynomial. (Here log 2 is clearly best possible as 2z shows.) The reader interested in this circle of ideas should consult, in addition to Boas' book, Volume 1 of Pdlya's Collected Papers [28]; on p.771-773 of this volume, Boas surveys the literature in this area. For some problems of this sort, the best possible value of the type is not yet known. Note 3.1 1. Some cautions are in order. (1) f can be entire, non-constant, and bounded along every ray through the origin. In fact, one can even require that f + 0 along each such ray. By Theorem 3.8, such a function clearly cannot be of finite order. An example (due to Malmquist , E(z) is defined as follows. and Mittag-Leffler) is the function ~ ( z ) e - ~ ( ' )where Let S be the open half-strip {z : Re z > 0, -7r < I m z < n ) and L its boundary,

&&

.c

where L is described clockwise. For a E G - (SU L) let E(z) = g d C . By considering successive half-strips {z : Re z > cr > 0, -n < I m z < n), E(z) can be analytically continued into the whole plane. This function is discussed in PblyaSzego (op. cit. I11 158-160 and IV 184. Indeed, even more can be shown (loc. cit. IV 185, 186). On the other hand, given an arbitrary subset of the rays through the

3.3. The Phragme'n-Lindelof Indicator

101

origin, it is not necessarily true that there is a non- constant entire function going t o 0 precisely along those rays and along no others as r + co (there are 2C such subsets and only C entire functions). If one only asks for boundedness along every ray, an example which can actually be presented in a first course a t just the right time "to confuse students who have just struggled to comprehend the meaning of ezttWtdtas discussed by D. J. Liouville's Theorem" is g(z in/2) where g(z) = Newman [174]. Further information on such functions can be found in two papers of Harald Bohr reprinted as numbers Ell and El2 of Volume I11 of his collected works [29]. The first of these originally appears in [31] and constructs such a function by Runge's method of "pole-pushing" . (2) There do exist entire functions of finite positive order whose PhragmknLindelof indicator function is identically constant. An easy example originates in the theory of elliptic functions (see Theorem 8.3.6). (3) Although an interval in which h(8) < 0 must be of length 5 nip, it is possible for h(8) to be positive (in an interval then of length 2 nip) and also zero in an interval of length nip. Furthermore, for any order p > 1, an entire function of that order can be found exhibiting this phenomenon. Such an example is given by "Mittag-Leffler's Function" :

Jr

+

>

which (by Stirling's Theorem) is of order l/a. A proof can be found in Cartwright's Integral Functions [42], pages 50-52.

Note 3.12. In our discussion of Phragm6n-Lindelof indicator functions h(B), the assumption h(8) < co was always necessary. If f is of finite positive order p

9

and infinite type, then Er,, = co at least for some 8. A Phragm6nLindelof indicator theory can also be constructed for such functions by replacing the "comparison function" r P by a more general "proximate order". The existence of proximate orders and this theory is discussed in detail in Cartwright, Integral Functions [42].

Note 3.13. For functions of growth (1, T), T < co, there is an extensive theory involving h(8) originating in a famous paper of P6lya [192]. Many of the results of this theory appear in Boas, Entire Functions [27] beginning in Chapter 5. Other aspects of it are discussed in Chapter VIII of Cartwright's Integral Functions. In this theory, the function f (z) = Cr=oanzn is associated with the function F ( z ) = CrZo*(= f (t)e-ztdt). f is of growth (1, T), T < co, if and only if the series defining F converges for some finite z. If D is the smallest closed convex set outside of which F is analytic, it turns out that h(-8) is the supporting function of D. This is the start of a fruitful theory. For functions of finite order p, p > 1, A.J. Macintyre [151] replaces F by the function F ( z ) = Cr=oanI' I-"-', and shows that a similar theory can be developed in some aspects in this case (here

Jr

(y)

3. Entire Functions

102

the role of F is played by .F(zllp)). Macintyre also finds it useful to consider the function

Note 3.14. The reader who wishes to penetrate the basic theory of entire functions (of finite order) more deeply in the vein of this chapter, should examine the books by Boas and Cartwright, already cited several times. These are largely complementary, and Boas concentrates attention on functions of growth (1, T), T < 00. One can hardly fail also t o recommend the relevant sections of the book by Pdlya and Szego, some of which have already been mentioned.

3.4

Composition of entire functions

While the order and type of the sum or product of two entire functions are reasonably controlled by the functions involved, this is no longer true of composition of functions. In fact, the usual case is that exemplified by eez, namely the order of an entire function which is the composition of two transcendental functions is usually infinite. This is made precise by the following Theorem of Pdlya (Journal London Math. Soc. 1, 1926, 12-15).

Theorem 4.1 (P6lya). If g(z) and h(z) are entire functions, then g(h(z)) can be of finite order only if either (a) h is a polynomial and g of finite order or

(b) h is of finite order, not a polynomial, and g is of zero order. Theorem 4.1 can be proved in a t least two ways. Pdlya's original proof is connected with some of the ideas of Chapter 2, and is presented here. We will need t o use the following result first proved by Harald Bohr.

Theorem 4.2 (Bohr). Let p, 0 < p < 1, be given. Suppose f is analytic on B(0, I), f (0) = 0, and maxl,l,p If (z)l = 1. Let r = rf be the radius of the largest circle, all of whose points are taken on by f in B ( 0 , l ) . Then for all such f , rf > C > 0 where C = C(p) is a constant depending only on p. Proof of Theorem 4.2. Suppose there is a function g satisfying the hypotheses of the theorem such that for all r 2 A > 0, there is a point C, with I<,[ = r such that g(z) # $ for all z E B(0,l). In particular then, there are points, say, a = Aeis, Then h is and, /3 = 2Aei4 such that g(z) # a , # /3 in B(0,l). Let h(z) = analytic for lzl 5 1, h(z) # 0, # 1 in B(0, l ) , and

e.

9.4. Composition of entire functions

103

So taking w = 1 in Theorem 2.2.5, we get that for all z E B(O,p), Ih(z)l 5 +(I, p). IP-al+(l,p), and so Ig(z)l A+3A$(l,p) for all z E B(0,p). That is Ig(z)-a1 But by hypothesis, maxl,l,, Ig(z)l = 1, and so A 2 ,+,&,,,), and taking C less than this constant, we get the theorem.

<

<

Proof of Theorem 4.1. We may suppose, with no loss of generality, that h(0) = 0. (If h(0) # 0, consider instead the functions hl (z) = h(z) - h(0) and gl (w) = g(w h(0)). Let f (z) = g(h(z)) and let F ( r ) = maxl,l=, (z)l, G(r) = maxl,l=, H ( r ) = maxl,l,, Ih(z)l. In order to use Theorem 4.2, consider the function k(z) = for any r > 0. Then maxl,l,l/z Ik(z)l = 1, k(0) = 0, and k is analytic on H(f-12)'

+

If

*

B ( 0 , l ) . (Any fixed positive number < 1 will do as well as 112). By Theorem 4.2, there is a positive constant C and an R > CH(rI2) such that h(rz) takes on every value w with Iwl = R for a E B(0,l). Hence h(z) does so in B(0, r ) . By the maximum modulus theorem, there exists at least one point on C(0, R) at which g(z) takes its maximum in B(0, R). Call such a point wo. By our choice of R, then there is a zo in B(0,r) such that h(zo) = wo. Hence, by the maximum modulus theorem,

If f is of finite order, then F ( r ) < Kern for some a < cm. Let a, be any non-zero coefficient of the power series for h(z) (note that m 2 1, since h(0) = 0). Then H (g), and so, by the maximum modulus by Cauchy's inequality, larn[($), theorem once more, if f is of finite order, then

<

G(cIamI

(f),)

5 G(CH

(i))5

F(r)

< ~ e ' . for all r > 0 .

Replacing r by rllm we see that the order of g(z) is 5 a l m . Hence, if h(z) is not a polynomial, we may let m + cm and the order of g(z) is 0. Similarly, if f has finite order and g(z) = Cr=,bnzn, then G(r) Ibnlrn for all n 2 1, and so Ibnl(CH(r/2))n G(CH(rI2)) < Kerm;whence the order of h is 5 a < co.

<

>

Note 4.1. Theorem 4.2 was proved by Bohr [31] for application to some problems in several complex variables, where it seemed easier to use than Landau's Theorem 2.2.3. The best value of C was determined by Hayman [103] to be for p = 112.

i,

Note 4.2. Actually, in Theorem 4.1 we have proved that (writing M (r, h) for the maximum modulus of the entire function h on B(0, r)), that if g and h are entire transcendental functions, then M ( r , g o h) 2 M(AM(r/2), h), g). However, from the statement of Theorem 4.2 and its use in proving Theorem 4.1, it is clear, as P6lya remarks, that for any p, 0 < p < 1, there is a C(p) such that

3. Entire Functions

104

v.

Using the result of Hayman mentioned in Note 4.1, the above proof can be elaboHowever, there is also an entirely different rated t o yield (4.1) with C(p) = proof (not using Theorem 4.2) of (4.1) with this value of C(p) and so also of Theorem 4.1, by Clunie [46]. Clunie also shows that C(p) = is best possible, and considers extensions of P6lya's result to meromorphic functions (where the maximum modulus function is replaced by the Nevanlinna characteristic-See Chapter 4). He also considers various variants comparing the growth of g o h with that of g or h.

Note 4.3. The case (b) of P6lya's Theorem 4.1 can actually occur. Let g(z) = CEO2-k2zk and h(z) = ez. Then

The order of f can now be found from Theorem 2.1. By partial summation (for r > 0)

Hence, as r

+ m, letting t = ( & ) ' I 2 ,

a, = (1

+ o(1) (i--2r +logl )2! Jm

2-t2t'+2dt

+C)

It now follows from Stirling's formula and Theorem 2.1 that the order of f is 2. In fact, using Theorem 2.2 with p = 2 and Stirling's formula, we find that v = whence the type of f is &. Clearly there are many similar examples.

Example 4.1. As an example of the use of Theorem 4.1, we prove the following theorem by Thron [228]. Theorem 4.3. Suppose f(f (z)) = g(z) where g is entire of finite order, not a polynomial, and takes some value w only finitely often. Then f is not entire.

Proof. By Theorem 4.1, if f were entire, f must be of order 0 and not a polynomial (since g is not a polynomial). Consider the set of points {z,) at which f (2) = w, ~ the ) set of points a t which f takes on the value z,. and for each m, let { z ~ , be Then g(zk,,) = f (f (zk,,)) = f (zm) = w. Hence, by hypothesis, there are only finitely many distinct points among the {zk,,). Hence each point {t,) is taken on

3.4. Composition of entire functions

105

only finitely often by f . Hence by Picard's Theorem 2.2.6, (compare Note 2.2.10) the set of points {z,) is either empty or contains one point, say zo. In this latter ( z ) , h is entire of order 0, and never 0, whence case, f (z) - w = (z - ~ ~ ) ~ h where by the Hadamard Product Theorem 1.3, h is constant and so f a polynomial, contradiction. Similarly, if the set {z,) is empty, f is constant, again a contradiction.

Note 4.4. A particular case of Example 4.1 is, of course, g(z) = eZ. On the other hand, there is a real-analytic function f such that f (f (x)) = ex. The construction of such a function is difficult, but was demonstrated by Hellmuth Kneser [128]. Thron's observations show that Kneser's f cannot be entire. Example 4.2. Nathanson [I661 considered "multiplication rules for polynomials" and asked for solutions of

The proof of Pblya's Theorem 4.1 shows immediately that there are no entire solutions of (*) other than quadratic polynomials z2 bz c. For suppose F is entire, transcendental, and satisfies (*). Let G(z) = z F(z), and by the proof of Theorem 4.1, writing M ( r , f ) for maxl,l,, If ( z )1, we have that there is a constant C > 0 such that for r > ro,

+ +

Log M ( r , F o G) But for r

> rl

>

+

Log M ( C M ( r I 2 , G), F ) .

(4.2)

> r + 1. Since by

the Hadamard Three Circles

say. C M ( r I 2 , G)

F.

is an increasing function of r , we also have, Theorem (see Appendix), og for r > r9. -, Log M ( C M ( r / 2 , G ) , F ) > Log M ( r + l , F ) . Log (CM(rI2, GI) Log ( r 1) ' and so,

+

Log M ( C M ( r I 2 , G), F ) Log M ( r 1, F )

+

>

Log C Log (r 1)

+ +

Log (r12) Log M(r12, G) Log ( r 1) Log (r/2)

+

But by Cauchy's Inequality, since G is entire and transcendental, m as r -+ m; hence comparing with (4.2), we get LO M ( r FOG)

'

-+

og (r/2) -+ m as r -+ m. If (*) holds, then for all sufficiently large r, Log M ( r , F o G) 5 Log M(r, F ) + Log M ( r 1, F ) and so (by the maximum modulus theorem), LO M ( r F o G ) 5 2, which is a contradiction. Hence F must be polynomial, whence by (*) (if it is non-linear) its degree d must satisfy d2 = 2d. Hence d = 2 or 1. If d = 2, one immediately sees from (*) that F must be monic. That z2 ba c = F ( z ) satisfies (*) for all b and c is an easy verification. That there are no linear polynomial

+

+ +

106

3. Entire Functions

solutions (other than the constant solutions F 0 and F 1) is also easily verified. One small point, perhaps worth noting, is that it is necessary to use M ( r 1, F) instead of M ( r , F ) , since it is not true, in general, that is bounded as r + oo (consider F ( z ) = exp(exp(exp 2)))).

+

Note 4.5. There has been some interest in the question of what meromorphic functions can be represented in the form f o g(z) where g is entire and f meromorphic, with neither f nor g linear. The reader interested in questions of this sort, should consult the book by Fred Gross, Factorization of Meromorphic Functions [91].

Chapter 4

Introduction to Merornorphic Functions Rolf Nevanlinna's theory of meromorphic functions which dates t o 1924 has been called by Walter Hayman, the most important occurrence in function theory during the twentieth century. It can be viewed as an extension to meromorphic functions of the sort of theory discussed in the preceding chapter for entire functions, where the logarithm of the maximum modulus, log M ( r , f ) is replaced by the Nevanlinna characteristic T ( r ,f ) . However, Nevanlinna's theory when applied t o entire functions does not necessarily reduce to the previous theory (as a simple example, the The Nevanlinna theory of meromorphic functions "Nevanlinna type" of eZ is represents a profound deepening of ideas associated with Picard's theorem, the concept of "deficient" for f if the equation f (z) = a has "relatively few" (though perhaps infinitely many) solutions. It turns out that the number of deficient values is always countable. The theory has a number of striking consequences. For example:

i).

If fi (z) and f2(z) are meromorphic in the plane; let El (a) = {z : f l (2) = a} and E2(a) = {z : fi(z) = a}. Then, if for five different values of a , El (a) = E2(a), either f i (z) fi(z) or both are constant. (Theorem 3.3)

=

The derivative of a meromorphic function assumes all finite values except a t most one. (Theorem 3.4) These and several other similarly striking results appear below. Throughout this chapter, we shall occasionally have use for the notion of Lebesgue measure.

4.

108

Meromorphic Functions

Nevanlinna himself gave expositions of the theory in Le Thkoreme de PicardBore1 et la ThQorie des Fonctions MQromorphes (Paris 1929) and Eindeutige Analytische Funktionen (Springer, Berlin, 1936). The second edition of the latter has been translated as Analytic Functions (Springer 1970). Another excellent source for the theory is Hayman, Meromorphic Functions [96]. This chapter is indebted to Hayman's book. Nevanlinna's "Second Fundamental Theorem" involves deficient values; his "First Fundamental theorem" is essentially a rewriting of the Poisson-Jensen formula, and it is there we begin.

4.1

Nevanlinna's Characteristic and its Elementary Properties

Poisson's formula for the real part of a function analytic in a disk and Jensen's formula (Theorem 3.1.1) can be combined and extended to meromorphic functions:

Theorem 1.1 (The Poisson-Jensen Formula). Suppose f (z) is meromorphic in B(0, R ) and analytic on C(0, R); that a,,p = 1'2,. . . , m are the zeros of f and b,, u = 1'2,. . . ,n are the poles of f in B(0, R). If for an r, 0 5 r < R, f (reis) # 0, # co,then

l

1 Log ~f(reis)[ = 271

2"

Log

If

R2 - r2 (Reim)lR2 - 2Rr cos(8 - ))

+ r2 d6 ,-

- \

Proof. (i) Since Re Log f (z) = Log If (2)1, if f has no zeros or poles, applying Poisson's formula (see Appendix for a proof) to Log f (z), we have

(ii) Consider the case f (z) = z - a, la1 < R. Then it is necessary to show that 1 Log [reie - a1 = 271

1

2K

Log

lReim

-

R2 - r2 R2 - 2Rr cos(8 - ))

+ r2 ddJ

4.1. Nevanlinna's Characteristic and its Elementary Properties

109

= I R ~ ~ @ ' - al, this is just Poisson's formula for the function But since IR - i%ei@'l

(iii)

Similarly, if f (2) =

5 ,one verifies that the formula holds for

I=-=1

Log

~oglre"-bl.

(iv) Multiplying f by a finite number of factors to cancel the zeros and poles and using (i), (ii), (iii), the theorem follows. The reader should note that if r = 0 (and there are no poles), then (1.1) reduces to Jensen's Theorem (Theorem 3.1.1).

Note 1.1. If the meromorphic function f has a zero of order k, say, a t 0, then is analytic and non-zero a t 0 and has the same zeros and poles a t 0 as f and the same modulus on C(0, R). Hence, if c = lim,,o Theorem 3.1.1 yields

y,

Log Icl = -

in

ln

Log

If

(rei@)ldq5

+

. . - k log R .

The case of a pole a t 0 can be treated similarly. Such modifications can always be made when necessary; explicitly recognizing them becomes a bit tiresome. Hence, it will always be assumed that the formula (1.1) makes sense, (i.e. f (0) # 0, # co) knowing that these exceptional cases, if they occur, can always be treated in a trivial manner. To rewrite (1.1) following Nevanlinna, we need some definitions.

Definition 1.1. For x real I n other words,

> 0,

Log + x =

Log +x = max( Log x, 0)

f oorrxO" < x < l .

I t is worth noting that Log x = Log +x - Log +$, and so

and this prompts

4. Meromorphic Functions

110 def. Definition 1.2. m ( R ,f ) =

& soZpLog +( f ( ~ e ~ @ ) .) j d $

Let r l , . . . , rN be the moduli of the poles b l , . . . ,b~ of f in B(0,R) arranged in non-decreasing order. We make

Definition 1.3. I f f is meromorphic in B(0,R) for 0 < t the number of poles of f in B(0,t ) . Then N

121= x N

Log v=l

R Log - =

v=l

n(t,f

Tv

Jd

< R, define n ( t ,f )

to be

Log (R/t)dn(t, f) =

) ~ o (gR I ;~1 )+

J

n(t,f

)

(1.2)

0

Equation (1.2)prompts

Definition 1.4. I f f is meromorphic in B(0,R ) ,

In a manner analogous to (1.2),we find that if al,. . . ,aM are the zeros of f in B(0,R), then they are poles of and so

5

(

1)

N R , -

M

-

.I:/

= ~ ~ ~ ( ~ ; ' l " Log d t = x p=l

Thus, the Generalized Jensen Formula (the case r = 0 of (1.1)) becomes in this notation:

Theorem 1.1 (a). If f is meromorphic in B(0,R ) , analytic on C(0,R ) , and f (0)# 0, # 00, then

Definition 1.5. The Nevanlinna characteristic T ( R ,f ) of a meromorphic function f is defined b y T ( R ,f )

def

m ( R ,f )

+ N ( R ,f ) .

Thus the Generalized Jensen Formula, Theorem 1.1 .(a), becomes

4.1. Nevanlinnals Characteristic and its Elementary Properties Theorem 1.1 (b). T ( R ,f ) = T ( R ,j)

111

+ Log I f (0)I .

Theorem 1.2 (Nevanlinna's First Fundamental Theorem). If f is meromorphic in B ( 0 , R ) , analytic on C(0,R ) , then for each complex number a ,

+

and, in fact, the O(1) can be replaced by - log If ( 0 )- a1 E ( a ,R ) , where E ( a ,R ) log+ la1 log 2.

+

<

Proof. Trivially, for any positive integer p, and a , any complex numbers,

Hence

<

Also, clearly, since the order of a pole of C;=, f v ( z ) at a point a0 is the sum of the orders of the poles of the f , at 20, N ( R , C;=, f v ( z ) C;=l N ( R , f v ( z ) ) . Hence

<

Taking p = 2, fi ( z ) = f ( z ) ,f 2 ( z ) = a , a complex constant, gives

T ( R ,f

+ a ) < T ( R ,f ) + log'

la1 + log2 (since N ( R ,f 2 ) = 0 ) .

Replacing f ( z ) by f ( z ) - a in (1.4) gives

while replacing a by -a in (1.4) gives

T ( R ,f - a ) By Theorem l.l.(b),

and so the theorem follows.

< T ( R ,f ) + log'

la1

+ log2 .

4.

112

Meromorphic Functions

Note 1.2. Products may be treated similarly to sums, viz: For p a positive integer

whence m (R,

nE=,fv(a)) <

m(R, fv(a));while once again clearly

and so

Notational Convention: Often one is dealing with only one function f at a time. In this circumstance, it is usual to write T(R) for T(R, f ) and, for finite N (R, n (R, ; and a , m(R, a ) , N(R, a), n(R, a) instead of m (R, m(R, oo), N(R, oo), n(R, oo) instead of m(R, f), N(R, f ) , n(R, f). (Thus, for entire functions f , n(R, 0) has the same meaning as n(R) in Chapter 3). This notation will be used for the rest of this chapter whenever only one function is being dealt with. If it proves necessary to distinguish between several functions under consideration at the same time, we will write T(r, f),m(r, a, f ) etc. In this notation, Theorem 1.2 becomes

&),

&)

&),

Theorem 1.2 (a). For any two values of a, the sum m(R, a) + N(R,a) is the same or differs by at most a bounded amount as R + oo; viz: m(R, a) N(R, a ) = T(R) O(1).

+

+

)

Note 1.3, n ( R , a ) isan "averagesize" of Log ( I f ( n e f o ) - a l on arcs where 1 f ( ~ e " a1 is small, whereas N(R, a) is a sort of "logarithmic average" of the number of zeros of f (a) = a in B(0,R). Example 1.1. Suppose f is a rational function, say

There are several cases: (i) p > q. Then for a finite,

4.1. Nevanlinna7s Characteristic and its Elementary Properties

113

But, since p > q, lim,,, f (reie) = co and so, for T sufficiently large, say r 2 TO, If (reie) - a1 > 1 and so m(r, a) = 0. On the other hand, if a # co, f (z) = a has exactly p roots and so n(t, a ) = p if t > to say. Hence

+

and so by Theorem 1.2 (a), T(T) = p log r O(1) as r + co. (ii) If p < q then, if a # 0, similarly m(r, a) = O(1) and N(r, a ) = qlogr+O(l), whence T ( r ) = q log r 0(1), as r -+ co. (iii) If p = q, and a # Ic, then similarly again m(r,a) = O(1) and N ( r , a ) = p l o g r 0 ( 1 ) , and T ( r ) = plogr O(1) as r + co.

+

+

+

Example 1.2. f (z) = eZ. Writing z = reie = r cos 8 + i r sin 8,

(In the interval, ( n / 2 , 3 ~ / 2 ) ,cos8 < 0, and so eTCoSe < 1). Clearly n(r, co) = 0 and so N(T,co) = 0, and def T(T) = ' m ( ~ , c o ) N ( ~ , c o = ) TIT.

+

(1.5)

+

2+

It follows from Theorem l.l.(b) that for f ( z ) = eZ,m(r, a ) N(r, a) = O(1) for all a E C, and there is some interest in investigating m and N separately. If a = 0 this is easy, for n(r, 0) = 0, so N ( R , 0) = 0; while

+

If a # 0, then one easily sees similarly that n(t,a) = f O(1); whence N(r,a) = O(logr), and so (by Theorem 1.2(a) and (1.5)), m(r,a) = O(1ogr) as r + co; however, with care, one can get better results which illustrate the sort of computations which may be made in a specific case. The concept of Lebesgue measure is briefly needed. Suppose a # 0, # co.Then

2+

4.

114

Meromorphic Functions

<

Now, since a # 0, # co, if T is sufficiently large, e-T la\ 5 er, or, in other words, 1. Set l o r a = cosa where 0 5 a 5 7r; then

I+1

<

:I"

4 7 , a) = -

log+

I,T

1 cos e - er cos a

l

dB

1

1

(1.6) dB.

- 11 > 1/21) and El = [0, 7r] - E . If B E El, then Let E = {B E [O, 7 r ] : leT(COSe-COSa) 112 < er(cose-cOsa)- 1 < 312 and we h&e two subcases: 1, then cos B - cos a 0 and so (i) If er(cose-cosa)

>

ler(cos 0-cos a) -

>

T

1 n= 1

(COSB - cos a ) n!

2 T(COS 0 - COS (2) ;

If 112 < er(cose-cosa) < 1, then (cos B - cos a) < 0, and 0 < ~ ( c oas log2 < 1, and so l e ~ ( c O ~ e - c " ~ " ) - 11 = 1 - e ~ ( ~ ~ s e - ~ ( C O 8-cos S a)(er(c~s a-cos 8) - 1) > - T (COS a - cos B)er(cos8-C0S a) > $r(cosa-COSO). From (i) and (ii), for B E E',

(ii) COSe) <

Since, by definition T cos a = log la/, (1.6) becomes (where "meas" indicates Lebesgue measure) m(r, a )

(h)+ fS,+: S,, (h)+ 7 fS,, ( 1 case )

5 log+ +

5 1%'

- cosal

log+

log 2

meas (E) (1.7)

dB.

It is necessary to examine the integral in (1.7) when 8 is in the vicinity of a . Hence, given E , 0 < E < min(a, T - a ) for a # 0, # co, and T > 1 m ( ~a,)

< log+

(+i) + :/

+ Jff+' logf a-C

log 2

7r

[o,a]-[a-c,~+c]

cos 8 - cos a1

cos B - cos a1

)d8

~ ~ ~ a )

4.1.

Nevanlinna's Characteristic and its Elementary Properties

115

Now cos 8 is continuous and so for E sufficiently small, using the definition of log+, /a+e a-e

log+

(I

2 cos 8 - cos a1

)

=

/_: log ( 1 cos(8 + a2)

= O(1)

(in fact

- cosal

+ 0 as E + 0 ) .

Hence, if f (z) = ez also, for a # 0, # co, m(r,a) = O(1) as r Theorem 1.2(a), and (1.5) N(r, a) = r l x O(1) as r + cm.

+

+ co and

so by

Note 1.4. It is worth noting that for entire functions, even in such a simple case as f (2) = eZ, the ratio of T(r) to log M(r) is not necessarily 1 as r + co (being in this case It is also worth observing that for f (2) = eZ the weaker result N(r, a ) = f + O(1ogr) is easily computed directly, while the stronger result above most easily comes from consideration of m(r,a) and using Theorem 1.2(a).

a).

$&

Example 1.3. Suppose g(r) = where a , b, c, d are complex constants, ad bc # 0, f (z) is meromorphic in B(0, R), f (0) # cm, g(0) # cm. Suppose c # 0, then, as r + m, T(r, f dlc) = T(r, f ) O(1) (since by Theorem 1.2.(a), m(r, dlc) N(r, dlc) = m(r, 0) N(r, 0) 0(1)), and T ( r , cf + d) = T(r,c(f = T(r, f d/c) O(1) = T(r, f ) O(1); also T (r, = T(r,cf d) + O(1) = T(r, f ) O(1) by Theorem l.l(b); hence by Theorem 1.2(a) again T r, cf + ) = T r, c cf +dl 9 ) = T (r, + 0(1) = T(r, f ) + 0(1). Hence T (r, g) = T(r, f ) O(1). If c = 0, even fewer steps are needed. Actually, T ( r ) has an interesting and useful expression as an average of N(r, a ) over a E C(0,p).

+

+

+ 4))

( ~

(

+

+

+ +

+

+

+

+

+

+

&)

-)

Theorem 1.3. I f f is meromorphic in B(0, R), then for 0

< r < R,

Proof. By Theorem 1.1, if g is meromorphic in B(0, R)

where a,, p = 1,.. . ,m are the zeros of g and b,, v = 1,.. . ,n are the poles of g. Taking g(z) = f (z) - peis for each 8 E [O,27r] we get

4.

116

Meromorphic Functions

(compare discussion preceding Theorem l.l(a) and the notational convention following Note 1.2). Integrating both sides from 0 to 27r, we have

kJd2"

Log

If

(0) - peie[dO=

1 1

Jd2" JdZrr dO&

Log ) f ( ~ e ' @-) peield4 (1.8)

r2rr

+

If

Now, Log (Rei@)- peie[ 5 Log (If (Rei@[ p) which is > 0, integrable in [O, 2 ~ ] and independent of 8, hence the double integral is absolutely convergent and we can interchange the order of integration on the right in (1.8). So it becomes necessary to evaluate & J ~ Log ~ " la - peield6 for a = f (0) (for the left side of (13 ) ) and for a = f (Rei@)(for the right side of (1.8)).

& s:"

Log la - peie\dO = Log p

I

1:

+ & J:"

Log - - eie dO and applying Theorem 3.1.1 (Jensen's Formula), to the function g(z) = - z, z = reie, we have But

where the a, are the zeros of

lLrr 2~

(if a

# 0), or

- z in B(0, r).

Taking r = 1 we get,

~ o -ge i e l d O = l ~ g I - { o ' log

%

,

if % > I if % <1

(by continuity)

and this clearly holds for a = 0 as well. (Alternatively one can use the argument in the proof of Jensen's Theorem.) Hence (1.8) becomes

Since the poles of f are the same as those of f /p, the result now follows from the definitions of m and T.

4.1. Neuanlinna's Characteristic and its Elementary Properties

117

Note 1.5. Theorem 1.3 in the case of p = 1 has an interesting geometric interpretation. Writing out the definition of N(r, eie) and interchanging the order of integration, we have

The inner integral represents the total length, say L(t) of arcs of C ( 0 , l ) covered by values of f for z E B(0, t), each arc being counted as often as it is covered. And, 2K J2" o n(t, eie)d6 = engt o c(o,l)= "Average length" of such arcs. Hence

'

d Average length of arcs of C ( 0 , l ) T-T(T) = dr covered by values of f for z E B(0, r) A similar argument leads to Theorem 1.4. For a given meromorphic function f , T ( r ) is an increasing convex function of log r . Proof. Take p = 1 in Theorem 1.3. Since N(r,eie) is an increasing function of r, so is T ( r ) by Theorem 1.3. Also by Theorem 1.3, and the definition of N ,

and since

& J":

n(ex,eie)d9 is an increasing function of x, the theorem follows.

Note 1.6. By the Hadamard Three-Circles Theorem (see Appendix) for f entire, log M ( r ) is an increasing convex function of log r . This is one of many respects in which for entire functions T ( r ) and log M ( r ) have similar behavior, even though they are different (see also Theorem 1.7 below). Nevanlinna's original proof of Theorem 1.4 uses the Poisson-Jensen formula for an annulus instead of Theorem 1.3 (see Le ThkorBme de Picard-Bore1 et la ThQorie des Fonctions MQromorphes, 1928, p.9). Note 1.7. Clearly N(r, a ) def. = SoT yn t,ad t is also an increasing convex function of r (since n(t, a) is non-decreasing). However, m(r, a ) need be neither increasing nor convex. For z E B(O,1/2), by the maximum moduFor example, consider f (z) =

a.

4.

118

Meromorphic Functions

> 2. Hence

Since f (112) = -f (z), the same inequality is also true for lzl

But f (1) = f (-1) = oo and log' lzl is a positive continuous function of z, hence m(1,oo) must be > 0. In general, as Example 1.2 illustrates, N(r, a) is usually the "primary contributor" to T ( r ) and m(r, a) relatively small. That this is true "on the average" for a E C(0, p), is a direct consequence of Theorem 1.3 as shown by

Theorem 1.5. For a given meromorphic function f ,

+

+

Proof. By Theorem 1.2, T(r) = m(r, peie) N(r, peie) +log+ If (0) -peieI E(p, O), where IE(p, 8)1 5 log' lpeiel log2 = log+ Ip] log 2. So, arguing as in the proof of Theorem 1.2 (with T ( r ) = T(r, f ) ) ,

+

+

If (0)l + log p + + log+ P in m(r, peie)dO

+ T(r, f lp) + logp +

la

E(O)dO

&1

2a

E(O)dO ,

by Theorem 1.3.

By Example 1.3, T(r, f lp) = T(r, f )

+ 0(1), and so we get the theorem.

Note 1.8. A more careful analysis of the bounded error term in Theorem 1.5 shows that in fact

kiff

m(r, peis)dO 5 I log lpll

+ log2 .

Note 1.9. Theorem 1.3 is a result of Henri Cartan. Much more is known about the "small" size of the set of a on which m(r, a) is "large". In fact, Ahlfors in a + €all) well-known paper [3] has shown that given E > 0, m(r,a) = ~ ( ~ ( r ) l / ~for a except for a E E, where the possible exceptional set E has zero capacity. The notion of capacity originates in potential theory; however, Szego showed that it is equivalent to a concept originating with Fekete called the transfinite diameter of a set, which may be defined as follows. Given a non-empty set S in the plane and a point (' E S , let 21,. . . , z, be arbitrary points in S, and dn=

min m a x I I ~ ~ - z j l fa,n d d = lim 6, n-+ca ...,z,ES CES j=l

%I,

4.1. Nevanlinna's Characteristic and its Elementary Properties

119

Then 6 is called the transfinite diameter of the set S. (Fekete's original definition is different from the one above, but he shows that both definitions are equivalent). The concept of capacity (or transfinite diameter) plays an important role in various areas of analysis. Fekete's paper [71] is Math. Zeit 17 (1923, 228-249); Szego's [225] in the same journal (1924), 203-208. In 1931, P6lya and Szego [201] published a comprehensive treatment of the relationship between potential-theoretic ideas and Fekete's in two and three dimensions (where Fekete's definitions and potentialtheoretic capacity again turn out to be equivalent). If f is entire, then T ( r ) and log M ( r ) are both logarithmically convex increasing measures of the growth of f , and it is worthwhile to consider their relationships more closely.

Definition 1.6. The Nevanlinna order of a meromorphic function f (z) is

KG

W = logr

k

,

O
Definition 1.7. The Nevanlinna type of a function of finite positive Nevanlinna order k is T(r) = v lim r-iw

rk

Theorem 1.6. Suppose f is analytic in B(0, R) and, as usual, M ( r ) = maxl,l,, Then, for 0 r < R,

<

If (z)l

Proof. Since f is analytic in B(0, R), N(R,cm) = 0, and so T ( r ) = m(r, cm) = T ;

lo-.'

Log

+If

(reie)ldO

<

Log + ~ ( r . )

On the other hand, for M ( r ) > 1, by the maximum modulus theorem, there is a xo = reie, such that If (zo)l = M(r). Since there are no poles in B(0, R), Theorem 1.1 gives for M ( r ) > 1, Log M ( r ) = Log +M(r) = Log

If

(reieO)l

Theorem 1.7. If f is an entire function, then its order equals its Nevanlinna order, and if it is of type 0 or cm, its type equals its Nevanlinna type; otherwise, both type and Nevanlinna type are positive and finite, but not necessarily equal.

4.

Meromorphic Functions

Proof. Take R = 2r in Theorem 1.6. Then T (r) 5 Log +M(r) 5 3T(2r) .

(1.10)

Hence

- logT(r) < lim l0gr - T+CC

T+oO

Log ( Log +M(r)) log 7-

T+03

logT(2r) log2r 1 0 g 2 ~ 1 0 g ~'

and so the Nevanlinna order k = the order p of f . Furthermore, if 0 < k = p < oo, from (1.10) again,

and, so if v is the Nevanlinna type o f f and r its type,

v = 0 if and only if r = 0 v = oo if and only if r = oo 0 < v < oo if and only if 0 < r < co

.

a,

Note 1.10. As Example 1.2 shows, for f(z) = ez, the Nevanlinna type v = while r = 1. However, the above proof shows that always v 5 r. In the other direction; taking R = (1 r in Theorem 1.6 where k is the Nevanlinna order of f , then for 0 5 r ,

+i)

+

and so, dividing by rk and letting r + oo, r < (2k 1)ev. Actually, one can do better. If k 2 112, then there is an entire function such def. zn that r = nkv, namely Mittag-Leffler's function Ellk(z) = for

2,

xp=om,

while log M ( r ) rk (see Cartwright, Integral Functions[42] op. which T(r) cit. p.50). One may note that El(z) = eZ,and Ez(z) = c o s h ( 4 ) . This shows that one cannot do better, in general, than r 5 nkv, and this bound is, in fact, true having been first proved by N. V. Govorov [87]. It follows from a result of Wahlund [241] that if k is the order o f f and 0 < k < 112, then r 5 &v. This result was also obtained independently by Valiron [233]. See also section 26 of Valiron [236], N

4. I .

Nevanlinna 's Characteristic and its Elementary Properties

121

Fonctions Entibres. The function f (z) = n F = l ( l + ~ n - l / ~ )0,< k < 112, has order k, and T = &, v = showing that this result is also best possible. These results were refined by V. Petrenko [I881 who replaced order by lower order in the theorems, and studied more closely the relationship for a given complex and T ( r ) ; see also note 3.14. number a between logt maxlZlzT

i,

(

A) If

(z)-al

< m , and of finite positive Nevanlinna & > 0. However, if p = m , then it is

Example 1.4. If f is entire of order p, 0 < p type, Theorem 1.7 implies that G,,,

-

%

possible that lim,,, = 0. Consider f (z) = eez. Note first that if a # 0, # m, and g(z) = e Z , then O(1) (since eZ = a if and only if z = Log a 2kin, k an integer). n(r, a, g) = We now compute n(r, a, f ) , for a # 0, # m . We have eez = a if and only if

+

eZ = Log a

+

+ 2kin ,k an integer.

(1.11)

(Log, as usual, indicates the principal branch of the logarithm.) For each integer k, (1.11) has r / n + O ( l ) solutions in B(0,r). Hence, from (1.11) we get that

and so

To compute the integral in (1.12), we have

Letting 0 = arcsin fi in the last integral we get

To evaluate this last integral, we have

4.

122

Meromorphic Functions

For the first integral on the right in (1.14), we have

and

as r -+m , since the last integral converges to r(3/2) as r + m . Also, similarly e-2Tuu-112du = 1 e-u~-112du = ($)lI2 (1 + 0(1)),

~d~~

since have

, Ji = (2~)

e - " z ~ - ~ / ~ d= u I'(1/2) =

fi. Hence for the first

integral in (1.14), we

For the second integral in (1.14) we have

Substituting (1.16) and (1.17) in (1.14), then (1.14) in (1.13) and (1.13) in (1.12), we get that for f ( r ) = e e z , and a # 0,# m , n ( r , a ) = & (9)'l2 2eT(1 o(1))

-

+ O(T)

+

eT (&)'I2

as T

7 m.

Hence dt-

eT (2$.)1/2 as

-

+

and so by Theorem 1.3, 2x

~ ( reis)d8 ,

+ Log +If

(0)l

N

e as T (2r3r)'I2

-+ m

On the other hand, clearly if f (2) = eeZ,log M ( r ) = eT;hence, for this function, a + ~ a s r + m . Example 1.4 raises the question for an entire function of infinite order of how much larger than T ( r ) can log M ( r ) be.

4.1. Nevanlinna 's Characteristic and its Elementary Properties In this direction, we have

Theorem 1.8. Suppose f is a non-constant entire function. Then for every e > 0, lirn r+m

log M T T ( r ) ( l ~ g A ! ) ) l += ~

O.

Proof. Since f is non-constant and entire, log M ( r ) -+ oo as r -+ oo (by Liouville's Theorem) and so T(r) + oo (by Theorem 1.6 with R = 2r). The idea of the proof is to put R = rg(r) in Theorem 1.6 where g(r) > 1 with g(r) + 1 as r + oo is chosen suitably so that T(rg(r)) < (1 e)T(r). In order to do this, we need first the following technical:

+

>

Claim: If lc is a real-valued function positive for x xo, and bounded in every finite interval (to the right of xo) but unbounded as x -+ oo, then given E > 0, there is a sequence x, such that k(x) < (1 ~ ) k ( x n )for all x in the interval (x,, xny(x,)e*) where y(x,) = max(1, erp((1og L(x,))-~-')).

+

Proof of claim: Suppose not. Then there is an E > 0 such that for all sufficiently large x, there is an J in the interval (x,xy(x) exp&) such that k(E) (1 + e)k(x). Suppose this is true for x 2 XI. We now define a sequence by induction: J1 = X I , and supposing has already been defined, define such that E [En, Jny(En) exp*] and k(S,+1) (1 + e)k(E,). Since k(x) is unbounded, we can assume that lc(J1) > 1 + E . Then

en

en

>

>

and so k(&) -+ 00 as n -+ oo. Then by the hypothesis, the sequence {en) must be unbounded, and since lc is unbounded, we can also assume y(J,) # 1. But (1.18) n log(1 E), whence also implies log k(J,)

>

+

Hence CTZllog (,+I - log J, converges, and so the sequence {log En) is bounded above, whence so is {En), a contradiction which proves the claim. To prove the theorem, let k(r) = T(r), and x, be the sequence of the claim. 0. Since x, -+ m as n -+ m , we can assume y(x,) # 1, and also log M(x,) By Theorem 1.6 and the Let r = x,, R = x,y(x,) E (x,,x,y(x,) exp*. claim, as r -+ oo through the sequence x,, we have (since T ( r ) -+ oo as r -+ oo),

>

5

(2 + o ( l ) ) ( l + e)T(r) (log T(r))-I-'

-

124

4.

Meromorphic Functions

Hence, i 0 as r i m through the sequence r , and replacing r by €12 throughout, the theorem follows.

Note 1.11. Theorem 1.8 is a result of Shimizu. Let E ( z ) be the function discussed in Note 3.3.12 (analytically continued throughout C). If f (z) = E1(z), then it can er and T ( r ) whence 1 e be shown that for this function Log M ( r ) cannot, in general, be replaced by 1 in Theorem 1.8. However, if f has a Picard exceptional value, then Hayman and Stewart [107], have shown that in Theorem 1.8, 1+ r can be replaced by 112 + E ; and Example 1.4 shows that in this case, this too is best possible.

-

- 5,

+

Nevanlinna's Second Fundamental Theorem

4.2

Nevanlinna's Second Fundamental Theorem has two useful forms and represents a far-reaching deepening of Picard's Big theorem. The first of these can be stated now, and the second is an immediate consequence, subject to certain definitions. This second form introduces the notion of the "deficiency" of a value a for a meromorphic function f (for example, if a is never taken on, it turns out to have deficiency I ) , as well as the index of multiplicity of a value a. Definition of these ideas is postponed until later (Section 3). The proof of the Second Fundamental Theorem involves somewhat complicated estimations, but the utility of the result more than amply repays the work involved. One expression of Nevanlinna's Second Fundamental Theorem is

Theorem 2.1. Suppose f is meromorphic and non-constant in B(0, Ro), where O < R o < m . A s s u m e f ( O ) # O , # m , f ' ( O ) # O . L e t O < r < R o , a n d l e t a l , . . . ,a,, where q 2, be distinct finite complex numbers such that la,, - avI 2 6(0 < 6 < 1) for 1 p < v q . then, for f ,

<

>

m(r, m )

<

+

9

m(r, a,) v=l

< 2T(r) - NI (r) + S ( r )

where + 2 N ( r , f ) - N ( r , f') is non-negative and S ( r ) is an error term such that

(A) if Ro = ca and f is of finite Nevanlinna order;

(B) if & = m and f is of infinite Nevanlinna order, outside a set G of finite measure depending only on T ( r ) , and not on the number or values of the a,;

4.2. Nevanlinna's Second Fundamental Theorem

125

( C ) if Ro < m and GT+.i;,,no = m , then the limit is to be taken through a suitable sequence {r,} which does not depend on the number or values of the a,. Clearly, suitable modifications can be made for the cases f (0) = 0 or m , fl(0) = 0. Proof. Throughout, the letters Ck or Ck( ) will denote constants or functions depending only on the indicated parameters. Initially, we break the proof in two parts. First, we prove (2.1) with an explicit expression for S(r). The estimation of S ( r ) in the cases (A), (B), (C) proves far more complicated. The easy estimates

and

and the similar inequalities for T ( r ) and log+ will be used frequently (compare Theorem 1.2 and its proof and Note 1.2). (I) The expression (2.1). Let F ( z ) = EL1 -. Our first task is to relate m(r, F ) and Cz=,m(r, a,, f ) and then to bound m(r, F ) above. Let K: be a positive quantity to be chosen later. Suppose for all v, 1 5 v 5 Q, If (z) - avl 2 K, then

2 v=l

log+

1 < Q log+ 1 K:' If (2) - avl -

On the other hand, suppose for some fixed vo,

1

< v < q, by the definition of 6,

Hence, assuming 0

If

(z) -av,/< K:, then for v

< K < 6, for v # vo, 1 < v 5 q, we have

# vo,

4. Meromorphic Functions Hence, assuming K

< 619, we have

>

Now, for v # vo, 1 5 v 5 q, and assuming K < 6/q, q 2 2, If (z) - a,[ 612 for otherwise if, say for vl # vo, I f (z) - a,, I < 612, then, lavo - a,, I I. If (z) a,, I I f (z) - a,, I < 612 $ I. 6 contradicting the definition of 6. Furthermore, if K < S/q, since q 2, a similar argument shows that If (z) - a,, I < K can hold for at most one value a,, of the a,. Hence (2.3) becomes

+

Or, choosing

>

+

K = 6/29,

if vo exists, while if no such v, exists, and K = 6/29, then by (2.2), (2.4) is trivial. Hence (2.4) holds in all cases, and integrating we have

We now need to bound m(r, F ) above. To do this, we use the following decomposition:

Now, by Theorem 1.1.(b)

127

4.2. Nevanlinna7s Second Fundamental Theorem So, we have (using Theorem 1.1 (b) again, this time on f ) ,

Also, by Theorem l . l ( b ) N (r,

$) - N f ) (r,

= m (r,

$)

f )+ LO^ l mfl '(0)

- m (r7

lml f'(0)

+

f (reiB)

=

1

2"

~ o g l f ( r e ' ~ ) l d O -LogIf(0)I Log lfl(reiB)dO- Log 1f1(0)/)

.

By the Generalized Jensen Formula (the case r = 0 of Theorem 1.1), this last expression equals

and substituting this for N (r,

$) - N (r, 8)in (2.7) gives

Substituting (2.8) in (2.6) now gives m(r, F )

5 m (T, $) +m(r, f ) - N (r, 5 )+

+ ~ o 1g + m(r, f l ~ =) ~ rf) -, m(r, f ) - ~ ( r +) m(r, f ' ~ +) m (r, $) + Log 1 &1, by the definition of Nl (r). Hence, by (2.5), m(r,co) + CE=lm(r,a,) 5 2T(r, f ) - Nl(r) + S ( r ) , where S(r) = m(r, f l F ) + m (r, $) + Cl (p, 6). This proves (2.1). ~ ( rf l,)

4.

128

Meromorphic Functions

To see that Nl(r) = N (r, $) +2N(r, f ) - N ( r , f') is non-negative, observe that if C is a k-fold pole of f , then it is a k 1-fold pole of f', so that 2n(r, f ) - n(r, f') counts this pole k - 1 times. Also, n (r, counts a k-fold zero of f exactly k - 1 times (as well as other possible zeros of f'). Thus (assuming f (0) # 0, # m ) Nl (r) = dt, where n l (t) is 2 0 and, in particular, counts each k-fold zero or pole of f in B(0, t) exactly k - 1 times. We now need to estimate m (r, and rn (r, , and clearly the second of these reduces to the first.

+

+)

$)

11.

(

Estimation of m r,

L,

-9

$

We will need an expression for derived by differentiating the Poisson-Jensen Formula (Theorem 1.1). If f (reis) # 0 or m; a, are zeros, b, the poles of f in B(0, R), and 0 < r < p R, then by Theorem 1.1,

<

Log

If

1 (reis)[ = 2r

2"

Log

If

p2 - r2 (peimm)lp2 - 2 r p c o s ( ~- 4) + r2 d4 p(reis - b,)

+

Now, writing z = reis = x iy, in some neighborhood of any point where f # 0, # m , Log f (z) is an analytic function of z, say Log f (z) = u(x, y) iv(x, y), and Log If (reis) 1 = Re( Log f (reis)) = u(x, y) is a harmonic function. So applying - i & , we get by the Cauchy-Riemann equations, the operator

+

&

(g g) -i

~ o lf(z)l= g u, - iu, = u,

Similarly, with z = reis = x

+ iv,

f'(z) =-

f (z)

'

+ iy, since

Hence (with z = reie), we obtain on applying

-i

& to both sides of (2.9)

4.2. Nevanlinna's Second Fundamental Theorem

Since f (2) # 0, # cm,if we let 6(z)=

min

(lz-apl,lz-bVl)then6(z)>O.

P>" la,l<~,lb~l<~

+

Also define, for the purpose of this proof only, n*(p) = n(p, f ) n (p, j) = the total number of zeros and poles of f in B(0, p). Then with r = rei6, for la,[ < p, 5 ~ p 2 - l ~ z l p-T ,= and similarly,

&1$

. Also, similarly

Hence, from (2.10) we get on taking the modulus of both sides, with z = rei6 and 0 < r < p,

or using Theorem 1.1(b), since m(r, g) 5 T(r, g) for any meromorphic function g,

Hence,

Integrating with respect to 6' from 0 to 2n gives,

l

4.

Meromorphic Functions

We need to estimate

and n * ( ~.)

(ii)

For (i) by definition, S(reie) is the minimum distance from reie to any of the E B(0,p). These a, and b, together consist of n* points, say z,, v = 1,2, . . . , n* = n*(p) where clearly we may assume n* 1 (otherwise the term vanishes). Let

a, or b,

>

n

T

E, = ( 8 : Ireie - z,1 < --},

E=

U E,,

and

v=l

We estimate the integral in (i) separately over E and E'. For the estimate over E, it is convenient for x real, to put log*(2) = log x

x {i't

> n*(> 1)

herw ise.

Then, for reie E E , G(reiO)< r l n * and so

Hence,

To estimate JEv there is no loss of generality in assuming z, real and positive. Let 6 = $ 7~ and E; be E, "shifted down" by n;then

+

and $ takes values in rl sin $1.

[-T,

TI.

+

Then, since z, is real and positive, Irei@ z,l

>

131

4.2. Nevanlinna's Second Fundamental Theorem

<

+ +

+ +

Also, if n/2 < 141 n, Irei@ z,l = (r2 z; 27-2, cos $)'I2 2 r , contradicting the fact that for 4 E E:, Irei@ z,l < r l n * . Hence, E: 5 [-n/2, n/2], and, in fact, [-do, do],where 40 is the smallest positive root of sin 4 = E: Thus, since 0 < $o n/2,

5.

<

/,.

log

r - zvl)

de 5 2

lrnO(A) JdrnO (6)

= 240 (log

5 , we have

d4

log

(&)+

1)

log

d4 (2.13)

.

5 n/2n*; and since 22 (log (&) + 1) is monotone increasing in [0, n/2], (2.13) yields JEv log ( , T , i ~ , u ,) 5 $ ( l + logn*), and substiSince sin 40 =

$0

tuting in (2.12)

J,log+ (&)

do

5 n ( l + log n*) .

On the other hand, trivially,

and so we have for (i)

It remains to estimate (ii) n* (p). For R

and similarly,

Hence, by Theorem 1.1(b) ,

> p,

4. Meromorphic Functions and log+

( ) < log+ (n*(p) + log+ (3+ log 2 < log+ (:) + log 2 + log+ R + log+ -

Substitution of (2.14) and then (2.15) and (2.16) in (2.11) produces

Finally, choosing p = increasing

(since 0

< r 5 p 5 R), we get, since T ( r ) is monotone

To prove the results (A), (B) and (C) of Theorem 2.1, we need to choose R in various ways, and, thereby, estimate S ( r ) m +m (r, +CI (q, 6). (cf.the conclusion of the proof of I).

(x:=,6) 5)

(A): Claim: If f is a non-constant meromorphic function in C which is of finite Nevanlinna order, then S(r) = O(1ogr) as r + oo. Proof of Claim: If we let 4(z) = C:=,(f (z) - a,), then

Also,

4.2. Nevanlinna's Second Fundamental Theorem

133

Since by hypothesis, for some finite k, T(r, f ) = O(rk) as r -+ m , l o g + ( ~ ( rf ,) ) = O(1ogr) as r + m . Thus taking R = 27 in (2.17) gives m r, = O(1ogr) and

( $-0

m r, a = O(1ogr) as r -+ m , and consequently the claim.

(

To show that

# -+ 0 as r + m , consider first the case in which f is not +m

rational. If we can show that the claim, we will have

as r -+ m , then since S ( r ) = O(1og r ) by

# -+ 0 as r -+ m . However,

T ( r ) 2 N ( r , m) =

lr

w td

t2

1 dt 2 n ( f i , m ) - log r 2 .

%

Hence, if = O(1) as r -+ m , then n(r, m ) = O(1) as r -+ m and so f has only finitely many poles. = 0(1) as r -+ m , then f By Theorem l . l ( b ) it follows similarly that if has only finitely many zeros. Since f is meromorphic in @, it is the quotient of two entire functions which may be written in Weierstrass Product form

%

Since f has only finitely many zeros and poles

is a rational function and =

&,

R ( z ) say. Writing h(z) = hl ( z ) - h2(z), eh(') = f (2). where h is non-constant (since f is not rational). Therefore, if T ( r ) = O(1ogr) as r -+ m , T ( r , e h ) T(r, f ) T (r, $) = O(1ogr) clogr O(1) = O(logr), as r -+ m .

+

+

-

<

+

= 0, and so eh(4 is an entire function of zero NevanHence lim,,, linna order, hence of zero order (by Theorem 1.7) and so by the Hadamard Product Theorem (Theorem 3.1.3), h must be constant, which contradicts f not rational. -+ m and consequently' T ( T )-+ 0 as r -+ m if f is not rational. So I f f is rational, then f (2) = P, Q polynomials, so f l ( z ) - P'(z) QQ'(z) > 0

a,

as r

-+m .

as r

-t

T

Similarly

-

p(z)

Q(,)

$$#-+ 0 as r + m . Hence m (r, $) -+ 0 and m (r, f ) -+ 0

m ; so, in fact, if f is rational S ( r ) = O(1) as r -+ m , and so

# -+ 0 as

-+ 00.

(B):Claim: If f is a non-constant meromorphic function in C of infinite order, then S(r) = O(logT(r)) O(1ogr) as r -+ m outside a set G of finite (Lebesgue) measure. The possible exceptional set G depends only on T ( r ) and not on the number or values of the zeros and poles of f . To prove this claim (and a similar one for (C)), we first need a Lemma essentially going back to mile Bore1 and interesting in its own right.

+

4. Meromorphic Functions

134

Lemma 1. (a) Let K ( r ) be a continuous increasing function of K ( r ) 1 for all r T O . Then, for r 2 T O ,

>

>

T,

with

a set whose measure is 5 2.

>

(b) If K ( T )is continuous and increasing, and K ( T ) 1 for all r E Ro < m, then for TO 5 r < Ro

outside a set E such that &

[ T O ,Ro],

& 5 2. ~ K ( T ) }I f. there is no such r , :K r + ' ) > (

Proof of (a): Let T I = inf{r > T O we are done; otherwise T I > ro and define inductively a sequence {r,) as follows. I f r, is defined, let rh = r, and r,+l = i n f i r rh : K K(r7.)

>

+

>

Let E = { r : K 2 K ( r ) ) . B y definition of r,+l, E b y the continuity of K , r, E E ; and so

n (rh,r,+l)

+ &) > >

=

4;

Now, by definition of T ; , K ( r b ) = K ( T , 2K(rn) (since rn E E ) , and so since r,+l 2 rh and K is increasing, K ( T , + ~ ) 2K(rn) for all n 1. Hence, K(r,+l) 2,K(rl) 2 2n (since K ( r ) 2 I ) , and so, by the definition of rk, and (2.20), 1 meas ( E ) 5 r; - rn =

>

x CY)

CY)

n=l

n=l

>

which proves (a). Proof of (b): Note that putting r = Ro - e-P (so p = log

(& O-T))'

K(Ro - e - P ) = Kl ( p ) , say,

<

satisfies the conditions of part (a)for po p < m. SO by part ( a ) ,Kl ( p 2 K l ( p ) ,except for a set El of values of p(> po) of measure 5 2. So,

+ h) <

4.2. Nevanlinna's Second Fundamental Theorem

135

It remains to show that (2.19) holds in the complement of the set El so defined. Suppose r E [ro,Ro] - El. Define r' = r (ro - r ) ( l - e-l/K(T)); the value 1 of p, say p', corresponding to r' is p' = - log(Ro - r') = - log(Ro - r ) = K(T) 1 ' Hence, since p is in the complement of E l , Kl ( p ' ) < 2Kl (p); p + -K(T) =p+~l(p). that is K ( r l ) < 2K(r) for r in the complement of El. But, by the Mean Value Theorem, for x E [ O , l ] , there is a 6 E [O,1] such that 1- e-" = xde-l)" 2 x/e, and so (since K ( r ) 2 I ) ,

+

+

and since K is increasing this proves that (2.19) holds in the complement of E l . We can now prove the claim of (B). Let @(z)= n:=, f (z) -a, as in the proof of (A), and note as there that T(r, 4) 5 qT(r, f ) O(1). In (2.17), take R = r Then

+ h.

+

+

and by the Lemma, the right side is O(logT(r)) O(1ogr) as r -+ m outside a set of finite measure. Similarly, m r, $ = O(logT(r)) O(1ogr) as r -+ m outside a set of finite measure, and this proves the claim. That + 0 as r -+ m now follows by the argument used in proving (A) (since T ( r ) + m as r + m ) .

(

'1

+

%

(C): Claim: I f f is a non-constant meromorphic function in B(0, Ro) where Ro < m , then S ( r ) = O(log+ T ( r ) + log ) as r -+ Ro outside a set G such that

(&)

S&,

<m.

w.

Proof of claim: Once more, let @(z)= n:=, f (z) - a,. Now in (2,17), take R = r Then

+

and by part (b) of the Lemma proved under (B), the right side is

+

O(1og T ( r ) ) O(1og T(r)

1 1 + log ) + O(1) = (logT(r) + log -) Ro - r Ro - r

1

4. Meromorphic Functions

136

& < 00.

( '1 = 0 ( l o g ~ ( r+ ) log (A))as r + ROoutside a set St such that L, & < 00. as r

+

Ro, outside a set S such that Js

Similarly, m r, $

To complete the proof of (C), by the hypothesis of this case,

lim

r+Ro

T(r) = 00, -log(Ro - T)

T(pn) hence there is a sequence {p,), limn,, p, = Ro, such that - log(Ro-pn) -+ co as n -+ CQ. On the other hand, if ro < p < Ro and a > Ro then for any function K continuous and increasing in [TO,Ro], with K (r) 2 1, (2.19) holds for some r in the interval p < r < a , since then JPu = l o g e > 2 and so by part (b) of the Lemma proved under (B), some point in [p,a] (in fact, some set of positive measure contained in [p,a]) must not belong to the exceptional set E.

9,

&

9,

-)

Since RO> ROand p, < Ro, there exists an rn E pn, Ro outside the exceptional set G and as pn + Ro, rn + Ro, whence for r, sufficiently near Ro,

and so

(

+ 0 as i n + Ro.

This completes the rather complicated proof of all cases of Theorem 2.1. However, before we reformulate it in terms of Nevanlinna's notion of "deficient value", some remarks are in order.

Note 2.1. The exceptional set in part (B) of Theorem 2.1 can actually occur. This follows from a construction of Hayman [I041 designed to settle a different problem.

Note 2.2. Results like the Lemma used in proving part (B) of the Theorem show the utility of non-trivial results involving "general" real-valued functions of a real variable in proving theorems in complex analysis. This point is also made by the paper of Hayman and Stewart mentioned in Note 1.11. Bore1 himself used a version of the Lemma (Acta Math., 1897) in discussing the growth of entire functions.

Note 2.3. The proof given of Theorem 2.1 is a so-called "elementary proof" and is due to Rolf Nevanlinna. There is another proof of the theorem using results from differential geometry due to Frithiof Nevanlinna [171].

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

4.3

137

Nevanlinna's Second Fundamental Theorem: Some Applications

Theorem 2.1 proves to have many interesting consequences; often it is cast in a somewhat different form (also called the Second Fundamental Theorem) which contains slightly less information, but which may sometimes be more easily used. To do this, we need some definitions.

Definition 3.1. f is said to be "admissible for Nevanlinna theory" if it is either non-constant and meromorphic in @ or non-constant and meromorphic in some T T disk B(O,~ o and ) limT+n0 - loa:R~-T) = m. Henceforth, in this chapter only, admissible functions f such that T ( r ) -+co as r + 6,0 < Ro 5 ca will be considered. It is also convenient at this point to redefine the function N(t, a ) to allow for the possibility that f (0) = a. It is clear that if we make the

Definition 3.2. N ( r , a ) = Sor n ( t , a ) - n (0,a ) d t + n ( O , a ) l o g r , t h e n i f n ( O , a ) = O ( i . e . f (0) # a), we have Definition 1.3 (with the notational convention of p. 112); while if n(0, a ) # 0, then n(t, a ) - n(0, a) = 0 for all t E [0, tl] for some tl suficiently small, and so n(t, a ) - n(0, a)

dt

+ n(0, a ) log tl .

However, n counts multiple roots multiply, and as already observed following Note 1.2, N1 has something to do with the number of such roots, hence we make

Definition 3.3. ii(t, a ) = number of distinct roots (i.e. multiple roots are counted singly) off (2) = a in B(0, t)).

Definition 3.4. N(r, a) = SoT

fi(t,a)-it(0,a)

dtffi(0,a)logr.

By Theorem l.l.(b), m ( r , a ) + N ( r , a ) = T ( r , f)+0(1) a s r and we make

+ &, 0 < & < co,

#

m ra Definition 3.5. 6(a) = 6(a, f ) = bT+Ro = 1 - KT+Ro N ( T 1 a ) . &(a) is T(T) called the deficiency of a for f .

Note 3.1. If a is a Picard exceptional value, then N(r, a ) = 0 and so 6(a) = 1. In any case, since 0 5 m(r, a) T(r), we have 0 6(a) 5 1, and S(a) > 0 means that there are "relatively few" (though usually infinitely many) values of the argument z such that f (z) = a; as appears explictly below, Theorem 2.1 says that this cannot happen for "too many" values a.

<

We also make the analogous

<

4.

138

Meromorphic Functions

w,

Definition 3.6. @(a) = @(a,f ) = 1-K,+R,N Definition 3.7. @(a)= lim,+n,

N r,a -N r,a ( $(,) ( )

r,a

and

.

0(a) is called the ramification index or index of multiplicity of a .

Note 3.2. Thus 0 5 @(a)5 1, and 0(a) > 0 means that there are "relatively many" multiple roots of the equation f (2) = a. Theorem 3.1. (Nevanlinna's Second Fundamental Theorem-second version). If f is admissible in B(0, Ro), then the set of values a for which @(a) > 0 is at most countable, and S(a) +0(a) 5 @(a) 5 2 . a Q(a)>O o(,~)>o Proof. First note that, by the definitions of 0(a) and &(a),given E > 0, for all r sufficiently close to Ro (i.e. r large if Ro = m , and Ro - r small if Ro < m )

Hence, for all such r ,

and consequently, @(a) 2 S(a)

+ 0(a). Now, by Theorem 2.1,

%

where + 0 as r + Ro (where it is understood that the limit is taken through a suitable sequence r, in case Ro < oo). Using Nevanlinna's First Fundamental Theorem (Theorem 1.2 (a)) and adding EL=, N(r, a,) to both sides of (3.1), we obtain (q - l ) T ( r ) 2 -Nl(r)

+ N(r, m ) +

9

+ o(T(r)), as r + Ro .

~ ( ra,), v=l

Replacing Nl by its definition (see statement of Theorem 2.1), gives (q - l ) T ( r , f )

5 N(r, f') - N (r,

$) - N(r, f ) +

9

N (r, v= 1

i + o(T(r, f ) ) f - a, ,

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

(since N ( r , co) = N(r, f') - N(r, f ) ; f has a pole of order k 2 1 if and only if f ' has a pole of order k 1). Furthermore, if is a zero of f (a) -a, of order p, then it is a zero of order p - 1 of f'(z) and hence, contributes exactly once t o n t, )-,(,,$);hence

<

+

( &

where Q counts those zeros of f' which may occur elsewhere than a t zeros of f (z) - a, for some v = 1 , . . . . Hence, from (3.2) we get

and so dividing by T ( r )and taking the lim as r (1- @ ( m ) ) ,whence

+ Ro, q - 1 5 CP,=,(l-

@(a,))

+

Now q was an arbitrary integer 2 2. Also, (3.4) shows that given a positive integer K, @(a,) can be >_ for at most 2K - 1 distinct values a,. Hence, the values of a for which @(a) 2 0 can be arranged in a decreasing sequence taking for example, first the finite number (if any) for which @(a) = 1, then the finite number (if any) 113, and so for which 1 > @(a) 112, then those (if any) for which 112 > 0, forth. Let the resulting sequence be called {a,). Then (3.4) says that

>

for any finite q, and so, if {a,) is an infinite sequence, letting q Theorem.

>

+ co gives the

Example 3.1. Picard's Little Theorem is an immediate consequence of Theorem 3.1. For if a is an excluded value, S(a) = 1. Hence, there can be a t most two such values of a. (A similar deduction can be made directly from Theorem 2.1.) In fact, Theorem 3.1 shows that there can be a t most two values of a for which N ( r , a ) = o(T(r)) as r + m . In fact, Theorem 3.1 shows that there can be at most two values of a for which 6(a) > 213 (i.e. for which Kr 7'm < 113).

4. Meromorphic Functions

140

Example 3.2. If f is entire, S(co) = O(co) = 1, (N(r, m) = N(r, co) = O), and so from Theorem 3.1

C

a(a) + e(a) 5

a finite O(a)>O

C

@(a) 5 1

a finite O(a)>O

and, hence, 6(a) > 112 for at most one finite value of a.

Example 3.3. Suppose a is a value such that f (z) = a has only multiple roots. Then 2fi(t,a) 2 n(t,a), and so

< <

whence

Thus, if f is entire (and so @(co) = I), it follows from Theorem 3.1 that there can be at most two values for which f (z) = a has only multiple roots. Either sin z or cos z and the values 1 and -1 for a show that two such values can occur. Note also, that in this case necessarily, @(I) = @(-I) = 112. Similarly, if f is meromorphic, from Theorem 3.1, there can be at most four values of a for which f (z) = a has only multiple roots. The Weierstrass elliptic function P ( z ) (see Chapter 8) has only double poles and satisfies a differential equation of the form

and so shows that such a function exists.

Example 3.4. Suppose p is the elliptic modular function of Chapter 2, and let

Then F is analytic in B ( 0 , l ) and maps it onto @ , - {0,1, co) and so 6(0) = 6(1) = 6(co) = 1. Hence, by Theorem 3.1, F cannot be admissible for Nevanlinna theory and so it follows that lim

,-+I-

T(T' F ,

- log(1 - r )

is bounded.

Note 3.3. A theory, which in some ways is parallel to Nevanlinna's theory of T ( r ) was developed by Ahlfors. If f is non-constant and meromorphic in C, and A(p) is the area of the image on the Riemann sphere of B(0, p), then

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

141

(cf. Definitions 2.2.2 and 2.2.3). By an argument involving Green's formula (see Hayman, Meromorphic Functions [loo], p.10-11; Nevanlinna, Analytic Functions [169], p. 171-173; Spencer [223]) one can show that

By rotating the Riemann sphere, cm can be made to coincide with any previous (with F, = f ) and clearly point a ; this amounts to replacing f by F, = does not alter A(p). Thus, d A(p) = p~

2"

Log (1

+ I ~ ~ ( p e " 1 ~ ) +d B4an(p, a ) .

(i.e. A(p) is the portion of the surface of the Riemann sphere Let A(p) = covered by the image). Then

lr

y d p = N(r,a)

iTh""

+-

+

Log (1 I ~ ~ ( r e " ) 1 ~ ) d B

Now, clearly for any f ,

& st"

Hence Log (I+[Fa(rei8)12)d9differs from m(r, a) by only a bounded amount; and so fy d p differs from T(r) by only a bounded amount.

Jl

y d p is called the Ahlfors-Shimizu characteristicof f . Since it differs from T ( r ) by only a bounded amount, often it makes no difference which characteristic is used. However, Ahlfors made it the starting point for a very deep and geometrical theory in which, instead of asking questions about points taken on, questions are asked about the number of times the image of a subdomain of B(0, r ) under f maps onto a fixed domain D of the Riemann sphere. For expositions of Ahlfors' theory of covering surfaces, see Chapter XI11 of Nevanlinna's Analytic Functions [169], Chapter V of Valiron's Fonctions entiires d'ordre jini et fonctions mdromorphes [236], and Chapter V of Hayman's Meromorphic Functions [loo]. N o t e 3.4. A meromorphic function may have a value with deficiency 1 which is not Picard exceptional. Consider r ( z ) which has its poles a t {-n : n a non-negative integer). Here clearly n(r, m ) = r O(1); whence N ( r , cm) = r O(1ogr). Using Stirling's theorem, one can show that m(r,co) as r + m , and hence as r + m . Hence 6(m) = 1 KT,, = 1. T ( r ) .Arguing further, since r ( z ) has no zeros n(r, 0) = 0 and hence N ( r , 0) = 0, and 6(0) = 1. Thus by Theorem 3.1, E Z,, # 1 if a # 0, m, and so if a # 0, m ,

+

lim,,,

=

!.

-

+

~w-

Actually even more refined results can be obtained for l?(z).

4. Meromorphic

142

(B)

(ii)

Figure 4.1: Alternating bands and lines. Let f(z) :

e 2 ~ i e ze2Kie-z

-

Functions

1

Zeros and

poles are distributed according to two main rules: first inside black and white bands respectively (together with uncountably many approximating values for both cases), alternating in the left and the right half-plane. Secondly, as blown up in the right figure, each horizontal straight-line nin includes poles and zeros located symmetrically to the left and the right of the imaginary axis.

Note 3.5. Although Nevanlinna's concept of deficiency is extremely useful, it does have one drawback for some functions for which T ( r ) grows rapidly; namely, the value of the deficiency is not independent of the choice of origin. The following example is due to Daniel Duguh [60]. Let f (z) =

e2niez

e2nie-;_11

; let rl(z) = e2"iez- 1; and let fi ( z ) = f (z - 1).

+

Then f has zeros at the points {logm nin : m , n integers, m > 1) in the right half-plane and poles at the points {- logm nin : m, n integers, m > 1) in the left half- plane (note that f(nin) = 1). Hence, since the poles and zeros lie symmetrically with respect to the imaginary axis,

and so 6(0, f ) = 6(m, f ) .

+

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications Furthermore, m(r, m , f ) =

al

2n

143

Log +If (reie)ld8

And clearly, n ( r , m , f ) = n(r,O, f ) = n(r,O,v) - 1 - 2[r] (the difference coming from the zeros of q on the imaginary axis). Hence,

So, from (3.5),

I m(r, m,~ ) + m ( r0,, v)+N(r, 0,q) I 2T(r, ~ ) + 0 ( 1,)

T(r, f ) = m(r, 00, f )+N(r, m, f

(3.7) by the First Fundamental Theorem (Theorem 1.2 (a)). From (3.6) and (3.7)

and so, since q is entire of infinite order, using Theorem 1.7,

i(1

that is, 1 - 6(0, f) 2 - S(0,q)) . But 6 ( - 1 , ~ ) = 1 and 6(m, q) = 1 since -1 is a Picard exceptional value for the entire function 7. Hence, by Theorem 3.1, 6(0, '1) = 0 and so

On the other hand, the zeros of fi(a) = f (a - 1) occur at the points (1 log m r i n : m, n integers m 1) and the poles of f l at the points (1 - log m r i n ; m , n integers m 1). So

+

>

>

Hence N(r, 0, f l )

e N(r, 0, f)

as r -, m ,

+ +

4. Meromorphic Functions

144 Also, since T ( r , f l )

-

T ( r , f ) as r

-+

m (by Theorems 1.2 and l.l(b)), we have

Hence 1 - 6(0, f l ) = 41 - 6(0, f ) ) ;

so, if 6(0, f l ) = 6(0, f ) , then they both must equal 1 in contradiction to (3.8).

-+ 5 as r -+

Similarly,

m and so 6(m, f l )

# 6(m, f )

also.

Hayman [lo51 has given an example of an entire function (of infinite order) for which the value of the deficiency of a point is not invariant under change of origin. He shows, in fact, that if

and fl(a) = f ( a + a ) where la1 2 1 1 2 , then 6(0, f ) = 0 (in fact, 6(a, f ) = 0 for all finite a), but ,-40

It seems not to be known whether this phenomenon first noticed by Dugu6 can take place for meromorphic (or entire) functions of finite order. Note 3.3 is somewhat disturbing, but not greatly so, as in a three page note immediately following Duguh's, Valiron proved:

Theorem 3.2. Suppose f (z) is a non-constant meromorphic function and T(r) its Nevanlinna characteristic. Let, for any complex b, fb(z) = f (a + b). Then, if + 1 as r + m, 6(a, fa) = 6(a, f ) for any a E C,, and any b E C. Proof. Note that for any complex number b, and r > Ibl, B ( 0 , r - Ibl) c B(b,r) c B ( 0 , r Ibl). Hence, if nb(r,a) denotes the number of a-points of f ( a ) in B(b,r), then n ( r - Ibl, a ) 5 nb(r, a ) n(r Ibl, a). Defining Na(r, a ) analogously, we have

+

<

+

and taking a = eie, we have, by Theorem 1.3 with p = 1, since clearly nb(r,a , f ) = n(r1 a , fb).

(we may assume without loss of generality that f (0) # 0 and f (-b) Hence, from (3.9) and (3.10), (since Nb(r, a ) = N(r, a , fb)),

# 0.)

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

145

where cl(r),f2(r) -+ 0 as r -+ m . But if w l + 1 as r -+ m , then -+ 1 as r + m (clearly, under the hypothesis, this need only be proved for Ibl < 1, but 5 = ) and so we have taking throughout then 1 in (3.11), that 6(a, fb) = &(a,f ) .

w-&

<

Note 3.6. Given arbitrary sequences of real numbers (6,) and (8,) such that 0 6, 8, 5 1, and C:==' , 6, 8, 5 2, and an arbitrary sequence {a,) of complex numbers, there exists a meromorphic function f such that f has deficiency 6, at a, and ramification index 8, at a,. The complete solution to this formidable "inverse problem of Nevanlinna Theory" was given by David Drasin [58].

< +

+

We now turn to some further consequences, beginning with a famous result of

R. Nevanlinna. Theorem 3.3. Suppose fl and f 2 are two functions meromorphic in C, not both constant. Let El (a) = {z : f l (z) = a ) and E2(a) = { z : f2(z) = a ) . If El (a) = E2(a) for Jive distinct values of a, then f l (z) f2(z).

=

Proof. Suppose a l , . . . ,a5 are distinct and El (a,) = E2(a,) for v = 1 , . . . ,5, where fl and f 2 are not identical and not both constant. If f l were constant, by hypothesis f2 is non-constant and so must omit four values contradicting Picard's Little Theorem; hence neither f l nor fi is constant. where the equality holds by Let Nv(r) = lV (r, = N r , -1, hypothesis. By equation (3.13) of the proof of Theorem 3.1, with q = 5, we have

&)

Hence

+

By hypothesis f i $ f2; hence T(r, f i - f 2 ) 5 T(r, fl) T(r, -f 2 ) since T(r, -f2) = T(r, f2), we have by Theorem l.l(b) and (3.12),

+ log2, and

4.

146

Meromorphic Functions

On the other hand, if f l (zo) = f 2 (20) = a, then zo is a pole of by hypothesis, El (a,) = E2(av), v = 1,. . . ,5, using (3.3), yields

and

SO

since,

5

5 (213 + o(l)) So,

5

C Nv(r) + O(1) -

v=1

< +

+

Nv(r) 3 o(1) O(1) = O(1) , v=1 and, consequently, Nv(r) is bounded for v = 1 , . . . ,5, that is each of 5 values is taken on by fl and f 2 only finitely often. But this contradicts Picard's Theorem.

Note 3.7. The same argument shows that if f l and f 2 are admissible in B(0, Ro), Ro < co, then by Theorem 3.1 outside a set E such that J, < co, we have

&

a

Hence, Nv(r) = O(log(&)), v = 1 , . . . ,5, outside E , and so t 0, v = 1,. . . ,5; j = 1,2, as r + Ro through a suitable sequence (since f l and f 2 are f 2 in the case of a disk of admissible). But this contradicts (3.12). Hence f l finite radius also.

=

Note 3.8. The functions fl(z) = eZ,f2(z) = e-" show the number 5 in Theorem 3.3 is sharp (El (a) = E2(a) for a = 0, co, 1,-1). In a sense, this is the only sort of exception. R. Nevanlinna has shown that if El(av) = E2(av) for v = 1,2,3,4, and f l f f2, then (with an appropriate indexing) El (al) = El (a3) = 0; the cross-ratio of the points (a1,a2, as, a4) = -1; and fi = S(f2)where S is a linear fractional transformation such that S(a2) = a2, S(a4) = a4 and S(al) = as. (In the above case, S(z) = In fact, Nevanlinna has shown that if E l ( a v ) = E2(av),v = 1,2,3, and f l f f2, then

i.)

and

+ b ~ 2 ~ - d ' ( z+) c ~ 2 e - ~ ( z ) f2

= kA2e-@(z)+ eBZe-"(.

+ mC2e-~(z)

where 4, +,x are entire; a, b, c, k, e, m are constants such that 2 m = a3 (with appropriate indexing); and

= al,

= a2,

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

147

Proofs of these results may be found in Nevanlinna's book Le The'ortme de PicardBorel. [I701

Theorem 3.4. Suppose f is not rational and meromorphic in @, then for every integer C 1. 1, the C'th derivative f(e)(z) takes on every finite value except at most one infinitely often. is true. I n fact, the stronger result CafiniteO(a, f(')(z)) 5 1 O(a)>O

+&

Proof. If f has a pole of order p a t zo, f (e)(z)has a pole of order p p L 2 e 1. Hence

+

+

Hence

- N(r, co, f

O ( w , f(e)(z))= 1- lim

r-tm

T (r,f (e))

>I--

+ C a t zo and

1 e+l

and so from Theorem 3.1,

Now, suppose f assumes a finite value a only finitely often. Then N ( r , a , f (e))= f -+ m (this is proved in O(logr), and since f is not rational by hypothesis, ~ >(9 0 and so + 0, the course of proving Theorem 2.1(A)). Hence T!>N(T(r,f'e') as r + 00. Thus @(a,f(e)) = 1, and since C 2 1, by (3.14) this can happen for a t most one value of a.

,

jy;bjf;)t;)

Note 3.9. If f ( z ) = t a n z , f l ( s ) = sec2z and fl(z) omits the value 0, and so 0 ( 0 , f') = 1. Also, sec2 z = 1 if and only if tan2 z = 0. But, if tan2 zo = 0, then I

I

= 0; hence sec2 z = 1 has only multiple roots,

=2tanzsec2zl

$(tan2z)l z=zo

z=zo

>

+&

and so (cf. Example 3.3) @(I) 112. So (at least for C = 1) the number 1 in the theorem is a sharp bound (and 0 ( l , sec2 z) = 112). It does not appear to be known whether the bound is sharp for C > 1.

Note 3.10. The function tan z also shows that the hypothesis that we are dealing with a derivative of a meromorphic function is essential as tan z omits both i and -i as values. Note 3.11. It can, in fact, be shown that the only possible omitted value for the derivative of a meromorphic function, which has a Picard exceptional value, is 0.

4.

148

Meromorphic Functions

This depends on results of Milloux, which concern functions of the form $(z) = -+ 0 as r + m. It can be shown that

~ t =a.(z) , f ("'(z), l an integer 2 1, where

and T(r, $1 I (e + 1+ o(l))T(r,f

1

and that the counting functions for certain roots of f ( z ) = a can be replaced by similar functions for certain roots of $(z) = b. For these results, see Hayman's Meromorphic Functions[lOO] Chapter 111, pp. 55-62. We may note that thus, for example, we have that e" az, a # 0, assumes every finite value infinitely often. In fact, it is now known that iff is non-constant and meromorphic in C, f (z) # 0 and f(e)(z) # 0 for some one value of e 2 2 for all z, then either f (z) = eaz+bor f (2) = (Az B)-n, n a positive integer. This was conjectured by Hayman and after many partial results, finally proved by Gunter Frank [72]. The proof depends on a result of Hayman (op. cit p. 74) that if F is meromorphic and non-constant in C and for some l > 2, N(T,F) N(r, N(r, &) = o(T(r, then F ( z ) = ea"+bl as well as some results on differential equations with entire functions where F ( z ) = n;==, F'(z) as coefficients; in particular, an estimate for m(r, and W is the Wronskian of the F', j = 1,. . . ,n. That no such theorem is true for l = 1 is shown by f (2) = ee' . Nevanlinna Theory can also be used effectively to study the behavior of functional iterates or other compositions of functions (compare Chapter 3, Section 4) for some other results and literature). A particularly striking result concerning fixed points was obtained by I. N. Baker [16]. Before proving this, we need a definition.

+

+

+

s) +

g)),

T)

Definition 3.8. Suppose f is entire. Let fl(z) = f (2) and define f,+l(z) = f (f,,(z)) If f,(zl) = zl, then zl is called a fixed point of order v. If zl is a fixed point of order v , but of no lower order, then zl is called a fixed point of exact order V.

Theorem 3.5. Let f be a transcendental entire function, then f has infinitely many fisced points of exact order n, except for at most one value of n. Proof. We need some ideas from Chapter 3. From the proof of Theorem 3.4.1 (See Note 3.4.2), we have that if g and h are entire transcendental functions, and if M ( r , f ) denotes the maximum modulus of f on B(0, r), then there is a constant C > 0 such that

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

149

<

Hence, since by Theorem 1.6, if k is entire, for 0 r < R , E T ( R , k) 2 logf M ( r , k) T(r, k), taking k = g o h, and R = 2r, we get from (3.15)

>

for all r sufficiently large (since M ( r ) + co as r + co). By Cauchy's inequality, since h is entire and transcendental, log + ca as r + co, hence for all r sufficiently large and any fixed integer N 2 1, CM(rI2, h) > (2r) N. Hence, since T is increasing (Theorem 1.4) we get from (3.16)

E:!21h1

where k = g o h, for all r sufficiently large and any fixed N 2 1. Now is increasing for all r sufficiently large (Theorem 1.4, and, in fact, by Theorem 1.6 + co as r + co). Hence T((2rlN,g) >-T(2r, g) l o g ( ( 2 ~ ) ~-) log 2~ for all r sufficiently large, and so from (3.17) we have (replacing r by 7-12) that if k = g o h, then T ( r , g)

31N qqq

for all r sufficiently large and any fixed N 2 1. Hence letting r N + co, we get

+ co and

then

T ( r g) = 0 . lim -

r--too

T(T,k)

Now, suppose f has only a finite number (possibly 0) of fixed points of exact order m; say they are [I,. . . ,(,. Take n > m. Then fm(Ci) = Ci, i = 1 , . . . , p , and so fn(Ci) = fn-m 0 fm(Ci) = fn-m(Ci). On the other hand, if fn(zo) = fn-m(zo) for some 20, then

<

and so fn-,(zo) is a fixed point of exact order m of f . So, if fn(zo) = fnWm(zo)= w, say, then either w is one of the Ci, or w is a fixed point of exact order j of f where j < m. In the latter case, fj(w) = w and so fn-m+j(z~) = fn-m(zo), and 1 j I m - 1Hence,

<

4.

150

Meromorphic Functions

By Nevanlinna's First Fundamental Theorem (Theorem 1.2), and since ~ ( r I ) T ( r ) , the right side of this equation is m-1

5

n-1

P

C T(r, fn-m+j - fn-m) + C T(T, fn-m) + O(1) = 0 (CT(r, fc))

.

Now from (3.18), with k(z) = fn(z),g(z) = fe(z), h(z) = f n - e ( ~ ) ,we have for l
NOW,let m(z) =

fn-'fnLl-,

2 -2

. Then (3.19) says that

Also, by the Second Fundamental Theorem (cf. equation (3.3) of the proof of Theorem 3.1)

Thus, taking q = 2, a1 = 0, a2 = 1, we have by (3.20), (1 + o(l))T(r,4) I N(r, 074) + n ( r , CO,4) + o(T(r, fn)) I N(r, 0, fn(z) - z) + N(r,0, fn-m(z) - z) + o(T(r7fn)) I N(r, 0, fn(z) - z) + T(r, fn-m(z)) + o(T(r,fn(z)) = N(r, 0, fn(z) - 2) + o(T(T,fn(z)))

on using (3.18) with k = fn and g = fn-,. T(r,+) < lim T ( r 7fn) - T+,

T+,

So

N(r,o, fn(z) -2) T(T,fn)

and we want to show that the left side of (3.21) is 5 1, thus establishing that has infinitely many fixed points for every n 2 m. But

(

')<

+

'"(') T(r, fn(z) - 2) T fn-m(z) - 2. = T(r, fn(z) - z) + T(r, fn-m(z) - z) + O(1) 5 T(r, fn) + T(r, fn-m) + O(logr) = T ( r , f n ) + o(T(r, fn)) ;

T , ) =T

fn

fn-m(z) - z (3.22)

4.3. Nevanlznna's Second Fundamental Theorem: Some Applications and T ( r , fn)

I T(r, fn(z) - 2) + O(logr) = T(r, $(z)(fn-m(z) - z)) + O(logr) 5 T(r, $) T(r, fn-m(z) - 2) O(logr) = T(r, $1 + T(r, fn-m) + O(1og r) = T(r, 4) + o(T(r, fn))

+

+

9

<

+

since T(T,FG) T(T,F) T(r, G); using Theorem l.l(b), Theorem 1.6 (since -1 +- ca as r +- m) and (3.18). From (3.22) and (3.23), lim,,, ,(.T(r94) log r , ) - , and so, from (3.21), we have . I

N ( r , O , f n ( ~-)z ) lim 7

,

T(r, f n )

21

Now, note that

by Theorem 1.6 and (3.21) again. Taking sight of (3.24), it follows that fn(z) - z has infinitely many zeros which are not zeros of f k (z) - z for any k 5 n - 1. Hence, fn(z) - z has infinitely many zeros of exact order n for every value of n except at most one (if p < m, and there were only finitely many zeros of exact order p, then there would be infinitely many of exact order m by the above, contradicting the hypothesis).

Note 3.12. The exceptional value of n can occur. If g is entire and f (z) = e g ( " ) +z, then clearly f has no fixed points of order 1. It does not seem to be known whether 1 is the only possible exceptional value in Theorem 3.5. In the case of polynomials, Baker has also shown that if f is a polynomial of degree 2 2, then f has at least one fixed point of exact order n for every integer n 2 1, with at most one exception, and here again, the exception can occur as the example f (z) = z2 - 2 shows (the only fixed points of order 2 are 0 and 2 and these are both fixed points of order 1). For particular functions ad hoc methods sometimes suffice to demonstrate the existence of fixed points. For example, to prove that sinz has infinitely many fixed points, one might observe that - 1 was entire, of order 112, and so by Theorem 3.1.6 has infinitely many zeros. Or, one might note that sinz was periodic, and hence, by Picard's Little Theorem, if sinz - z assumed the value 0 only finitely often, then it must assume the value 2n infinitely often, and so sin(z + 2n) - z - 2n = sin z - z - 2n = 0 infinitely often. Clearly, this last argument shows that any periodic entire function has infinitely many fixed points.

9

4. Meromorphic Functions

7

Figure 4.2: Zeros a n d critical points. Two methods superimposed: the values distribution, showing the zeros of f (z) = z2 - z at the origin and at z = 1 Oi (center and right black discs), and the existence of a critical point c = 112 + Oi where the nested equipotential curves intersect.

+

N o t e 3.13. A study of the iterates of a transcendental entire function was initiated by Fatou [70] Fatou connected this study with Montel's normal families, (see Section 4 of Chapter 1) in the following way. Let us say that a family of entire functions {fk(z)) is normal at a point p, if and only if, it is a normal family in the sense of 1.4 in some neighborhood of p. Consider an entire function f (z) and its set of iterates: fi (z) = f ( z ) , fn+1 (z) = fn(fl (z)), n = 1 , 2 , . . . . Let F ( f ) be the set of all points p E C , at which the family of iterates {fn(z)) o f f is not a normal family. If f is entire transcendental, then (i) 3 (f ) is a non-empty perfect set. (ii) 3 ( f n ) = 3 ( f ) for every integer n

> 1.

(iii) If a E F ( f ) , and f(P) = a ,then /3 6 3 ( f ) ; also then f ( a ) E F ( f ) . (iv) If y 6 C is a fixed point off of order n , and I fA(y)l while if IfA(y)l < 1, then y $! F ( f ) .

> 1.Then y E F ( f ) ;

(v) 3 ( f ) is the derived set of the set of fixed points of all orders of f . Thus the complement of 3 ( f ) is an open set whose connected components are the maximal domains of normality of the family of iterates {f,), and a major question is how does F(f ) divide the plane.

4.3. Nevanlinna's Second Fundamental Theorem: Some Applications

153

I. N. Baker has shown that for any constant A > 0, there is an entire transcendental f such that F ( f ) C { z : Im(z) > A, R e z < 0 ) but, on the other hand, for no entire transcendental f is F ( f ) contained in a union of finitely many straight lines (Journal London Math. Soc. 40, 1965). The construction of the function f mentioned above involves the function E ( z ) already mentioned in Notes 1.11 and 3.3.12. At the end of his paper, Fatou mentions that it would be interesting to obtain an example of a function f such that F ( f ) = @, and suggests that eZ may be such a function although "je n'en ai pas de preuve rigoreuse". This was finally proved, over 50 years later by Michael Misuriewicz.

Note 3.14. A central fact used in proving Theorem 3.5 is equation (3.18). This clearly is related to the last section of Chapter 3. Clunie has completely settled the relationship between the growth of the composition of two meromorphic functions and the growth of the composing functions (cf. Note 3.4.2). Thus Clunie shows, among other results, that for f meromorphic transcendental and g entire transcendental, then lim T(r, f g) = m ; T(r, g)

T+oo

lim T(r, f +

g)

T(r,f)

= m ,

but there exists a meromorphic function f and an entire function g, both necessarily of infinite order, such that lim -

r--too

T(r, f g) = 0 . T(r, f )

Note 3.15. The "deficiency" &(a,f ) = l h T + , defined by Nevanlinna is not the only one possible. For example, one might consider the quantity A(a, f ) = lim m(r, a, f = 1 - lim N(r, a, f ) T+T(r,f) T= T(r,f)

'

This is known as the "Valiron-deficiency" of a with respect to f . If A(a, f ) is called Valiron-deficient.

> 0, a

Nothing like Theorem 3.1 is true of Valiron-deficiencies; in fact, a function may have uncountably many Valiron-deficient values. However, in a famous paper of Ahlfors (cf. Note 1.9), he proves the following: Let E be the set of all Valiron-deficient values of a transcendental function f meromorphic in B(0, Ro). Then if lirn,,~, T(r, f ) = m , given E > 0, and an arbitrary continuous decreasing function h such that &,o l h t dt is finite, there is a B(O,pn) and Cr=,h(p,) < E . sequence of disks B(O,p,) such that E C Ur=3=, In particular, if & = oo, then this is always true.

4. Meromorphic Functions

154

In fact, Ahlfors showed even more sharply, that if E* is the set of complex numbers a such that m(r, a, f ) # O(T(r))1/2f' as T + 00, then E* has the above property. (See also, Nevanlinna, Analytic Functions [169], pp. 272-276.) Hayman has proved that given an F,-set S which is "small" (of capacity zero), there is an entire transcendental function f (of infinite order) which has S as its set of Valiron-deficient values. ([104].)

Note 3.16. Similarly Petrenko has introduced the "deviation" of a value a for a function f meromorphic in B(0, Ro) as P(a,f)= lim log+ M ( r , a, f r+Ro T(r,f)

&

where M(r, a, f ) = maxl,l,, if a # 00 and M ( r , 00, f ) = maxl,l=, If @)I. Petrenko obtained results analogous to the ones of Ahlfors cited in the previous note (dealing with what might be called the a-deviation ,&(a, f ) in which T(r, f ) is replaced by (T(r, f))" in the above definition, and 0 < a 5 1). See, for example, Izv. Akad. Nauk. USSR 33, (1969) 1330-1348, and 34 (1970) 31-56).

Note 3.17. Nevanlinna Theory has applications in areas not at all adumbrated in this chapter; for example, to differential equations or Riemann surface theory. The interested reader might begin by consulting Nevanlinna's book, Analytic Functions [I691 and the book by Wittich [251]. By consideration of an appropriate Riemann surface, Wittich, for the first time, constructed a function with irrational deficiencies and ramification indices (op. cit p. 127-128) (cf. Note 3.4). Nevanlinna theory is an active field with a huge literature, the gradual solution of old problems, and the present introduction of promising new techniques. The present chapter will have served its purpose if some readers are interested enough to pursue further aspects of the theory.

Chapter 5

Asymptotic Values An asymptotic value of an entire or meromorphic function is a complex number a such that, as lzl + w along a specified path, f ( z ) + a. In this chapter, we consider two famous results involving such considerations. The first is Julia's Theorem concerning the behavior of entire or meromorphic functions in unbounded angular regions, to which allusion has already been made in Note 2.2.7. The second is the Denjoy-Carleman-Ahlfors Theorem limiting the number of asymptotic values an entire function of finite order may have. As will become clear, there are connections to ideas in previous chapters, for example, normal families, Picard's Theorems, the Phragmkn-Lindelof principle.

5.1

Julia's Theorem

Definition 1.1. Let a E C,, and f be a meromorphic function. If f ( z ) + a as la1 + w along some continuous path y, then a is an "asymptotic value" of f ( z ) . If a is finite, y is called a "path of finite determination". If a = oo, y is called a '$path of infinite determination". If If ( z ) ]is bounded on y, but limlZl,, f ( z ) does ZEY

not exist, y is called a "ath of finite indetermination". All definitions hold i f f is meromorphic in C - B(0,p ) for some p instead of the whole plane. Theorem 1.1. A Picard exceptional exceptional value of a meromorphic function (2.e. a value taken on only finitely often) is an asymptotic value.

-*

Proof. Let a be a Picard exceptional value for a non-constant meromorphic function f and suppose first that a is finite, and f entire. Then g(z) = is also entire and non-constant, and so by Liouville's Theorem g is unbounded. If there were a continuous path y such that lim 1g(z)l= cm, lzI+m 2E-r

5. Asymptotic Values

156

then a would be an asymptotic value of f and conversely. We now construct such a path y for g(z). Since g(z) is entire and non-constant, M ( r ) = maxlZl,, 1g(z)1 is an increasing function of r and lim,,, M ( r ) = cm. Let {p, : n = 1,2,. . . ) be a strictly increasing sequence of real numbers such that pl = M ( r l ) for some rl > 0 and lim,,, p, = cm. By Liouville's Theorem, there exists a point [ in the complement of B(0, r l ) for which lg(<)l > pl = M (rl). Consider the set of all such points S = {[ : lg(<)1 > pl ). By the maximum modulus theorem, if [ E S, > r l . The set S is open and consists of one or more connected open sets in C on whose boundary curves Ig(z)l = pl. Choose one such connected component; call it Dl. Now D l must be unbounded (for otherwise we have a finite region such that Ig(z)l = pl on its boundary and Ig(J)I > pl for some point ,€ in its interior, contradicting the maximum modulus theorem). Furthermore, g is unbounded in D l ; for otherwise, since (g(z)l = pl on the boundary of D l and for z E D l , lzl > r l , applying Theorem 3.3.1 with a ( z ) = 7 , we get that 1g(z)1 5 pl in D l , contradicting the definition of D l . Hence, there is a point [I E Dl such that lg(&)I > p p Repeating the above construction, we arrive a t an open- connected set Dz E Dl such that 1g(z)1 > p2 for all z E Dz and 1g(z)1 = p2 on the boundary of Dz (hence BdDz n BdDl = 4). As above, D2 must be unbounded, and g must be unbounded on D2. Continuing in this way, we get (formally by induction) a sequence of regions

such that on the boundary of D k , Ig(z)l = pk, and such that each of the D k is unbounded. Choosing a point on the boundary of each D k and connecting these points in sequence by a continuous curve such that the curve connecting a point on D k with one on Dk+1 lies entirely within D k , gives the desired path y. If f is meromorphic and cm is a Picard exceptional value (in particular if f is entire) the above arguments applied to f instead of g, show that cm is an asymptotic value of f (the only change necessary is to appeal to the Casorati-Weierstrass Theorem, instead of Liouville's Theorem, to guarantee that there is an unbounded sequence of points (5,) on which 1 is unbounded). If f is meromorphic and a is a finite Picard exceptional value, we may again argue with g(z) = , using the Casorati-Weierstrass Theorem as indicated above.

If

Example 1.1. eZ has two asymptotic values, 0 and cm, and the corresponding paths are the negative real axis and the positive real axis respectively. Example 1.2. t a n z has two asymptotic values i and -i, and the corresponding paths are the negative imaginary axis and the positive imaginary axis respectively. Example 1.3. Let F ( z ) = J : q d t where k is a positive integer and the integral is taken along a straight line.

5.1. Julia 's Theorem Then, writing z = reie, t = peie, t9 fixed, we have

Taking t9 = y , m = 0 , 1 , 2 , . . . ,2k - 1, we see that as lzl argz = y ,

+ cm along the line

Thus F(z) has 2k distinct finite asymptotic values, and they are approached along straight-line asymptotic paths of finite determination. Since F is entire, cm is also an asymptotic value by Theorem 1.1. Taking 0 = $ in (1.1), we see that this line is a path of infinite determination for F. We may note that t9 = $ is, for example, another path of infinite determination, and if k is large, there are many more such.

9,.

Example 1.4. Suppose k is a positive integer and g(r) = Then, if ym is the path { z : argz = 71,m = 0,1, ..., 2k- l,g(z) - + O as lzl + cm o n r . Hence g has (at least) 2k paths of finite determination, but the asymptotic value for each of them is 0. Example 1.5. eeezhas 0 as an asymptotic value, and a path of finite determination I' along which 0 is approached is

I' = { Log ( t + i7r) : t 2 0) . Example 1.6. Suppose a > 0, then F ( z ) = ei" is bounded outside some B(0, p) (for any p > O), but there is no path y to cm such that limlZl,, F ( z ) exists. 2E-Y

Note 1.1. Clearly the proof of Theorem 1.1 holds good for functions meromorphic in the exterior of some B(0, p). Also, clearly for a meromorphic function which is analytic a t 0, we may assume that all paths have initial point the origin (so that in Example 1.5, we might equally well take the path to be the union of the straight line from the origin t o the point Log 7r i7r/2 with I?).

+

Consideration of situations like Examples 1.3, 1.4, and 1.6, together with Theorems 3.3.3 and 3.3.4 leads to the following

Definition 1.2. Suppose yl and y:! are two continuous paths to cm with the same initial point zo, but otherwise non- intersecting, and suppose limlZl,, f (z) = zFr~

limlzl+mf (z) = a, a finite. Then yl U yz determines two unbounded 'regions in ZE72

the plane. Suppose f (z) + a uniformly in the closure of one of these regions. Then

5. Asymptotic Values

sin(zk) there , exist 2k paths Figure 5.1: Paths of finite determination. For 7 z where this function is determined: see Figures A, B, C for k = 2,3,4 respectively, where there are 4,6,8 radial paths respectively. Between any two such paths, there exist regions (the white ones) where points are mapped to m . All such regions, as well as all paths, meet together a t m , as shown in the three views over the Riemann sphere (figs. D,E,F). Figure F offers a blow-up of the neighborhood of m : one notices the same value distribution as about the origin appearing here again, together with a flowered-shape region surrounding such a neighborhood.

yl and 7 2 are called '%ontiguouspaths of finite determination" and the closure of the region in which f(z) + a uniformly, is called a 'Yract of determination7'. Similarly one can define "contiguous paths of infinite determination" as similar paths yl,yz such that If (z)l + m as lzl + m along n and yz and If (z)l + m uniformly in the closure of one of the two unbounded regions determined by yl U72, and "contiguous paths of finite indetermination" similarly, if If (z)l is bounded in the closure of one of the two unbounded regions formed by two paths of finite indetermination. In all other cases paths of the same kind are said to be %on-contiguous". (Paths of different kinds are not compared.) N o t e 1.2. In the terminology of Definitions 1.2, the paths in Example 1.4 are noncontiguous, since if lc is a positive integer and e, is the line {z : argz = E(m $)},

rn = 0 , 1 , . . . ,2k - 1, then as lzl

+m

along em,

lql+

+

m.

5.1. Julia's Theorem

159

Theorems 3.3.3 and 3.3.4, suggest for an entire function of finite order k, there can be a t most 2 1 r / ~=~2k distinct asymptotic values or 2k non-contiguous paths of finite determination which are straight lines. Examples 1.2 and 1.4 show that the value 2k can be realized and that the two statements are different. We will return to this question in Section 2 of this chapter.

Note 1.3. Since a Picard exceptional value is asymptotic, the question naturally arises after Chapter 4, whether a value deficient in Nevanlinna's sense is an asymptotic value. The answer is no. H. Laurent-Schwarz showed that the function 1+z4-" g(z) = (l-,4-.)'-2'n has Nevanlinna order ) and 6(0,g) = 6 ( ~ , >~ 0,)

n:.=,

but neither 0 nor oo is an asymptotic value [218]. The first example with an entire function, seems to have been given by Hayman [105] who showed that if

f (z) = n r = l ( 1+ (e ) 3 n ) 2 n , where la[ 2 112, then 6(0, f ) > 0, but 0 is not an asymptotic vake for f (see k s o Note 4.3.3). Hayman's example has infinite order, but he remarks that it is possible to construct similarly more complicated examples of finite order.

Theorem 1.2. Between any two non-contiguous paths of finite determination, there is a path of infinite determination. Proof. Suppose yl and 7 2 are the paths (with common initial point zo, say). If f is bounded in one of the two regions whose boundary is yl Uy2, and limlzl,, f (a) = a; '

lirnlzl+, f (z) = b; then if yl and y2 are straight half-lines, it follows from Theorem ~ € 7 2

3.3.4 that a = b, and f(z) + a uniformly in the region in which it is bounded, whence yl and 7 2 are contiguous, contradicting the hypothesis. But, it is easy t o see that the argument of the proof of Theorem 3.3.4 does not depend on the shape of yl and 7 2 (cf. also Note 3.3.5). Thus f ( t ) is unbounded between yl and 72. The argument used in Theorem 1.1 now produces the desired path.

Note 1.4. It is perhaps worth pointing out here that the examples of Note 3.3.11 show that a non-constant entire function f may + 0 along every ray through the origin yet not approach 0 uniformly in a region determined by two such rays. Indeed, by Theorem 1.2, there is a path 'I such that If (t)l -+ oo as Izl + oo along r in a t least one of the regions determined by any two such rays. Asymptotic paths are closely related to a deepening of Picard's Theorems for meromorphic functions due to Gaston Julia. Theorem 1.3 (Julia's Theorem). Let f be meromorphic in C, not rational, and have an asymptotic value a (finite or oo). Let y be a simple Jordan path extending ) 0. Then there is a zo to oo, and suppose y = { ( ~ ( t :) 0 5 t < oo), where ( ~ ( 0 > and an 6 > 0 such that f takes on every value in C,, except possibly two, infinitely often in B(zoa(t),cla(t)l), for 0 5 t < co. (Iff is entire, then oo is an omitted

160

5. Asymptotic Values

value, and so the conclusion is f takes on every finite value except possibly one, etc.) Proof. This proof uses Montel's theory of Normal Families, (cf. Theorems 2.2.42.2.6). Consider the family of functions

< <

Suppose 0 < A < B < oo, and consider the annulus A = {z : 0 < A lzl B). If T were normal in A, then there would be a sequence of functions iftn(z)) (where {t, : n = 1 , 2 , . . . ) is a strictly increasing sequence and limn,, t, = oo) such that { ftn (z)) converges uniformly in A either to a meromorphic function F ( z ) say, or to oo. Suppose for now {ftn) converges to F. Let r be an asymptotic path from 0 to oo such that limlZl,, f (z) = a. Consider the circle C(0, R) where 0 < A < R < B . z €.r Then, for each positive integer n , there is a C, E C(0, R) such that Cna(tn) E I?, and hence + co . ft, (Cn) = f (Cna(tn)) + a , Let B E C(0, R) be a limit point of {Cn}. Then given E > 0, there is a 6 > 0 such that for IB - C',I < S, 1 ftn ( 8 ) - ftn (Cn)l < 6, and consequently F(B) = limn,, ft, (B) = a. But R was arbitrary, 0 < A < R < B ; so on each circle C(0, R), 0 < A < R < B , there is a point a t which F takes the value a; consequently F(z) a in A. But then limn,m f (za(tn)) = limn+, ftn (z) = F ( z ) r a for all z E A and this contradicts the Casorati-Weierstrass Theorem (since ca is an isolated essential singularity of f). Similarly, if { f t n ) converges uniformly to oo in A, we again contradict the Casorati-Weierstrass Theorem. Hence, T is not normal in A, and consequently, given a sufficiently small tl > 0, there is a point zl in A, and a closed neighborhood B(zl, €1) C A of z1 such that { ftn ) is not normal in B(zl, €1). Consider the set

The function f must take on every value except a t most two infinitely often in P,, , for otherwise, the argument of Theorem 2.2.6 shows that the family {ftn (z)) = {f (zg(tn))) is normal in p,, which we know is false. (Theorem 2.2.6 was proved for analytic functions, however, the extension to meromorphic functions, allowing one more omitted value, requires no essential change in the argument and can be safely left t o the reader.) 61, such that We now consider a strictly decreasing sequence { E , ~ 0} ,< 6, E, + 0 as m + oo, and repeat the preceding argument. This produces an infinite

<

5.1. Julia's Theorem

161

sequence {z,) such that f ( z ) takes on every value in C, except a t most two, infinitely often, in the set

Let zo be a limit point of {z,}. It follows that there is an 6 > 0 such that f (z) takes on every value in C,, except a t most two, infinitely often in B(zoa(t),clo(t)l), 0 5 t < oo. (Note that zo E A is not 0 and that no sequence iftn)can be normal in A.) Taking y = {a(t) : 0 5 t < oo) in Theorem 1.3 to be a ray through the origin, Theorem 1.3 has an immediate corollary.

Theorem 1.4. Under the conditions of Theorem 1.3, there is at least one ray R through the origin such that in every angular region bisected by R , f takes on every value except at most two infinitely often. Definition 1.3. Simple Jordan curves extending to co contained in the union of the disks B(zoa(t),tla(t)l),0 5 t < oo of Theorem 1.3 are called "curves of Julia". Rays having the property of Theorem 1.4 are called 'Yines of Julia" or "directions of Julia". Note 1.5. Since for a transcendental entire function, oo is an omitted value, by Theorem 1.1, oo is an asymptotic value, and hence every transcendental entire function has a curve of Julia. Clearly there are similar theorems (obtainable by a suitable non-singular linear fractional transformation) for the neighborhood of any isolated essential singularity. Also, clearly a similar theorem holds for functions meromorphic in the exterior of some disk B(0, a ) (one simply assumes a(0) > a and a < A < B ) . Example 1.7. ez has two lines of Julia: the positive imaginary axis {z : argz = 7r/2), and the negative imaginary axis {z : argz = -7r/2} (this result can, in fact, be deduced from the periodicity of ez). Note that the family of functions {ezt : 0 5 t < oo) is normal a t every point z except those on the imaginary axis. Hence, these are the only two lines of Julia. Example 1.8. The function -omits the values 1 and -1. Arguing as in Example 1.7, the positive imaginary axis and negative imaginary axis are the only lines of Julia. Here again, the asymptotic paths for the excluded (and hence by Theorem 1.1 asymptotic) values are the positive real axis and negative real axis. Note 1.6. It is worth noting that (as is to be expected) different choices of ~ ( t ) give different points a t which the family f (zu(t)) fails to be normal, (but, of course, the curves of Julia for f remain the same). For example, taking f (z) = eZ and a ( t ) = teie, 0 < t < co,O fixed, we see that the family comprising the functions

5. Asymptotic Values

eZ - 1 Figure 5.2: Example 1.8. The function - The superimposed graphic level ez+l' of equipotential curves shows that curves are dilating about the the origin, which is a repelling fixed point.

+

f (u(t)) is normal everywhere except a t points with argument n/2 - 8 kn, k an integer (apply Theorem 2.2.4 and Definition 2.2.3). (Though Theorem 2.2.4 is stated for analytic functions, it is easily seen to be valid for meromorphic functions as well.) Hence, we get once more the result of Example 1.7 (where the argument was with 8 = 0). It may also be noted that the family of functions {ft(z)) defined by ft(z) = f (zu(t)) is never normal a t 0 (for any f analytic a t 0).

Example 1.9. Let 4(z) be any non-trivial elliptic function (see Chapter 8). Then 4 has two independent periods and a "period-parallelogram" which tiles the plane and in which the values of 4 are repeated. Also, 4 always has a t least two poles (perhaps one double pole) and two zeros in every parallelogram which is a replica of the period-parallelogram. If C is any point, then B(Cu(t), +(t) I) will, for any simple Jordan curve extending to m traced by a(t), and t sufficiently large, contain an arbitrarily large number of such replicas; hence infinitely many zeros and infinitely many poles of 4. Thus any subsequence of the family {+t(z) = $(zu(t))) which converges must converge both to 0 and to m , a contradiction. It follows that &(z) is nowhere normal, and thus that every ray through the origin is a line of Julia (in fact, every simple Jordan curve from 0 to m is a curve of Julia).

A fairly simple necessary condition for a function to have at last one line of Julia is Theorem 1.5. I f f is meromorphic in @ and has a line of Julia, then there is a

5.1. Julia's Theorem

@,

sequence {z,) such that lim,,,

z, = oo, and

Proof. Iff has a line of Julia, given E > 0, there is a sequence {z,) and a correspondlz,I = oo, and ing sequence of disks {B,) where B, = B(z,, ~lz,I)such that lim,,, f takes on every value except a t most two in the union of any infinite subsequence of the {B,). Let

Then the functions of any infinite subsequence {f,; (w)) take on every value, except a t most two in each disk {w : Iwl < 6). Thus {f,(w)) is not normal at 0 (since one cannot extract convergent subsequences). Hence, by Theorem 2.2.4

And so, bv (1.2),

Example 1.10. If f (z) = c o s ( G ) , then

and this goes to oo as z integer.

+ oo through the sequence z = ( k r + ~ / 2 ) k~ a, positive

Example 1.11. If g(z) =

z2

then

Now, sinz has infinitely many fixed points, (e.g. since the function

5,

z2

- 1 has

order cf. Note 4.3.10); call this sequence of fixed points {ck). ~ h only k limit is oo. Consider in (1.3), z = c;. Then point of

which

{ck)

+ oo as k + oo.

164

5. Asymptotic Values

nr==,

Example 1.12. Let F ( z ) = -. We prove F has no line of Julia. Writing, as in Chapter 2, (Definition 2.2.3)

&J,

(x(F(-z)) = we see that (cf. proof of Theorem 2.2.5) since F(-z) = x(&) = x(F(z)); so in estimating x(f(z)), we may assume with no loss of generality, that Re z 0. Writing z = x iy, we have,

>

Now, for x = Re z

so for Re z Rez 2 0,

+

> 0, we have

> 0, IF(z)l 5 1. Since

is increasing for 0

5 u < 1, we get for

Now assume

k a positive integer. We break the last expression in (1.4) into four parts. For n = k, we have (since Rez 0)

>

<

2k+2

- 22k + 1212 For n = k

\

<

2 4 - < - , by (1.5) and (1.7). 2k - IzI

+ 1, similarly, since R e z > 0, we have by (1.5) and (1.7),

-,

5.1. Julia's Theorem For the first part of the sum, we get, using (1.5) and (1.7),

Similarly, for the "tail" of the sum we get,

Taking (1.8), (1.9), (1.10), (1.11) together in (1.4), and since k was an arbitrary positive integer, we get that there is a positive constant A such that

But, since x ( F ( z ) =

m,

by Theorem 1.5 we have that F has no line of Julia.

Note 1.7. Thus, although every entire function has a line of Julia (see Note 1.5), not every meromorphic function does. If T ( r , f ) is the Nevanlinna characteristic of a function such as that in Example 1.12, for which tX(f (teie)) is bounded as t -+ m , then by Note 4.3.1,

and so,

Furthermore, the converse of Theorem 1.5 holds: If f (z) is meromorphic in @, and there is a sequence {z,) with lim,+, z, = co such that lim,,, Iz,lx(f (z,)) = m , then for every E > 0, f takes on every value except a t most two in the union of any infinite subsequence of the disks B(z,, tlzv\). (For a proof, see Lehto [141], Theorem 3.) Hence, if a function has no line of Julia, then its Nevanlinna characteristic must satisfy (1.12). Actually Ostrowski [I821 in 1926, already had characterized very explicitly all functions meromorphic in @ which have no line of Julia, showing that they are the ratios of two Weierstrass products each of order 0; satisfying certain auxiliary

5. Asymptotic Values

166

conditions on their zeros and poles. Functions of this sort are called "Julia exceptional". The characterization by the growth of ~ ( f is) a result of Marty [154], and independently, Lehto and Virtanen [142]. This memoir by Marty, essentially his dissertation, directed by Montel, seems to have been overlooked until comparatively recently. For example, Theorem 2.2.4 first appears there, though some authors have credited it to Ahlfors, following his 1959 text. Ostrowski also considered "Julia sequences", i.e. sequences {u,) with la, 1 + co as v + oo such that the family of functions {f,(z)) defined by f,(z) = f (u,z) is not normal in C, from which one obtains a union of disks B(zOu,,clu,l), 0 2 v < oo such that f takes on every value, except at most two in this union. Such sequences had already been considered by Julia, and Ostrowski shows that questions about existence of curves of Julia can be reduced t o questions about the existence of Julia sequences. It is worth noting that commonly today (for example, in Lehto and Virtanen's papers) the "spherical derivative" is kX(z). For Marty, the "spherical derivative" of f a t a point to is k(1 Izo12)x(zo) in our terminology. These differences in definition generally make little difference, provided one is aware of them. Lehto and Virtanen in the cited papers have, in fact, shown that if f is a transcendental function meromorphic in some neighborhood of oo, and co is an essential singularity of f , then &,,, Izlx(f (2)) 1, and, in fact, this inequality is sharp in that there are functions meromorphic in C for which equality holds. An example of such a function is

+

>

Extensions of these results were obtained by Clunie and Hayman [47]. Connections between the ideas of Lehto and Virtanen and the concept of deficient values were discussed by Anderson and Clunie [9]. In particular, they show that a function with a Nevanlinna deficient value (see Definition 4.3.5) has a line of Julia. (Note that by Note 4.3.3, in general a Nevanlinna deficient value is not necessarily an asymptotic value; however, Anderson and Clunie show that for functions of slow growth; in ~ ) is , true.) particular for which T ( r ) = O ( l ~ ~ r )this One may further note, that if a transcendental function f meromorphic in C has no line of Julia, then by Theorem 1.3 it has no asymptotic values, and so by Theorem 1.1, no Picard exceptional values. Hence, it must take on every value infinitely often in every neighborhood of oo. This illustrates the distinction between Picard's Theorems and Julia's Theorem. Another similar observation is that since every entire transcendental function has a line of Julia, it follows by Theorem 1.5 (also due to Lehto and Virtanen) that for every entire transcendental function f , lim Izlx(F(z)) = lim 211. lfl(z)l = o o . 2-t1 1 f (z)I2

-

2--iw

+

5.1. Julia's Theorem

167

Note 1.8. Theorems 1.3 and 1.4 provide another example of the utility of the idea of "normal families"; using the non-trivial Montel-Carathkodory-Landau criterion of Theorem 2.2.6 in a version for meromorphic functions. The proof given of that Theorem in Chapter 2 uses the elliptic modular function; however, as remarked (see Note 2.2.6) a proof is also possible which is based on Schottky's Theorem (Theorem 2.1.4) instead. Similarly, Theorems 1.3 and 1.4 alternatively can be proved directly from arguments based on Schottky's Theorem. For this sort of proof, see Cartwright [42], Chapter VII. In addition to Schottky's Theorem, an inequality of Carleman is also required. This approach goes back to Milloux's thesis published in Journal de Mathematiques [158]. Actually Milloux has proved a somewhat stronger result than the existence of lines of Julia as defined above. Define a ray argz = 0 to be a line of Julia in the sense of Milloux, if, given 6 > O,q > 0, there is a closed disk Vg = B(Reie,GR) such that f takes in Dg every value, except possibly values which under stereographic projection lie in two neighborhoods on the unit sphere each of radius 5 7 = q(6). Milloux proved, among other results, that every meromorphic function with an asymptotic value has a line of Julia in the sense of Milloux. The disks were called "cercles de remplissage" by Milloux because in them f "fills" more and more extensive regions of the plane. Actually, Milloux obtained detailed quantitative information about, for example, the relation between q and 6 or lower bounds on the number of times values are taken on in certain disks. In particular, the fact that the Nevanlinna characteristic of a "Julia exceptional" function (see preceding note) is 0(log2r ) also follows from Milloux's quantitative results. (See also, Acta Mathematica (1928), 188-255.) Note 1.9. There are, in fact, several other variants of lines of Julia; one can, for example, consider the concept of "Borel- exceptional" instead of "Picard-exceptional" (cf. Theorems 3.1.5, 3.1.8; Note 3.1.9) and make definitions, for example, in terms of the exponent of convergence of the moduli of the a-points, or n(r). See, for instance, Valiron, Directions de Borel des fonctions meromorphes [237]. Note 1.10. Julia's Theorem essentially says that, given a simple Jordan curve extending to oo, a "neighborhood" of that curve, and a function f meromorphic in @ with an asymptotic value a t oo; then there is an orientation of the curve cum neighborhood such that in the neighborhood f takes on every value except a t most two (one if f is entire). From this point of view, it is reasonable to suppose that connections may exist between the Phragmkn-Lindelof indicator of Chapter 3, Section 3, and Julia's Theorem. Such relationships have been studied, especially by M.L. Carwright and Valiron. For example, If f is entire of finite order and type, and 0 = y is the endpoint of an interval in which h(8) > 0, then argz = y is a line of Julia. Or If f is entire of finite order p > 112, then it has a line of Julia in every angle equal t o max(n/p, 27r - 7rlp) (and so at least two). These results mostly deal with directions of Borel which are a fortiori lines of Julia. See Cartwright [43], and Integral Functions [42], Valiron [237].

168

5. Asymptotic Values

Note 1.11. Suppose f is an entire function of infinite order in an angle V = {reis : a 2 e 2 ,B, 0 < r < m ) , (see Definition 3.2.4). Suppose L bisects V, then L is a line of Julia. For a proof, assume, on the contrary, L is not a line of Julia. Then f omits two finite values in an angle W bisected by L. Without loss of generality, we can assume that L is the real axis, that the angle W = {reie : 161 5 a n/2,O < r < m) of "opening" a n , and that 0 and 1 are the omitted values. maps B ( 0 , l ) conformally onto the right half-plane, Since the map g(z) = the function l+z mcz, = f 1-2

((-).)

is analytic in B ( 0 , l ) and maps it onto f (W). Hence omits 0 and 1 as values, and so by Note 2.1.7 (sharp estimates in Schottky's Theorem),

for all z E B(0,p),O 5 p < 1, where A = Hence, for z E C(0, p), 0 5 p < 1

If

(1)l # 0 and C is a positive constant.

where K I is a constant. Putting z =

w-+l

, we get from (1.13) that for 0 5

=p

< 1 , (and so

w E W),

where Kz is a constant. Furthermore, as lwl + m(w E W), p -+ I-. Thus, from (1.14) . , lim logloglf(w)l Iwl-tm loglwl a


wEW

contradicting the assumption that f was of infinite order in W. Combining this proof with Note 3.2.4, we have that given an arbitrary entire function of infinite order 00

there exists a sequence {c,),

E,

= 1 or -1 such that for the function

every ray through the origin is a line of Julia.

5.2. The Denjoy- Carleman-Ahlfors Theorem

169

This and other results about lines of Julia for functions of infinite order are due to P6lya in Sections 50-68 of Mathematische Zeitschrift [192]. Appropriate analogues for some of these for functions of finite order were found by Anderson and Clunie [lo]. However, the above result would not appear to have any analogue a t all for functions of finite order. For, consider eZ in a small angle V bisected by the positive real axis. eZ is of order 1 in V, but the positive real axis is not a line of Julia for eZ since eZ has as its only lines of Julia, the positive and negative imaginary axes (see Example 1.7).

Note 1.12. The large literature on functions analytic or meromorphic in the unit disk has largely been excluded from consideration, however, it seems worth remarking here that there is an analogue of the line of Julia, a so-called "radius of Julia", but here there seems to be no simple necessary and sufficient condition involving ~ ( (z)) f (cf. Theorem 1.5 and Note 1.7) for a radius of Julia to exist. See Lehto [141], p.203-205 for some results. Note 1.13. A natural question is what can be said about the relationship between lines of Julia for f and for f ' in general. A lengthy memoir on this subject from the point of view of directions of Bore1 (see Note 1.9) was published by Milloux [I591 in Journal d'Analyse Mathematique. This memoir again contains quantitative, as well as qualitative results, and deals with functions holomorphic in the unit circle as well as the plane. A striking qualitative result going back to Biernacki [26] is: Every entire function of finite positive order has at least one line of Julia in common with its sucessive derivatives and integrals.

5.2

The Denjoy-Carleman- Ahlfors Theorem

Suppose f is meromorphic in @. In the first section of this chapter, we saw that if f has an asymptotic value, and hence a path r of finite or infinite determination, then in an "angular neighborhood" W of ,'I f takes on every value except a t most two. I f f is entire, then rn is one omitted value and there can be a t most one finite omitted value. However, we have not discussed whether there is any limit t o the number of different regions like W, or to the number of asymptotic values a function may have. For entire functions of finite order, the Phragmkn-Lindelof theorems of section 3 of Chapter 3 indicate that such a limitation may exist, and the DenjoyCarleman-Ahlfors Theorem shows that not only does it exist, but provides a sharp estimate for the number of asymptotic values in terms of the order of the entire function. Two asymptotic values are to be considered different if they occur on noncontiguous paths, even if they are numerically the same. (Compare Examples 1.3 and 1.4 and Note 1.2.) As remarked in Note 1.2, Theorems 3.3.3 and 3.3.4 suggest if we restrict attention to asymptotic paths which are straight half-lines emanating from the origin, then a transcendental entire function of finite positive order p can have a t most 2p non-contiguous paths of finite determination. (For 0 < p < 112, Note 3.3.4 shows that f can have no paths of finite determination which are rays.

170

5. Asymptotic Values

"' -fi

and the positive real axis This is not true for p = 112, as the example of shows.) However, even proving 2p to be an upper bound for the number of straightline paths of finite determination says nothing about the situation when the paths are arbitrary Jordan curves. Clearly, i f f is an entire function of finite order p, and lim, > 0, then a 5: p. Thus the situation for arbitrary Jordan curves is settled by

,

9

Theorem 2.1 (Denjoy-Carleman-Ahlfors). If the entire function f (z) has n "distinct" finite asymptotic values (or, equivalently, n non-contiguous paths of finite determination), then lim logM(f-1 > , , + rn/2 where, as usual, M(r) = maxl,l,,

If

(z) 1.

Proof. Suppose a l , . . . , a n are the n finite asymptotic values of f . Let

and g(z) = f ("1

. Then

G = {a : Ig(z)l > 1) contains n disjoint regions G I , . . . ,Gn such that (i) g(z) is unbounded in G k ,

(iii) there is an ro

> 0 such that for all r > TO,

Less formally, removing the set {z : Ig(z)l 5 1) from the plane, divides it into disjoint regions a t least one of which is unbounded for each asymptotic value, 0 is in none of these regions, and in these regions g is bounded away from each asymptotic value. Let Sk,r= Gk n C(O,T),T > TO 2 o . Then the Sk,rare arcs in Gk with endpoints on the boundary of G k , and except for these endpoints, all Sk,,are disjoint. Also, clearly, logf Ig(reie)l is harmonic in 6, (where, as usual, for a > 0, log+ a = max(1og a , 0)). Put log+ lg(reie)l = u(reie) and consider the function

5.2. The Denjoy-Carleman-Ahlfors Theorem

171

The reason, as will appear, for considering a k ( r ) is that, on the one hand, we can get a differential inequality involving a k ( r ) and the maximum length of an S k , , C G k , which, when integrated, gives a non-trivial lower bound for a k ( r ) ;on the other hand, one easily sees that Ci=,ak(r) is bounded above by A JC(O,T)log' If (reis )I2do for some constant A. Combining these results produces a non-trivial lower estimate of the growth of log M (r). Clearly ak(r) is twice differentiable and we will obtain a differential inequality ((2.10) below) by using the fact that u is harmonic in G. Note first, for later use, that u(reis) is 0 at the endpoints of each Sk,,. Differentiating, we get for r > r o (where all integrals without indicated limits are over the circular arc Sk,,)

and az(r) = 2

I

+

( u , ) ~ uuTTd6.

(2.2)

However, u is harmonic in G and by the polar form of Laplace's equation

+

r2uTT ru,

+ uoe = O ;

hence (using (2.1))

Also, from (2.1), by Schwarz's inequality,

(;a: (r ))

<

1 / u2d6

u:d8 = a k (r)

1

ujd8

and so

Substituting this in (2.3) gives

But, integrating by parts, since u is 0 a t the endpoints of Sk,,,

,

5. Asymptotic Values and so from (2.4) we get

and we need to estimate J uide from below. To do this, it will be convenient to use the following well-known inequality for whose proof we digress momentarily.

Claim: If y(d) is a continuously differentiable function in (0, T) and y(0) = y(n) = 0,then

Proof of Claim: 0

5

JdT (2(@)) dB

sine

sinz Ode =

( ~ ' ( 0sine ) - ~ ( 8~) 0 ~ 8 ) ~ dB (sin 8)2

since the integrand in the second integral = - $ ( ( y ( ~ ) ) cot ~ 0) and y(0) = y ( ~ = ) 0. Returning now to the estimation of (2.5), we change variables in order to be able to use the result just proved to estimate J u28. Let /3 be the argument of the initial point of Sk,, and 1C, the size of the central angle subtending S k , , Making the change of variable 0 = ,B

+ %,

Hence

2

(5)2(f)

(&u

(r,p+ ~ ) ) ~ d + ,

and by the claim just proved, (2.6) becomes

Changing the variable back to 6 in the right hand integral in (2.7) gives, on letting

Lk (T) = length of Sk,,.

5.2. The Denjoy-Carleman- Ahlfors Theorem Substituting (2.8) in (2.5) gives

Since a

k

is clearly positive, (2.9) can be rewritten as

and so

or, finally,

integrating gives for r

> TO,and some constant C I ,

Log ak(r)

+

> 27r

Log r

1

dt

+ C1

and so, exponentiating and integrating again,

where C2 and C3 are constants, Cz > 0. Now, by definition,

Also, the curves Sk,,are disjoint except possibly for their endpoints, and by definition log' lgl differs from log' If 1 by an additive constant. Hence, there is a positive constant Cq such that

(Note that the union over k of the S k , , might not cover all of C(O,r).) Combining (2.12) and (2.11) shows that there is a positive constant A such that Jd21(logf 1 f ( r e " ) ~ ) ~2d A ~

ST 2 To

k=l

exp (2n

SU TO

i d t ) du Lk(t)

(2.13)

5. Asymptotic Values and we need t o estimate the right side of (2.13). By the inequality of the arithmetic and geometric mean,

By the inequality of the arithmetic and harmonic mean, and since 211t (the Sk,t are disjoint except possibly at their endpoints), we have

3 exp 2n

n

dt

>exp2mly

nL, < Lk,t

(E) n

&dt

=

Substituting (2.15) into (2.14), and then (2.14) into (2.13) gives

for all sufficiently large r , where B is a positive constant. But the left side of (2.16) is trivially 5 2n(log M(r))2 for all sufficiently large r, and so we get lim

->0

and the theorem.

Note 2.1. Theorem 2.1 has been phrased for entire functions; a natural question is what can be said about meromorphic functions. Suppose we consider for an entire function f , a branch of its inverse. It turns out that asymptotic values for f are exactly the non-algebraic singularities associated with branches of the inverse of f (or equivalently non-algebraic singularities on the Riemann surface for the inverse of f .) This result is due to Iversen [I211 although it had been suggested earlier by Hurwitz. Iversen also (following earlier work of Boutroux) established a classification of these singularities. In particular, if g is a branch of the inverse o f f , a transcendental (i.e. non-algebraic) singularity w of g is "directly critical" if there is a sufficiently small neighborhood B ( w , p ) of w such that in the image of B(w, p ) under g, f never takes the value w. For example, a Picard exceptional value of f is always a directly critical transcendental singularity of g. On the other hand, 0 is an asymptotic value of (consider the positive real axis), but clearly not a directly critical transcendental singularity of the corresponding inverse. For further information on this sort of investigation, see Nevanlinna [169], Chapter XI, or Valiron Fonctions entikres d'ordre fini et fonctions me'romorphes, Sections 29-39.

%

5.2. The Denjoy-Carleman-Ahlfors Theorem

175

A similar analysis holds for meromorphic functions. Ahlfors [4] in 1932 showed that if f is a meromorphic function of finite Nevanlinna order k (see Chapter 4), then the number of directly critical transcendental singularities of the inverse of f is a t most 2k for k 112 and 1 for k < 112. This generalizes Theorem 2.1; a t first sight it seems to say something weaker than is known for entire functions. For Theorem 2.1 says that the number of finite transcendental singularities (directly critical or not) of the inverse of an entire function f of finite order k is a t most 2k for k 2 112. However, by Theorem 3.3.4, an entire function f must be unbounded in the "angle" between two paths of finite determination, and hence since f never assumes CCI, the argument of Theorem 1.1 produces an asymptotic path of infinite determination, and thus, i f f h a s p finite asymptotic values, oo is a "p-fold" directly critical singularity of the inverse of f . Realizing this, Ahlfors' cited generalization actually allows for a slight sharpening of Theorem 2.1. For suppose the entire function f of finite order k, has exactly m finite asymptotic values which are directly critical transcendental singularities for the inverse, and, say, n other finite asymptotic values. Then co is a t least an L'm n-fold" directly critical transcendental singularity of the inverse of f and hence by Ahlfors' 1932 result, m (m n) 2k. Thus, we can also state that the inverse of an entire function of finite order k 2 112 has a t most k directly critical transcendental singularities which are finite.

>

+

+ + <

@

, then 0 and m are asymptotic values, hence For example, if f ( z ) = z7 transcendental singularities of the inverse of f , but oo is directly critical, and 0 is not. On the other hand, there does not seem to be any bound known on the number of finite asymptotic values of a meromorphic function of finite Nevanlinna order. The Russian mathematician, A. E. Eremenko has constructed a meromorphic function , as an asymptotic value. In his of zero Nevanlinna order with every value in @ book [170], Nevanlinna constructs, for any positive odd integer A, a meromorphic function fx of Nevanlinna order 1 which has 2X finite asymptotic values. Note 2.2. The value n/2 in Theorem 2.1 is sharp as is shown by Examples 1.3 and 1.4. Furthermore, by the remarks of the preceding note, none of the finite asymptotic values in these examples can be directly critical transcendental singularities of the respective inverse functions. The question arises naturally, however, whether if n is the number of finite < co, something asymptotic values of an entire function f and lim_ ,, more can be said. Heins [ I l l ] showed that, in this case, f in fact has order n/2. Heins' results were refined by Kennedy [127]. Both Heins and Kennedy state theorems for subharmonic functions in general instead of just for log If (z)l. Kennedy [127] showed that these and other specialized improvements of Theorem 2.1 due t o Ahlfors and MacIntyre were essentially best possible. f i r t h e r refinements are due to Hayman.

'w

Note 2.3. The inequality of the "claim" above is one of a type generally known as "Wirtinger's Inequality and related results" even though the particular inequality

5. Asymptotic Values

176

ascribed t o Wirtinger was known, and in more general form, some time before that ascription. For surveys of this kind of result, see Mitrinovic [160]; Beckenbach and Bellman [18]; Hardy, Littlewood, Pdlya [98]. The result of the claim is 8257 in this last book. Mitrinovic gives a history of such results. The proof given above of the claim is quick, but not well-motivated. Such inequalities are connected with so-called "isoperimetric problems", and another easy proof of the claim along these lines is: By the classical isoperimetric inequality, if L is the length of a simple closed Jordan curve C and A the area of the bounded region defined by C, then L2 47~A with equality holding only for a circle. Applying this to the function y(8) of the claim, considered as mapping [O,T] onto C,

>

Hence, by Schwarz's inequality,

and the result follows. Note 2.4. At first glance, quite a bit seems to be "thrown away" in the final inequality estimations of the proof of Theorem 2.1. It is perhaps worth noting, therefore, that the estimate (2.10) is really essential to the proof. If, for example, 0, one would get instead of one made the trivial estimate in (2.5) that j'uid8 (2.101,

>

and this leads only to the estimate

instead of (2.16). One may also note that the

term is essential to the truth of (2.10).

Note 2.5. Theorem 2.1 was conjectured by Denjoy in 1907; he proved it for straight line paths. In 1921, Carleman proved a theorem for arbitrary paths in which n/2 was replaced by and Ahlfors [5] finally obtained the result with n/2. In 1933, Carleman noted the Cartesian analogue of the inequality (2.10) and that it could be used to prove Theorem 2.1 [39]; replacing conformal mapping arguments of Ahlfors. The above proof is essentially a variant of this proof of Carleman's due t o Dinghas. Ahlfors' proof depends on an inequality which has proved very useful in analytic function theory and has become known as "Ahlfors' distortion theorem", although

9,

5.2. The Denjoy-Carleman-Ahlfors Theorem

177

it is only one of two "main inequalities" dealing with conformal mapping proved in his paper. For an English version of a proof of Theorem 2.1 along Ahlfors' line, see Chapter VII of F'uchs (781. Another proof of Theorem 2.1 was apparently found independently by Beurling in 1929 and published as part of his thesis [21]. However, in 1928, Grotzsch [93] also solved a conformal mapping problem which, as noted by MacIntyre [152], leads to a proof of Theorem 2.1 simpler than Ahlfors'. The ideas of MacIntyre's proof are quite different from those above, and presented here as one example of an approach to the theorem in which the use of conformal mapping is explicit, as well as providing an improvement on Theorem 2.1. The ideas of GrMzsch, Ahlfors, and Beurling, indeed were seminal for what is now known as the "method of extremal metrics" in geometric function theory which was developed by Ahlfors, Beurling and others, but this is not discussed in this book. (It would appear that the phrase "extremal metric" makes its first appearance in Beurling's cited dissertation.) Theorem 2.2 which follows is purely a result in conformal mapping, and might be in Chapter 1, but that it is somewhat technical and has no rationale there.

Theorem 2.2. Let { S k : k = 1,.. . ,n ) be a set of n simply-connected regions which are disjoint and contained i n the annulus {z : 1 < Izl < R < oo), bounded by Jordan curves, and such that B d S k has a non-degenerate arc i n common with each of the circles C ( 0 , l ) and C(0, R) (see Diagram 5.1). If the S k are conformally mapped onto the rectangles with sides a k , bk in such a way that the arcs S k n C ( 0 , l ) and Sk n C(0, R) map onto the sides of length an (where 0 is a vertex, and one side of length a k lies along the positive imaginary axis; one of the length bk along the positive real axis) then,

where L is a lower bound to the length of the curves which the pre-image of the sides of length bk map onto under w = logz.

5. Asymptotic Values

Diagram 5.1 Shaded regions are examples of regions S k .

Proof. Note first that univalent functions hk carrying out the proposed mappings with prescribed vertices for the rectangles exist by virtue of the Riemann mapping theorem and Schwarz-Christoffel formula (see Theorems 1.5.1, 1.5.4). Slit the annulus along the boundary of one of the S k ,and map the corresponding simply-connected region onto the w-plane by w = logz, w = u + iv. Then the bounding circles are mapped onto lines of abscissa 0 and log R, and the Sk onto "strips" S i between these lines (see Diagram 5.2).

I

I

log R

0

Diagram 5.2 Typical S i

>

u

5.2. The Denjoy-Carleman-Ahlfors Theorem

179

<+

Let gk be the inverse of h k . Write hk(z) = C = iq. Then gk maps the rectangle Rk : {< = J ill : 0 < J < b k , 0 < 11 < ah) onto Sk (see Diagram 5.3).

+

Diagram 5.3 and fk = log gk maps Rk onto S;. Let Ak be the area of the strip S;. The open strips S; are all contained in a rectangle with sides of length log R and 2w and they do not overlap (though two of them may have a common boundary). Hence

On the other hand, by Schwarz's Inequality,

and the theorem follows. To apply Theorem 2.2 to the problem of asymptotic values of functions of finite order, we need to specialize the ak and bk (as we may), and reformulate the problem in a way suggested by Theorem 3.3.4 (see also, Note 2.1).

Theorem 2.3 (ETheorem 2.1). Iff is an entire function bounded on n continuous curves which divided the plane into n unbounded simply connected regions, in each of which f is unbounded, then lim -

r+co

1% M ( r , f rnI2

> .

5. Asymptotic Values

180

Proof. In Theorem 2.2, we take (as clearly as we may) the bk to be all the same, say bk = logR*, C;=l ak = 27r, and L subject to the corresponding restrictions. We also take the Sk to be the G k of the earlier proof of Theorem 2.1, truncated by the circle C(0, R). Then Theorem 2.2 says that L2 < logR log R* Furthermore (see Diagram 5.2), L is a lower bound to the lengths of continuous curves extending from the line u = 0 to the line u = log R and so L 2 log R. Hence, with the above prescriptions, we get log R* 2 log R

.

(2.18)

Clearly (again by the Riemann Mapping Theorem and Schwarz-Christoffel formula), we can also consider with no loss of generality, the rectangular images of the truncated G k as "stacked" contiguously to a height of C;=,ak = 27r along the imaginary axis. If we do this, with the above prescriptions, we have mapped the annulus A = {z : 1 < lzl < R ) onto a rectangular region R with vertices at 0, log R*, log R* 2i7r, 2 i r , in such a way that the curves BdGk n A are mapped onto straight lines parallel to the real axis. If we now apply the exponential map, this rectangle in turn, is mapped onto the annulus A* = {z : 1 < lzl < R*), and the straight lines go onto radial lines. thus, composing the conformal map of A onto R with the exponential, we have a conformal map of the annulus A onto the annulus A* in which the curves BdGk n A have been "straightened". The crucial thing for us about such a map is expressed by (2.17), namely,

+

Now assume, contrary to the assertion of the theorem, that lim 7'-+w

1% M ( r ) = 0 . 7412

A contradiction will be obtained by showing that f is bounded in one of the regions.

Assume with no loss of generality, that If (z)l 5 1 on the n curves and on C ( 0 , l ) . (We may divided by an appropriate constant, if necessary.) The condition (2.18) means that there is an infinite sequence of circles C(0, Rj) such that log M ( R j )

< r j ~ ~, l 2

(2.20)

where {ej) is a decreasing sequence of positive numbers and r j -+ 0 as R j + co. We now map the annulus A j = {z : 1 < lzl < Rj) onto the annulus A; = {z : 1 < IzI < R;) as in the preceding paragraph. Clearly, for at least one of the regions G k ,say G, between two adjacent curves, there will be an infinite number of values of j for which the corresponding radial

5.2. The Denjoy-Carleman-Ahlfors Theorem

181

<

lines in the djform an angle 27rn. We now need an inequality on the modulus of a function analytic and bounded in a simply-connected region contained in a sector of B(0, I ) , which will then be applied to a sequence of functions where j now runs through the infinite sequence such that the corresponding angles are 27rn.

<

Claim: Suppose g is analytic in a region R entirely contained in the sector S = {w : 0 argw 7r/p, lwl 5 1). Let A be the circular arc part of BdS, and suppose R is such that A BdR. Suppose further Ig(w)l M for w E R , and Ig(w)l m < M for w E B d R - A. then for any w = reie E R,

< <

<

log lg(reie)1

<

5 log m + - log IT

($) Arctan (rp) .

(2.21)

<

<

Proof of Claim: C = $(w)= maps the semidisk {w : 0 Iwl 1,I m w 2 0) onto the quadrant {C : 0 a r g c 5 ~ / 2 )with the diameter of the semicircle going onto the positive real axis and the circular arc onto the positive imaginary axis. Replacing w by wp, $(wp) maps the sector S onto the quadrant. Furthermore,

<

(E)

ICil

i

= 1 if argC = 0, and ICiI = e-"I2 if argC = n/2. Hence ~ ( w = ) is analytic in S except at the "corners" and Ix(w)l = 1 on the radii (except for the endpoints), Ix(w)l = e-"I2 on the circular arc A (except for the endpoints). Hence, if $(w) = m(X(w))$lOg(E). Then I$(w)l = M on A - {endpoints of A); I$(w)l = m on BdS - A. Furthermore, with w = reie, 0 < r < 1, Log I$(reie)l = Log m

+ -T2 Log

= Log m

+ -7r Log

= Log m

+ -7r2 Log

( )

(1 1- ,,.pei0p 1 - r 2 p + 2irp sin $p

(z)arg( (z) ( ) (F) (s) ($) Im Log

J1-r~eie~12

2rp sin $p 1- 7 - 2 ~

It is easy to see that for fixed r, 0 < r < 1, this last expression has its maximum when cos$p = 0; SO,since M > m, when sin $p = 1. Hence, Log /$(reie)l 5 Log m

+ -2 Log

= Log m

+ -7r Log

since Arctan

(&)

7r

= 2 Arctan y.

Arctan

Arctan rp ,

5. Asymptotic Values

182

Finally, by hypothesis, on A, Ig(reie)l 5 M = J4(reie)l,and on BdR - A, Ig(reie)1 m l~#~(re") 1 (minimum modulus principle). Hence on BdR, 1, and so by the maximum modulus principle, throughout R,

< <

Log Ig(reie)l

5 Log Iq5(reie)l 5 Log m + - Log ?I-

(E)

Arctan (r")

,

which is the claim. In the claim, take p = n/2, w = z/Rj', and g(w) = f(R;w), where z is a fixed, but otherwise arbitrary point of G, and j runs through the infinite sequence of integers prescribed earlier. Then, by our assumptions, m 5 1, and so on, using (2.19) we have, by (2.20), Log 1 f (z)l

4 < ;sj~;"

Arctan

<

By (2.17), R j R: and so letting j + co through the prescribed sequence of integers, RT + m, ~j + 0, and so we get LogIf(z)lIO;

If(zIl51.

Since z was an arbitrary point of G, f is bounded in G, a contradiction which proves the theorem. Note 2.6. Only the trivial inequality L 2 log R was used in deducing Theorem 2.3 with the help of Theorem 2.2. However, if we make further conditions on the paths, then stronger results can be obtained, by the same proof. For example, suppose the asymptotic paths of finite determination yr, which form the boundaries of the regions Gk deviate so much from straight lines that along each one of them lim]zl+w 1 > 0. ZEY~

*

Then it is easy to see that L in the proof of the theorem can be taken 2 m Log R (if w = Log I, og I4 = whence R* 2 R('+*~)),and hence we get, repeating the above proof that

z,

d

lim 1% M(r) TTO T?(l+X2) O

>

from which it follows that the lower order and so a fortiori, the order of f is 2 (1 X2)n/2. This result is also due to MacIntyre, a similar result with lim in both places replaced by lim was found by Ahlfors. Kennedy (see Note 2.1) showed both of these results to be the best possible. Although entire functions of finite order p cannot have more than 2p finite asymptotic values, no limitation a t all can be placed on the finite, asymptotic values of a general entire function of infinite order, as the following example, due to Wilhelm Gross [92], dramatically shows.

+

5.2. The Denjo y- Carleman-Ahlfors Theorem

183

Theorem 2.4. There is an entire function for which every complex number is an asymptotic value.

Proof. Let g(z) = !j -$J e-c2d< - e-" + 1) (where the integral is along a straight line). We will need the fact (from the theory of the I?-function) that for a fixed 0

(

lz

Zlim -+OO arg Z=B

e -C2d<=

ir ( 15 ) -- Y25 , -

ify<e<;

(

ify
The function g has the following properties (which motivate its choice):

(i)

= 0;

(ii) If z = rei@,and 0 < E as z + m; (iii) If z = rei@,and 0 uniformly as z + m; z

< ~ / 4 then , for 141 5 7r/4 - E , g ( z ) + 1 uniformly

< c < n/4,

14 - nl

then for

< n/4 - E , ~ ( z+) 0

(iv) g(z) is real on the positive real axis and goes monotonically t o 1 as + m along the positive real axis.

(v) On the negative real axis, g takes on only real values whose absolute value is < 1 and they are all negative. As x

+ m along the negative real axis % J,"e-c2d(

0 to -1, and so does e-"'

- 1, and

% J,"e-c2d(

1

z = -- .) 6

Diagram 5.4

decreases monotonically from

-

e-l2

+ 1 has a minimum a t

184

5. Asymptotic Values

As a consequence of (iii) and (v), there is an region ro= {rei# : 14 - 7rI 5 $ ,O 5 r < m),

As a consequence of (ii), in the angular region

e > 3 such that

in the angular

rl = {rei@: 141 < $ ,O 5 r <

001

7

for some finite M. Let L? = .loand rl = rp) then, by (ii) and (iv), we can find a non-decreasing sequence of integers {e, : u = 0,. . . }, so that if I??) denotes the angular region

then for z E I??),

Setting correspondingly,

clearly in r t ) ,(2.22) continues to hold. For convenience, we now slightly change notation. Define ro Go,rl r GI. Each of Go and G1 contains an angular region which is mapped by

onto I'F). Call these angular regions Goo and Glo respectively. Similarly, each of Go and G1 contains an angular region mapped by K onto I'I1); call these Gol and Gll respectively. Each of Goo,Glo,Gal, G1l contains an angular region which is mapped by

.

K l ( z ) = zz

2'0+'1

onto I??); call these GOOO, Gloo, Golo, Gllo respectively. Similarly, each of Goo,Glo, Gal, Gll contains an angular region mapped by K1 onto I??), and we denote these by Go01, Gl01, Go11, G111. Continuing in this way, each given angular region GjOj,..,j,(j, = 0 or 1) contains an angular region which maps onto rr") under

5.2. The Denjoy- Carleman-Ahlfors Theorem and these are denoted

G3031 . . ...3hO . ;

and each GjOjl,.,jhcontains an angular region which Kh maps onto Fib+') and these are denoted Thus, we obtain a family Gh+1 of 2h+2 angular regions, each of which is marked by a distinct index consisting of a string of 0's and 1's of length h+ 2. Furthermore, every two regions with the same first h 1 members of the string are contained in the number of the family Gh indexed by that string of length h + 1. Taking the intersection of all such regions, we obtain a set of rays, each of which ...j,... is uniquely determined by a countably infinite string of 0's and 1's: if Sjojl is such a ray, then the portion jOjl j2 . . .jn of the string indicates that this ray is C Gjojl...j, E Gn. Further, note that if X is a positive constant, then replacing z by Xz, the system of families Go, GI, 62,. . . , and of rays S is mapped onto itself. 2'~+'l+"'+eh-l Consider now, the function g (iz ) in Gjojl...jh. 2'0+'l+"'+'h-1 e~+e~+...+t,,-~ E rp) , and so g iz2 If jh = 0, then iz )I
+

I(

)I

I

(

The functions gh have the following properties: (vi) gh(0) = 0 (from (i)). (vii) In an angular region E Gh whose indexing string ends in 0, gh(z) + 0 uniformly as z + co and, moreover,

(from (ii) and (2.22)). (viii) In an angular region E Gh whose indexing string ends in 1,gh(z) + 1 uniformly as z + co, and, moreover

(from (ii) and (2.23)).

(ix) We may choose Ah

SO

that lgh(z)l < 2-h in B(0, 2h) (by (vi)).

5. Asymptotic Values

186

{a,)

be any well-ordering of the numbers {t) and {it) where t is an Now let arbitrary member of a countably dense subset of (-1,l) e.g. t rational. Given an arbitrary complex number A Bi, there is a subset {P,,) of {a,) such that Cr=opv, converges absolutely to A B i (in fact, there are 2 N such ~ subsets). Finally, consider the function

+

+

Given a positive integer n , for z E B(0, 2n), since

< 1, we have by

(ix)

Hence, for z E B(O,2"), y(z) is representable as a uniformly convergent series of analytic functions, and so is analytic. Since n was arbitrary, it follows that y is entire. Now given a complex number A Bi, let

+

+

be an absolutely convergent representation of A Bi, and write the function y(z) as ~ ( z= ) 71(2) + 7 2 (2) where 00

+

where the indices vn in yl(z) are the same as in the representation of A Bi, and the indices tn are the others. Let jk = 1 if k E {vn : n = 0 , 1 , 2 , . . . ), and jk = 0 if k E {tn: n = 0, 1 , 2 , . . .). Then, on the ray Sjoj ,...j ,... = S*,say

and since (viiii),

Cr=olBu, 1 converges, the series converges uniformly on S*and hence by 00

On the other hand, similarly by (vii), on S*,

5.2. The Denjoy-Carleman-Ahlfors Theorem

187

and the last series converges since p < 1. Hence the series for y2 converges uniformly on S*,and so again by (vii) lim y2(z) = 0 . 1%1+03

Hence, the arbitrary complex number A

+ Bi is an asymptotic value for y(z).

CI

Note 2.7. Since y(z) is entire, oo is also an asymptotic value. In fact, it is clear that in the definition of y,we can define the sequence {e,) so that in (2.24) we may add the prescription 1 argg(z)l < -; whence in (2.25) we have arggh(z)l < 7r/4. Defining y in this way, we obtain as Gross goes on to show, a function for which every value E C, is an asymptotic value along uncountably many non-contiguous asymptotic paths. The function S : e-S2d[ was earlier used by Iversen to exhibit an entire function with 2 N distinct ~ asymptotic values. This proof is easier than the above, but contains the same basic ideas. For an exposition of Iversen's result, see P6lya and Szegi5 [200] Section IV, Problems 189 and 191.

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Chapter 6

Natural Boundaries In the preceding chapters, we have been usually concerned with functions analytic or meromorphic in the whole plane. In this chapter, we will largely be concerned with what might be termed the opposite phenomenon: functions which have no analytic continuations beyond the domain of validity of their original definition. However, just as in previous chapters, no hard and fast separation can be made and, as is to be expected, results about "non-continuability" are tantamount t o results about analytic continuation. Again, this is only an introduction. the reader who glances a t Bieberbach's Analytische Fortsetzung [22] or the notes t o the relevant papers in the first volume of P61ya1sCollected Papers [I931 will realize how much has unavoidably been omitted.

6.1

Natural Boundaries-Some

Examples

We recall

Definition 1.1. Suppose f is analytic in a region R, C E BdR, and f can be analytically continued over C , then (' is called a "regular point" of f . If f cannot be analytically continued over I , (' is called a "singular point" of f . Definition 1.2. Suppose C r = oanzn has radius of convergence r , and in B(0, r ) , 00 En,., anzn = f ( z ) . Then, if there is no point of C(0,r) over which f can be analytically continued, C(0, r ) is called a "natural boundary'' for the function. Definition 1.3. Suppose an analytic function is defined somehow in a region R. If there is no point of BdR over which f can be analytically continued, then B d R is called a "natural boundary" for f . Several examples will elucidate the definitions. When dealing with power series with a finite radius of convergence, clearly, we may without loss of generality, always normalize so that the radius of convergence is 1, and we will do this without further comment.

6. Natural Boundaries

190

Example 1.l. For z E B(0, I), let f (z) = Cr=ozn!. Then the series has radius of convergence 1. We now show that C ( 0 , l ) is a natural boundary for f . Let p, q be relatively prime positive integers, ( = e2"iplq, and suppose 0 < r < 1. Then

say. fi(r() is a polynomial, and so as r + I-, tends t o some finite limit. Also, for n 2 q, qln!, whence In!= 1. Hence, f2(r() + co as r + I-. It follows that f (rc) + co as r + 1- and so ( = e2"ip/q is a singularity of f (if f could be analytically continued over then by continuity, the limit would exist and be finite). Since the points {e2"ip/q} are dense in C(0, 1)) the result follows.

c,

Example 1.2. For a E B(0, I ) , let F ( z ) = Cr=,z2". Then the series has radius of convergence 1. We now show that C ( 0 , l ) is a natural boundary for f . Clearly, as z + 1- along the real axis, f (z) + co and so the point z = 1 is a singularity of f . Now f (z2) = Cr=lz2n+1= 00 z2 = f (z) - z2, or

From (1.1) since z = 1 is a singularity o f f , so are the points where z2 = 1, viz. 1 and -1 singular points. Replacing z by z2 in (1.1)gives

and so by (1.1), f(z) = f(z4)

+ z4 +

z2

,

and it follows that i and -i are also singularities of f . Continuing inductively in this way, we find that the points p odd, ic a non-negative integer are singularities o f f . But these are dense in C(0, I), which is thus a natural boundary. Although Example 1.1 uses a number-theoretic argument, and Example 1.2 a functional equation t o show that C ( 0 , l ) is a natural boundary, they both have many zero coefficients, and it might be thought that this is a characteristic of power series whose circle of convergence is a natural boundary. There is some truth in this (see Theorems 2.1 and 4.3); however, even one zero-coefficient is not necessary as shown by

(31,

6.I . Natural Boundaries-Some

191

Examples

Example 1.3. Let ~ ( n = ) Cdln 1 (that is r(n) = the number of positive divisors of n). For z E B(O, l ) , let C r = l f(z) = C r = l r(n)zn . Since 1 5 ~ ( n <) n , the series has a radius of convergence 1. ) Cdln 1= We now show C ( 0 , l ) is a natural boundary for f . Clearly ~ ( n = Cpv=n p,v 1 (the number of ways of writing n as a product of two factors). Hence

(since by absolute convergence in B(0, l ) , we may rearrange terms of the series a t will). Hence, for z E B(0, I ) ,

Let

c

= e2"ip/q, p and q positive relatively prime integers, and 0 From (1.2), we can write 03

f ('0 =

" C 1- ( r < ) ~ C u=1 u=l +

(rcIp 1-

= El

< r < 1.

+ C2 , say.

In E l , p = mq, m a positive integer, and so (rc)p = rm4;hence, since 0

< r < 1,

which tends t o m as r -+ I-. Hence lim,,l- (1 - r)E1 = m. If q 1. p, then

11

- (r()pI2 = 1 + r 2 P - 2rp cos

4

> 2rp

since q '( p, q 1. p, and sin2 x is periodic with period decreasing in (7r/2, 7r) .

.ir,

increasing in (O,7r/2), and

6. Natural Boundaries Hence, since 0

< r < 1,

so is bounded as r + KI. So (1 - r)(C1 C2) = (1 - r ) f (rc) + oo as r + I-. Hence, not only is every point e q a singularity o f f , but even more, f (r() O(&) as r + I-.

+

*

#

Related t o some aspects of Example 1.3 is

xr==,

Example 1.4. Consider the series $ -. In the first place, the series converges for z E B(0, I), since if lzl in Example 1.3

< 1, then

as

and the last power series has radius of convergence 1. On the other hand, if lzl > 1, then putting z = $, we have

Hence, the series also converges for lzl > 1. But, arguing as in Example 1.3, it is easy t o see that every point e2"ip/q is a singularity for the function represented by the series in B ( 0 , l ) . Thus the series Cr=l represents two distinct functions, one, say f l (z), analytic in B(0, I ) , and the other, say f ~ ( z ) analytic , in C, - B ( 0 , l ) . C ( 0 , l ) is a natural boundary for fl(z). Furthermore, if lzl > 1, fi(z) = -1 - f l ($).

&&

Note 1.1. Series of the form Cr=,an- "" which occur in Examples 1.3 and 1.4 are called Lambert series. Although named for an 18th century mathematician, the first serious investigation of them seems t o be a paper of Konrad Knopp [130]. They play a role in certain number-theoretic problems (as demonstrations in the Examples might lead one to guess).

6.1. Natural Boundaries-Some

Examples

193

It is easy to see, along the lines of Example 1.4, that a Lambert series 00 "" is convergent for z E B ( 0 , l ) if and only if C r = = ,anzn is converC n = l angent, and is convergent for z E C, - B ( 0 , l ) if and only if C= :, a, is convergent. See also, Example 5.3.

Note 1.2. Although C ( 0 , l ) is a natural boundary for the functions of Examples 1.1-1.4, this does not mean that the various series in question might not converge a t some point, say e 2 " i ~where y E (0,l) is irrational. This is because while divergence to co a t a point on the circle of convergence of a power series indicates a singular point, convergence does not necessarily indicate a regular point (e.g. Cr=, $ and z = 1). Neither, for that matter, does simply divergence indicate a singular point; (e.g. C r = = ,(-l)nzn and z = 1). I do not know whether, in fact, any of the series in question converge for some point e2niv, y irrational. In Examples 1.1 and 1.2, 1 is a limit point of the coefficients of the series 03 En==, anzn as n -+ co. In Example 1.3, co is a limit point of the coefficients. One might hope that if, for the coefficients {a,) of a power series with radius of convergence one, a, -+ 0 as n + co, then there might not be so many singularities on C ( 0 , l ) . Theorem 1.1. I f f (z) = Cr=oanzn has radius of convergence 1, and lanl n + co, then f (z) does not have a pole on C ( 0 , l ) .

+ 0 as

Proof. If f had a pole on C(0, I ) , say a t eie, then lim (z - eie)f (z) # 0 z+e"

I4<1

(the limit may be infinite). But, for 0 < r < 1, if lanl

If

(reie)[ =

+ 0 as n + co,

lo I

03

C anrneine < C lanlrn = o n=o

(A).

Hence, lim (z - eie)f (z) = 0 z-+e'@ radially

,

a contradiction which proves the theorem. However, poles are isolated singularities, and the problems we are discussing involve - non-isolated singularities. In fact, while Theorem 1.1 says that if limn,, lan/$ = 1 and limn,, a, = 0, then Cr==,anzn cannot have a "good" singularity, it turns out that the function represented by such a series in B ( 0 , l ) can have every point on C ( 0 , l ) as a singular point, as is shown by the following example.

6. Natural Boundaries

194

Example 1.5 (Lusin). For m a positive integer, let gm(z) = 1+ a + - . . + zm-l = -2nib imkgm (e-sz) . We may note that gm (for z # 1) 1-zm , and let hm(z) = is a polynomial of degree m - 1, and hm is a polynomial of degree m(m- 1)+m- 1 = m2 - 1. Define f (z) by

The motivation for the exponent of z is that it is

C:=<'

k2, and so

Hence, each polynomial in the definition o f f follows the preceding one with no overlapping of terms. Furthermore, each coefficient of a term of hm has modulus 1, since no two terms of any gm overlap in the definition of h,. It follows that lim lan/ = lim m-'l2 = 0 and

n+ w

m+m

lim

n+w

lanl$ = 1 .

We will prove that limn+o anein@diverges t o oo for every real q5 E [O, 2,). Suppose, on the contrary, that the sequence of partial sums C:=o anei@nwere bounded for some q5 as p + co,then there would be a constant M such that

We now chose a non-negative integer K , 0

5 K 5 m - 1, as follows:

(i) If q5 = 0, K = 0. (ii) If the nearest integer to (iii) Otherwise, K =

2 (i.e. [2 +

I

) is m, then K = m.

[$+ J] < m - 1.

We will show, that for k = K ,

contradicting (1.3) and so proving that every point of C ( 0 , l ) is a singular point of f.

6.1. Natural Boundaries-Some

Examples

To prove (1.4): In case (i), $ = 0, K = 0, and gm(l) = m > In case (iii), if the nearest integer t o is 5 m - 1, then since I sin21 x real,

2

By the choice of K, 0 5

I?

F.

( 4 - %)1 = n 1 2- [$+

for real x, I sinxl = sin 1x1 and

$1 I <

a/2. Hence, (since

% is monotone decreasing in -

(3) T,

< 121 for

[0,n/2]), the last

expression in (1.5) is 2 m = as claimed for case (iii). In case (ii) a similar argument works, the last expression in (1.5) being

1

sin

y (2n - $)

p ( 2 n - 4)

Sierpinski slightly modified Lusin's example t o exhibit a power series with radius of convergence 1 which converges a t z = 1, but diverges to oo a t every other point of C ( 0 , l ) : Example 1.6. Let f (z) = CrT0 anzn be the series of Example 1.5. For z E B(0, I), define g (z) by g(z) = (1- I) f (z2) =

C anzZn- C anzln+' = C

n=O

n=O

a T / 2 ~-r

r=O r even

CQ

Since 1a.l 4 0 as r -+ m, C z o ( - l ) r a [ g l converges. But if (1 - () C r = o an<2ndiverges to m by Example 1.5.

C

.

< # 1 and I<[

adzT

= 1, then

Note 1.3. Examples 1.5 and 1.6 raises the following question: For what subsets E of C ( 0 , l ) does there exist a function g(z) = C r = o anzn, analytic in B ( 0 , l ) , such that the series converges on E and diverges on C ( 0 , l ) - E? E cannot be an arbitrary subset of C ( 0 , l ) since there are 2" subsets of C ( 0 , l ) and only CNO = C power series. Staniszewska [224] has proved the following theorem:

Suppose E is a subset of C ( 0 , l ) which is a Ga, set. Suppose E c D where D is an F, set, and has logarithmic measure 0 (i.e. given 6 > 0, there is a countable

196

6. Natural Boundaries

*

cover of D by "open segments" of C(0,l) of lengths tj, tj < 1, j = 1,2,. . . , such that CP1 < f). Then there is a function H(z) analytic i n B(0, I), and continuous o n B(0, I), such that i f for z E B(0, I), H(z) = Cz=o anzn, then the series Cz=o anei+n converges for ei$ E E and diverges otherwise. Despite Examples 1.5 and 1.6, the condition a, += 0 is a useful one. For example, if f (z) = C= :o anzn for z E B(0,l) where limn,, lan\5 = 1, and lim,,l- f (reie) = oo, then eie is a singular point for f ; hence, if eie is a regular point for f , lim,,lf (reie) # oo. But, as already remarked, this does not mean that C:', anzn converges for z = eie (consider C:=o(-l)nzn, and z = 1). However, if an += 0 as n + oo, Canzn does converge at regular points of f on C(0, l ) as a famous theorem of Fatou and M. Riesz demonstrates. The Fatou-Riesz Theorem shows that if for z E B(0, I), f (z) = C r = oanzn, where a, + 0 as n += oo, and the series fails to converge at a set of points dense in C(0, I), then C(0,l) is a natural boundary. E B(0, I), f (z) = C r = o anzn, where limn,, la,l* = 1, and limn,, a, = 0. Then the series converges at every regular point of the function f on C ( 0 , l ) and, i n fact, uniformly on every closed arc of C(0,l) composed of regular points.

Theorem 1.2 (Fatou-Riesz). Suppose for z -

Proof. Suppose zo is a regular point off on C(0, I), then f is analytic in some small neighborhood of zo, and so in particular on some open arc of C(0,l) containing zo. length of R2B* = length of RIA* = q

Diagram 6.1 Let A and B be the endpoints of a closed arc I' of C(0,l) on which f is analytic. Then there is an open arc I" of C(0,l) with endpoints A', B', say such that I" > I',

6.1. Natural Boundaries-Some

Examples

197

and f is analytic on I". Consequently, there are points A* and B* on C ( 0 , l ) with A* between A and A', B* between B and B' such that f is analytic on the closed arc I?* with endpoints A* and B*. Then we can draw radii from 0 to A* and B* and extend them a small finite distance, say q to points R1 and Rz say, such that f is analytic on the closed circular sector defined by S = {z : 0 I lzl I 1 q, arg A* 5 arg z 5 arg B*). (See Diagram 6.1.) a n t n , and Let Sk(z) =

+

c:=~

Then gk is analytic in

S, and it is enough to show that lim gk(z) = 0 for z E B d S ,

k-tm

to prove the theorem. For, if (1.6) is true, then, by the maximum modulus theorem, lim gk(z) = 0

(1.7)

k-tcn

for z E I'. But, for z E I?,

where L is the positive minimal distance from lim

k-tw

If

r to B ~ S and , so we would get

(z) - Sk(z) 1 = 0 uniformly on I? .

To prove (1.7), since limn,, a, = 0, given 6 > 0, we may choose no = no(6) so that for n > no,lanl 5 6. Let M =max{lf(z)l : z E S ) . Clearly, limk,, gk(0) = limk + , ak+lA*B* = 0. Now there are three cases: (i) If z E either the line OR1 or the line OR2 and 0 < lzl < 1, then for Ic > no, lf(z) - Sk(z)l I Cz=k+l lanl lzln I 6 C r = k + l IzI - 1-lZ1 and so from (1.6),

c,

since in this case one of lz - A*/, lz - B*l will = 1 - lzl and the other will be <1+1=2. (ii) If z E the line ORlU the line ORz and 1 5 lzl 5 1 q , then for any k = 0 , 1 , 2 , ...

+

6. Natural Boundaries

198

+

+

where M * = M Czo lanl (1 rl)n depends only on M, laol, .. . , lano1, and 7. Hence, for these points z, from (1.6),

since now one of 1z - A*I, lz - B*I will = lzl - 1 and the other will be (1 7) = 2 277. F'urthermore, since 1 5 lzl 5 1 q in this case,

+

+

< (1 + 77) +

+

and so, we get

+

for z E the lines OR1 U OR2 and 1 5 [zl 5 1 77. (iii) If z E the circular arc between R1 and R2 (on which lzl = 1 q ) , then, with M * as above,

Furthermore, for such z, 1z - A*I and lz - B*I are both get from (1.6),

+

< 2(1+ 77) and so we

Hence, since 6 was arbitrary > 0, from (1.8), (1.10), (1.11) and our observation , gk(2) = 0, and so the theorem. about gk(0), we get that for z E B ~ Slimk+, Suppose for z E B(0, I), f (z) = C r = o anzn. The distinction between convergence of Cz=oanzn and regularity of f a t a point of C ( 0 , l ) is sharply expressed by the following example of Losch [I481 of a power series representing a function f analytic in B ( 0 , l ) which diverges to oo at a set of points dense on C(0,l) (and so C ( 0 , l ) is a natural boundary for f ) but also converges at a set of points dense on C ( 0 , l ) (none of which can be regular points of f).

Example 1.7. Define the power series

Cr=oanzn, by

6.1. Natural Boundaries-Some

Examples

199

Let Pn(z) = ( g ( l + z2")) . Since the degree of Pn(z) is (2n + 1)2~",and the least exponent appearing in the polynomial Pn+l(z) is 22"+1 > (2n + 1)22n for n 1, each term of the series comes from one term in one of the polynomials Pn. Furthermore, the largest coefficient of Pn(z) is 5 !(,$11)2-22n which on , C is a constant. Hence, if at, where using Stirling's formula is 5 ~ 2 - ~ " - 'where l= 22" ~ 20 5~ T 5 , 22", denotes a non-zero coefficient of the power series, then lime,, lael* 5 1, and so the power series has radius of convergence at least 1. k Let sk(z) = En=o anzn and Sk(z) = E nk = l Pn(z), then if 22" 5 .t 5 22Y+1- 1, and lzl = 1, we have 22n

>

+

Hence lime,, se(z)will converge or diverge for z E C(0,l) according as lim,,, S,(z) does. Now, if C is a 2,th root of unity where m is a positive integer, then 22"

= 1 n. for n 2 m, and hence S,,(C) = C:=, Pn(C) diPn(C) = ; ( zr(l + C a n ) ) verges to +m as v + ca. On the other hand, if q is a 3,th root of unity, 77 # 1, m a positive integer, then letting {r], : p = 1,2,. . . ,3, - 1) denote an ordering of the non-real distinct 3,th roots of unity, we have that r12" for each positive integer n equals some one of the r],. Hence

Thus, setting q, = 2/2(1+ cos 27r . 3-,)

4, we have

and so, since q, < 2, S,(q) = C:=l Pn(q) converges as v 4 oo. Thus C;=, anzn converges at a dense set of points of C ( 0 , l ) and diverges at a dense set of points of C(0,l).

Note 1.4. As will be seen in Section 2 in particular, functions with large stretches of zero-terms in the power series expansion formed as in Example 1.7 by expanded sequences of non-overlapping polynomials, play an important role in the considerations of this chapter. Other examples of the phenomenon of Example 1.7 of "power-series type" were found by L. Neder [167]. Note 1.5. Fatou originally proved only that the series in Theorem 1.2 converged in every regular point if an 4 0. The uniform convergence on closed arcs, as well as the above proof, is due to M. Riesz [210].

200

6. Natural Boundaries

A proof of an entirely different sort, involving Fourier series, was given by W. H. Young [252] (see Sections llff). Young was, in fact, able to show that the condition of regularity o f f (z) at zo E C(0,l) is stronger than necessary for the theorem to be true, and it is enough to suppose, say that for some a > 0, is bounded as z zo. A version of this proof, which uses ideas from Fourier series, can be found in Titchmarsh [229], 218-220. The connection between the Fatou-Riesz Theorem 1.2 and Ostrowski's Overconvergence Theorem 2.2 below, was indicated by Erdos and Piranian and proved by Noble to hold under still weaker conditions like those in Young's proof [253]. See also Note 4.8 below.

I

I

Note 1.6. Another question which examples of this section suggest is: if, for 00 anzn, and limn,, lan[+ = 1, for what O E [O,2n) does z E B(0, I), f (z) = lim,,lf(reie) exist? If we add the natural assumption that f be bounded in B(0, I ) , the following well-known theorem of Fatou provides an answer: Suppose f is analytic and bounded in B(0, l), then for all O E [O, 2n) except possibly for a set of Lebesgue measure 0, limT,l- f (reie) exists. Fatou proved this theorem in 1906 and it was one of the earliest applications of measure theory to complex analysis. Another proof was given by CarathBodory in 1912 in which all one needs to know from measure theory is the theorem of Lebesgue that a function of bounded variation is differentiable almost everywhere. This proof is the one that seems to have found its way into the textbooks. The theorem has been the source of considerable further work. In one direction, F. and M. Riesz showed that if f is analytic in B ( 0 , l ) and bounded there, and if limT,l- f(reis) = 0 for a set of O E [O,27r) of positive measure, than f e 0. Priwaloff removed the condition of boundedness in this theorem. In another direction, the condition that f be bounded in Fatou's theorem can be relaxed, say to the boundedness of J" : log+ If(reis)ld8. However, examples have been constructed to show one cannot eliminate some sort of boundedness assumption on f . Note again, that one should not confuse the existence of the limit with the convergence of the series. For example, suppose for z E B ( 0 , l ) we take

then, lim,,lf (re") exists for every 8 except on the real axis, i.e. for 19 E (0,2n). However, C z = o zn does not converge for any 0 E [O, 2n). For proofs and further information, see Bieberbach [23], Vol. 11, Sections 111.7 and 111.8, Caratheodory [37], Vol. 11, Sections 310-313, Nevanlinna [169], Sections VII.3 and VII.4, and references there.

6.2. The Hadamard Gap Theorem and Over-convergence

6.2

The Hadamard Gap Theorem and

Perhaps the most famous of all conditions for a function to have a natural boundary is in the title of this section. Even though the result is contained in a more farreaching one due to Fabry, to be discussed in Section 4, nevertheless there is some point to discussing it separately. One reason is that the theorem allows an extension in a different and initially somewhat surprising direction, namely the phenomenon of over-convergence which will also be discussed in this section. We give here a direct proof of this theorem due to Mordell [164] which unfortunately may seem at first somewhat unmotivated (but see Note 2.1).

Theorem 2.1 (Hadamard Gap Theorem). Suppose f(z) = CT=oanzn, limn,, lanl$ = 1, and a n = 0 except when n belongs to a sequence {nk) such that nk+l > (1 + a ) n k , where a > 0. Then C ( 0 , l ) is a natural boundary for f .

Proof, The series C = :o anzn has radius of convergence 1, and so f has at least one singular point on C(0, I ) , which (by rotation if necessary) we can take to be 1 without loss of generality. Let a be real, 0 < a < 1, p be an integer > lla, and,

+

Now, if lwl 5 1, then clearly lzl 5 1, and in fact, if lzl = 1, then 1 = IwlPla (1 - a)wl 5 (a (1 - a))lwl, and so Iwl = 1, whence 1 = la (1 - a)wl, thus 1 = a2 (1 - a)2 2 a ( l - a)Re w, and so Re w = 1; whence w = 1. Thus, if Iwl 5 1, then lzl < 1 except that z = 1 when w = 1. Let

+ +

+

+

x 00

$(w) = f (z) =

an(awP

+ (1 - a ) ~ ~ ' ' ) ~

n=O

Since Izl < 1 if lwl 5 1, except for w = 1, then 4(w) as a function of w is analytic in B ( 0 , l ) except possibly at the point 1. Replacing n by n k in (2.1), we get

n=O

k=o

T=O' '

'

03

=

an,Qnk (w) k=O

, say, where Qnk(w) is a polynomial.

(2.2)

6. Natural Boundaries

202

+

- 1 > pa > 1, and so Since p > l / a , and nk+l > (1 a)nk, we have p (p 1)nk < pnk+l. Hence, the degree of Qnk < exponent of the smallest power in Q,,,, , and so in rewriting the expression on the right in (2.2) as Cr=obnwn, each bn comes from exactly one term in one polynomial Q n k . It follows that the series for 4(w) must have radius of convergence 1 since otherwise it would be convergent for some real w > 1 and thus Canzn would converge for some real z = wp(a+(l-a)w) > 1, contradicting 1 being a singular point for f . Since the series CF=o bnwn for $(w) has radius of convergence 1 and is analytic on B ( 0 , l ) - {I), 1 must be a singular point for 4. Replacing w by eisw in all the above arguments, we see that every point of C ( 0 , l ) is a singular point for 4; hence every point of C ( 0 , l ) is a singular point for f , as was to be proved.

+

(

)

Note 2.1. The basic idea behind Mordell's proof is a familiar one for finding singular points for a function defined by a power series, where we may take the disk of convergence to be B ( 0 , l ) and by rotation, if necessary, consider the point under discussion to be 1, namely: expand the function in a power series around some point on the real axis between 0 and 1; then 1 is a singular point, if and only if the new disk of convergence is tangent to B(0,l) at 1. Often, however, as in Mordell's case, a preliminary transformation makes the resulting formula simpler to apply. For example, a well-known test for singular points may be obtained as follows. Suppose f (z) = Cr=oanzn, where limn,, lan[ = 1. Make the transformation z = then B ( 0 , l ) is mapped conformally onto the half-plane {w : Rew > -1121, with C ( 0 , l ) going onto the line {w : Rew = -112). Let

s;

Then F is analytic in the disk {w : lwl < 1/2), and indeed the only possible singular point of F on C(O,1/2) is w = -112, which corresponds to z = 1. Furthermore, 1 will be a singular point of f if and only if -112 is a singular point for F. But for Iwl < 112, with the usual notation for generalized binomial coefficients,

6.2. The Hadamard Gap Theorem and Over-convergence

203

Thus, since the radius of convergence of the series for F is at least

3, we have

and also, (ii) z = 1 is a singular point for f (z) = Cr=oanzn if and only if

This criterion (ii) can also be used to prove Theorem 2.1. Clearly many changes can be run on the above sort of argument. For some further developments, see Bieberbach [22], Analytische Fortsetzung, sections 1.61.8. F'unctions whose power series have Hadamard gaps also surprisingly enough have a Picard property, at least if the gaps are large enough. This was proved by Mary and Guido Weiss [246] who showed that. There is a q > 1 such that if F ( z ) = CEO=, ckznb is analytic in B(0, I), Ickl diverges, and nk+l > qnk, k = 1,2,. . . , then for every w E C, there exist infinitely many z E B ( 0 , l ) such that F(z) = w. They remark that q can be taken to be about 100; however, the best lower bound for q is not known and it is conceivable that q need only be > 1. Compare also, Note 4.11.

$.

Example 2.1. For z E B(O, l ) , let f (z) = Zr=D=, By Theorem 2.1, f has C ( 0 , l ) as a natural boundary, even though the series converges absolutely at every point of C(0,l). A natural question is how "regularly" must gaps of the sort hypothesized in Theorem 2.1 occur in order for f to have C ( 0 , l ) as a natural boundary. In this connection, we have Example 2.2 (M. B. Porter). Let pn be the maximum of the moduli of the coefficients of the polynomial (z(1- z ) ) ~ " . Then in each of the polynomials (z(1z ) ) ~ "the moduli of the coefficients is 5 1 and at least one of them is actually equal to 1. Consider the function defined by the power series

&

Since the degree of &(z(l - z ) ) ~ "is 2.4" whereas the smallest exponent appearing in the polynomial &(z(l - z))~"" is 4"+' power series comes from one term in one polynomial.

> 2 .4",

each term in the

6. Natural Boundaries

cxJ z 2 n

-.

Figure 6.1: Natural boundary. An example of the series n=l

n2

Since by construction then, laml 5 1 for all m and laml = 1 for infinitely many m, the radius of convergence of C ~ = , a m z m is 1. Hence, the sequence of partial sums formed by adding the polynomials in sequence, is convergent in B ( 0 , l ) . However, the transformation t ( z ) = 1 - z does not alter the polynomial L ( z ( 1 - z ) ) ~ "Thus, . the sequence of partial sums so formed, also converges in the Pn disk B ( l , 1) = { z : 11 - zl < 1 ) which lies partly outside B ( 0 , I ) , and thus effects an analytic continuation of f into B ( 1 , l ) . Hence C(0,l) is not a natural boundary for f . Actually, since pn in fact is the maximum coefficient in the binomial expansion of ( 1 + z ) ~ "and , this clearly is less than or equal to the sum of all the coefficients, and a t least as large as the arithmetic mean of all the coefficients, we have

Em,

converges for ] z ( l - z ) 1 < 2 and diverges for Hence, the series l z ( l - z)l 2 2. The transformation +(I - z ) = -u or z = $ $ (8u - 1)3 and

+

the Hadamard Gap Theorem (Theorem 2.1) applied to fact { z : l z ( l - z)l = 2) is a natural boundary for f .

Cr=D=, S u 4 * show that, in

6.2. The Hadamard Gap Theorem and Over-convergence

{z: Iz(1- z)I = 2)

Diagram 6.2

< <

Note 2.2. The series of Example 2.2 has large gaps, indeed, if 2.4n+l m 4 n + l , then the coefficient a, = 0. But there are also large segments where all the coefficients are non-zero, namely for 4n I: m I: 2.4"; hence Theorem 2.1 does not apply, since the condition nk+l > (1 a)nk requires gaps between every exponent. Porter's Example 2.2 was published in 1906 [205], and shows that a selected sequence of partial sums of a power series may converge in a region exterior to the disk of convergence of the original series. It seems to have gone unnoticed. The phenomenon was independently rediscovered by Jentzsch in 1914, and again, independently by Ostrowski in 1921 who developed a theory for such series. Porter and Jentzsch were interested in problems related to the zeros of partial sums of power-series with a finite radius of convergence. See Jentzsch, Acta Mathematica 41, (1918), 253-270, especially pages 263-270. The preceding paper in this journal, also by Jentzsch, contains his well-known theorem on the limit points of the zeros of such partial sums. We cannot, in this chapter, go into the connection between such results and the phenomenon exhibited by Example 2.2. For a summary of such results, see the discussion in Ostrowski [183], Theorems V-VIII.

+

206

6. Natural Boundaries

Definition 2.1. Let CF=o anzn be a power series with a finite positive radius of convergence. Suppose there is a sequence of partial sums of the series which converges in a region partly external to the disk of convergence. Then the power series is called overconvergent.

In studying overconvergence we may, of course, again assume the disk of convergence is B ( 0 , l ) . Clearly a power series can exhibit overconvergent behavior only in neighborhoods of regular points. Ostrowski generalized examples like Example 2.2 to Theorem 2.2 (Ostrowski). For z E B(O, I), let f (z) = C;=o Cnzn, where limn,, IC,I; = 1, and suppose there exist two infinite subsequence of integers, say { n , , : v = 1 , 2 ,... ) a n d { n L : u = 1 , 2 ,... ) s u c h t h a t C n = O f o r n , , + l < n < n ~ - 1 , n: 3 (1 a)n,,, a > 0. Define S,,, Y = 0,1,2,. . . by So= E:io Cnzn; S,, - En"+' n=nL Cnzn for Y 2 1. Then the series CEO S,, converges uniformly i n a region which includes every point of C ( 0 , l ) at which f is regular.

-

+

Proof. Since a rotation does not affect the appearance of gaps of the sort hypothesized, there is no loss of generality in taking a regular point for f to be z = 1. Then f is analytic in a neighborhood of z = 1, and so for all 6 > 0 sufficiently small, f is analytic on B(1/2,1/2 6). Choose such a 6 with 0 < 6 < 112. then if

+

applying the Hadamard Three-circles Theorem (see Appendix) to Rk(z) on the disks B(1/2,1/2 - 6), B(1/2,1/2 E ) , B(1/2,1/2 6), where 0 < E < 6, gives with M:", Mik), Mik) the maxima of Rk(z) in the respective closed disks,

+

+

to show that E:=o S,,(z) converges uniformly in a region containing z = 1, it is Mik) = 0. It is clear sufficient to show that for all sufficiently small E > 0, limk,, that lime,, M:~) = 0; however, for all we know limk,, M3(k) may be infinite. The hypothesized gap condition, however, will allow us to show that the term with "outweighs" the term with Mik)). Since limn,, IC,I = 1; given 7 E (0, I), there is a K = K(7) such that

MY) IcnI


(&)"a

6.2. The Hadamard Gap Theorem and Over-convergence Then we have k-1

where the constant implied by the last 0 does not depend on k. And similarly, letting M = maxl,-l/~l<1/2+6 (z)l,

If

+ oo where again the constant implied by the last 0 does not depend on k. Substituting (2.5) and (2.4) in (2.3) gives

as k

Using the hypothesized gap condition and choosing q E (0, 6/2), we note that < 1, while > 1 (since E < 6); so (2.6) becomes

'-d 1-ll

for some constant C.

6. Natural Boundaries

208

We now show that the expression in braces on the right in (2.7) is limk,, ~ z ( " = 0, proving the theorem. Note that as E + 0, r ) -+ 0,

Log (1 - 6) Log (1

+ 26)(1+ a ) -

Log (1

< 0; whence

+ 6) Log (1 - 26) . (2.8)

Hence, if this last is negative, we can pick the left side of (2.8) is negative. But, as 6 + 0,

E

> 0 and

r)

> 0 small enough

so that

+ 6) Log (1 - 26) = + 0(d2))(26+ 0(b2)) - (6 + 0(d2))(-26 + 0(d2)) = -2b2a + 0(d3) .

Log (1 - 6)(log(l+ 26))(1+ a) - Log (1

(1+ a)(-6

Hence, the right side of (2.8) is negative for all 6 > 0 which are sufficiently small, and this proves the theorem as already remarked.

Note 2.3. Interestingly enough, Theorem 2.1 is an immediate corollary of Theorem 2.2. To see this, observe that, under the hypotheses of Theorem 2.1, if f had any regular points on the circle, then by Theorem 2.2 with nut = nv+l the series Cp=oS, would converge at some point outside B(0,l); but each Sv is just one term of the original series and so this is impossible. Hence f has no regular points on C(0,l). Example 2.3. Once again, no kind of convergence on the boundary seems to be an adequate substitute for regularity in Theorem 2.2. Already in 1891 Redholm pointed out, in a letter to Mittag-Leffler, that if f (z) is defined for t E B ( 0 , l ) by f (z) =

C 2-nzn2 n=O

then the series converges absolutely on C ( 0 , l ) and is termwise infinitely differentiable on C(0, I ) , each differentiated series also converging absolutely; nevertheless, by the Hadamard Gap Theorem (Theorem 2.1), C ( 0 , l ) is a natural boundary for

f. An obvious question raised by Theorem 2.2 is how large can the region of overconvergence be? The following example of Ostrowski shows that it may be infinite.

Example 2.4. For k 2 2, let K = K(k) = 10lok,and for z E B(0, I), define f (z) by CO (-K2z)" f(z) = C a n z n = v! n=O k=2

6.2. The Hadamard Gap Theorem and Over-convergence For convenience in the following arguments, let

xE2

so (for z E B(O, I)), f (2) = $k(z). Note that since the degree of the polynomial qk(z) is K 4 K = 104.10k+ 10lok, while the exponent of the term of lowest exponent in $k+l (z) is that each term of each polynomial contributes one distinct term of CTz0anzn. K4 K2 Writing ak(z) = CvZO byzU,since Ibm+l 1 = ;n7iri.lbml, we have for v K 2 , Ib,I lbKz I. On the other hand,

+

>

K 2 ~ 2

<

>

K2 and this is clearly = lbKal (since for m < K 2 , lb,+ll = ;;;TTlbml Ibml). Hence, every coefficient of the polynomial @k(z)has modulus 1, and at least one has modulus 1; hence C:=o anzn has radius of convergence 1. But for lzl k, since trivially k K2/10,

<

<

<

by a weak form of Stirling's Formula, and since e2.' Hence, for Jz/5 k and Re z > - 1,

2

and so by Stirling's Formula, (since 6(1+

since trivially K > 3k2. Since (2.10) holds for lzl uniformly for Re z > - 1.

< 8.2.

h) < 3),

5 k, R e z > 2 - 1, we get that

qk(z) converges

210 Furthermore, if R e z 5 -1 and lzl 5 k, then lzl

6. Natural Boundaries

> 1 and so from (2.9),

and by Stirling's Formula,

Thus, CEO=, +k (z) diverges for all z with R e z 5 - 1. Hence, we have that although the power series C r = o a,zn for f (z) has radius of convergence 1, f can be analytically continued by overconvergence into the region { z : R e z > -11, and no further. 101Ok+l

Note 2.4. In Example 2.4, the coefficients an are 0 for lo4,10k+,olok
f.

The question arises naturally as to what regions % of the plane can, in fact, be "regions of overconvergence"; that is, regions for which there exists an analytic function f whose power series around 0 has radius of convergence 1, but f can be analytically continued into % by overconvergence and B d 8 is a natural boundary for f . This question was answered by Ostrowski [I841 who showed that any simplyconnected region % (with at least two boundary points) such that 0 € %, cm @ % can be such a "region of overconvergence". Theorem 2.2 can also be proved by the method used by Mordell to prove Theorem 2.1; this seems to have been noted independently by Crum and Estermann, see for example, Estermann [68].

Note 2.5. Theorem 2.2 also has a sort of converse first proved by Ostrowski which shows the intimate connection between overconvergent series and series with gaps. If a sequence Smk (z) of partial sums -of the series C z = o amzm which represents the function f (z) i n B(0, l ) , where lim,, lamlA = 1, is uniformly convergent i n a neighborhood of some point of C(0,l); then CZzoamzm can be expressed as the s u m of two power series, one of whch had radius of convergence > 1; while the other exhibits gaps of the sort described by the hypotheses of Theorem 2.2. Ostrowski's [I851 original proof is in Sitzungsberichte der Preussischen Akademie, and uses results of Faber on conformal mapping and on Tschebyscheff polynomials. There is another proof by Losch [149], considerably shorter, which interestingly enough proves the theorem via conformal mapping and a consideration of the phenomenon of overconvergence for power series in two complex variables.

6.2. The Hadamard Gap Theorem and Over-convergence

211

-that this fesult, together with Theorem 2.2, implies that if a It is worth noting series Cr=oa,zn, lim,,, la,l; = 1, represents a function f (z) in B ( 0 , l ) and is overconvergent a t some regular point of f on C(0, I), then it is overconvergent a t every regular point of f on C(0,l). A brief proof of Theorem 2.2 and one quite different from that already given, was noted by Zygmund [256]. Zygmund's proof connects overconvergence with Borel-summability (see Note 2.5 below). Because the proof is not too long, even after the necessary facts of summability theory have been provided ad hoc, and the idea is interesting, we give it below, even though none of the motivating context of summability theory will be presented. Theorem 2.3 (Ostrowski's Overconvergence Theorem (Theorem 2.2)).

<

6 C ( 0 , l ) is a regular point for f (z) = Cr=o cnzn, lim,,, ~c,l$ = 1. Then there is a neighborhood of C in which f is analytic. By performing a preliminary rotation, if necessary, we can assume without loss of generality that C = 1, and the neighborhood is B ( l , 6) say, where 6 > 0. Suppose z E B(1,6/2) - B ( 0 , l ) and let K be a simple closed rectifiable Jordan curve (oriented positively) lying in B ( 0 , l ) U B(1,6) containing 0 and z in its Jordan interior, and such that for w E K, Re(z/w) 5 1 - 6 < 1 (this last condition can clearly always be realized). Then, we have

Proof. (Zygmund): Suppose -

&

Jr

and since the iterated integral is majorized by ,S jdwl e-u6dv, which converges, we can interchange the order of integration, and obtain

Thus, since f is analytic on KU Jordan interior of K and eU"/" is analytic everywhere except a t w = 0 the residue theorem gives,

We may transform (2.13) as follows. Let Tk(z) =

c:=~ cTzT;then ITk(z)l =

6. Natural Boundaries

O((k + 1)(1 + ~ 5 ) ~ for ' " ) z E B(1,6/2) - B ( 0 , l ) and

x w

= lim e-. x+w

Tn(z)xn+l

n=O ( n + l ) ! Thus from (2.13), we arrive a t the result that for all z in a sufficiently small neighborhood of a regular point of f which is on C(0, I), lim e-. X+W

C Tn(z)xn+l (n + I)! = f (2)

n=O

where Tn(z) = C:=, cTzT. Equation (2.14) for z as above is the required and well-known result from summability theory we need. We now proceed to Zygmund's proof proper. Observe first, that for any J, r], 0 < J < 1,Q > 0,

For A(x, and hence

c), note that the terms of the sum are increasing in size (since x > r) A(x,O = e-"

x

xT < eWx[(l- J)x] r! -

x[(l-oxl

[(I - <)XI! T<(l-ox and an application of Stirling's Formula, together with the power series expansion for log(1 - u) shows that

where C1 is a positive constant. Similarly, in C(x, r ] ) , the terms of the series are decreasing in size, and in fact, times the preceding term, and so each term is <

&

6.2. The Hadamard Gap Theorem and Over-convergence

213

and once more Stirling's Formula and the power series expansion for log(1 - u) show that there is a positive constant C2such that

From (2.16) and (2.17), as x -+ co, A(x,<) (2.15), as x + co, B(x,<,q) + 1. Now (for z E B(1,6/2) - B ( 0 , l ) as above),

& and applying the argument leading to (2.16), we get with

Letting x = X=l-&

+ 0 and C(x,q) + 0; hence from

7

<

Now, as S + 0, X -+ > 0 and so for all 6 sufficiently small 2S + d2 - 1/2X2 < 0, and hence, we have from (2.18)

Similarly, we obtain by the argument leading to (2.17) that

Substitution of (2.19) and (2.20) in (2.14) produces lim e-" x-+OO

C (l-€)zSnS(l+o)x

Tn(z)xn+l (n I)! = f ( z ) ,

+

for all z in a sufficiently small neighborhood of a regular point on C(0,l). We now use the gap hypothesis (see the statement of Theorem 2.2). Let x = m,andchoose~<<< 1 - ( l + a ) - ; a n d O < q < ( l + a ) i - 1 ; wethenhave that (1 - <)x > (1 a)-+ 2 n, ,

+

a

214

6. Natural Boundaries

and Hence, in (2.21) with these choices of x, (, q, Tn(z) = Tny(z) for all n in the sum. Skin the notation of Theorem 2.2) So (2.21) becomes (since Tny(z) =

for all z in a sufficiently small neighborhood of a regular point of f on C ( 0 , l ) and so, since as already observed B(,(,q) + 1 as v + oo, the theorem follows. Note 2.6. Suppose C r = o an is an infinite series. If CQ

n

xn lim e - " E E a k 7 = L , x+m n. n=O k=O

then the series Can is said to be "summable (B)" or "summable by Borel's first method" to L (compare equation (2.14) above). If

then the series Can is said to be "summable (B')" or "summable by Borel's second method" to L (compare equation (2.13) above). The summability methods (B) and (B') are known to be regular; that is, they sum convergent series to the value to which they converge, and it is also known that every series summable (B) is summable (B') to the same sum, but the converse is false. The methods are clearly connected with the Laplace transform, and also with P6lya's theory of entire functions of growth (1,r) and its later developments (cf. Note 3.3.13). Borel seems to have been the first to investigate their use for analytic continuation, though Hardy was the first to give correct proofs for some of the results, and, as P6lya remarks, some of the ideas go back at least to the Italian mathematician, Pincherle in 1888. In particular, analytic continuation by summability (B) (or (B')) of a function f analytic at 0 holds exactly within the "Borel polygon" of f . If with an entire function f (z) = C r = o anzn, we associate the e-zt f (t)dt = Cr=o as in Note 3.3.13, then there are also function F ( z ) = connections between lines of Julia (see Chapter 5) for f and the Borel polygon of F. For these, see Cartwright [42], Chapter VIII. For more information about Borel summability, see Hardy, [94], Chapters 8 and 9, and Peyerimhoff [189], pp. 71-82. Ostrowski's Theorem is discussed in section 9.5 of Hardy's book. Peyerimhoff gives a proof of it along Zygmund's lines, but,

Jr

6.3. The Hadamard Multiplication Theorem

215

instead of Borel-summability, uses the method known as ( E , 1)- summability, which may be defined by

If we note that

5

m=O

(

y m + l E ( y ) a T = ~ a T e( y ) y m + l = e a T E m - r )yrnc1 T=O m=r r=O r=o m=r

6

and that y = 112 corresponds to = 1, it is not surprising that there is a relation between (E, 1)-summability and analytic continuation (compare also to Note 2.1 where w = - 9 ) . The E is for Euler who was the first to consider such transformations of series.

Note 2.7. For further information on overconvergence, in addition to already cited papers, see G. Bourion [33] and Ostrowski [183], 251-263. there does not seem to have been much new learned about the phenomenon since Bourion's book.

6.3

The Hadamard Multiplication Theorem

If a function f given initially by a power series has its circle of convergence as a natural boundary, then that information is somehow "hidden" in the coefficients in the power series. The basic question is: how? (cf. the examples of Section 1.) More generally, if C E C is a singular point of a function f , initially defined by a power series, (say around O), how is that fact reflected in the coefficients of that power series? The Hadamard Multiplication Theorem provides information about this question, and is a useful tool, however it requires some care in interpretation.

Theorem 3.1. Suppose f (z) = CF=o anzn has finite radius of convergence Rf > 0, and g(z) = Cr=obnzn has finite radius of convergence R, > 0. Then the series h(z) = Crloanbnzn converges in B(0, Rf R,).

Proof. This is immediate from the Cauchy-Hadamard Theorem. We have 1 1 1 1 lim lan[; = -and lim ]bnl" = - . ntw n+co Rf 8,

-

Hence, limn,,

lanbnl$

1 5Rf R , '

Example 3.1. Theorem 3.1 does not say R f R g is the radius of convergence of 00 anbnzn and this is, in general, not true. Consider, for example, 00

00

216 and

6. Natural Boundaries

-

w

w

z C bnzn = g ( z ) = = C zZntl 1-

in B(O,I ) ; Rg = 1 22 n=O n=O Then h ( z ) = C;=, anbnzn 0 ; while Rf R g = 1. Hadamard developed the simple fact expressed by Theorem 3.1 in a significant way; t o express his result precisely we need some definitions.

Definition 3.1. I f f ( z ) = C;=o anzn and g ( z ) = C;==, bnzn have positive radii of convergence, then we define the Hadamard product of f and g as the power series C r = o anbnzn, and denote it by f * g. Definition 3.2. A star with center 0 is a region 9 C C such that 0 E 9 , and every ray emanating from 0 intersects 9 i n a single line segment. Definition 3.3. The chief star of a function f analytic at 0 is the largest star with center 0 in which f is analytic. Each ray emanating from 0 may or may not contain finite singular points o f f . The singular point o f f (if any) on each ray which is nearest 0 is called a corner of the chief star. Note 3.1. The chief star for f is also often called the "Mittag- Leffler star" for f . Less formally it can clearly also be described as the region formed by drawing rays emanating from 0 through every singular point of f , and then removing from the plane those portions of the rays beyond a singular point. Example 3.2. The chief star of {I : I m z = 0 , R e z 1 ) deleted.

>

&

is the plane with the line segment

Example 3.3. The chief star of a function analytic in B ( 0 , l ) for which C ( 0 , l ) is a natural boundary, is B ( 0 , l ) . Definition 3.4. Suppose f and g are analytic at 0 and f * g is their Hadamard product. Let S f and S g be the chief stars of f and g respectively, and E f and E, the sets of corners of S f and S g respectively. Consider the set

and f o m a star with center 0 from the points of A on each ray emanating from the origin which are nearest 0. This star is called the product star of S f and S g and denoted S f @ S g . Theorem 3.2 (Hadamard Multiplication Theorem). Suppose f ( z ) = C r = o anzn and g ( z ) = Czzobnzn are analytic at 0. Denote the chief stars of f , g , and f * g by S f , S,, and S f respectively. Then

,,

Sf*, 2 S f

asg.

6.3. The Hadamard Multiplication Theorem

217

Proof. Let Rf and R, be the radii of convergence off and g respectively. Consider first a point t E B(0, R f R g ) and let p be any number such that

&

Then, since p > ; for lwl = p, we have f ( f ) = C r = o a n ( $ ) n converges uniformly and similarly since plzl < Rg,~ ( w z = ) CrZo bnznwn converges uniformly. Hence (with the integral, as usual, taken in the positive direction),

Thus, for z E B(0, R f R g ) ,

On the other hand, for all z for which f ($)g(wz)$ is analytic on a path I?, the integral

represents an analytic function. Thus, the right side of (3.1) provides the analytic continuation of f * g to all such z when I? = C(0, p). Furthermore, for a fixed z, one can deform C(0, p ) in (3.1) (as I?) without changing the value of the integral, provided only that the deformed path goes around each singularity of the integrand in the same direction as many times as the original did. The region of analyticity of f ( f ) , call it Tf, is the image of Sf under the map w + $; the region of analyticity of g(zw), call it T, , is the image of S, under the map w + .: Note that

&

@ - B(0, Tf and B(O,%) C T,. Then T = Tf nT, is the region throughout which, for a fixed z, one can deform C(0, p). In fact, starting with C(0, p) (oriented positively), one can deform the path of integration in any way one chooses in T provided only that 0 E the Jordan interior of the resulting closed rectifiable oriented path and it goes around each point where f ($) is not analytic once in the positive direction (since -I < p) and each point where g(zw) is not analytic once in the Rf

8).

negative direction (since p < If z varies, the region T varies. However, for any z, one can always find a contour which is a suitable deformation of C(0, p) such that

6. Natural Boundaries

218

(6)

all points where f is not analytic lie in its Jordan interior and all points where g(zw) is not analytic lie in its Jordan exterior, provided only that the boundaries of the corresponding T f and Tg do not intersect. If, however, this does happen, then, clearly, the corners of T f and T, (i.e. the appropriate images of the corners of S f and S,) intersect. So, if the corners of T f and T, have no point in common, such a contour of integration can always be found. But T f and T, will have a corner in common for some z if and only if, for that z there is a corner a of Sf and a corner P of So such that

Thus, the analytic continuation envisioned is possible for all z on the line segment joining 0 and ap (excluding the point a p ) . This proves the theorem.

Note 3.2. While the deformations described in Theorem 3.1 are "obviously" possible, the reader sensitive to planar topology may wonder how obvious is "obvious". A theorem guaranteeing the ability to perform the required deformations can be found in Whyburn [248]. Some cautions are in order.

Example 3.4. Clearly S f * , need not equal S f O Sg as Example 3.1 shows (in this case, S f = S, = C - {z : I m z = 0, [ R e zl 2 11, and so S f @ Sg = S f = S,, but Sf*, = C.) Example 3.5. Singular points of f * g are not necessarily corners of S f @ S,. For example, let 00

n=O

By Example 1.2, k(z) is analytic in B ( 0 , l ) and has C ( 0 , l ) as a natural boundary. Hence Sk = B(0,l). For z E B ( 0 , l ) let

say. Then an = 1 if n is not a power of 2 and S f = B ( 0 , l ) . Let

Then

an = 2 if

n is a power of 2. Clearly

6.3. The Hadamard Multiplication Theorem

and so Sg = B(O,2). Hence Sf @ Sg = B(O,2). 1 , Sf.g = @ - {z : I m z = However, f * g(z) = Cmo(f zn = 0, Re z > 21, and f * g has singularities a t z = 2 and z = 3. This example is due to Faber. Another example of the same phenomenon can be found in the next section (Example 4.4).

+ &)

&+

Example 3.6. Although Theorem 3.2 says that f * g is analytic in Sf@ Sg,it says nothing about the possible analytic continuations of f * g. For example, suppose for z E B(0, I), 00

f ( 4 ) = g(z) =

zn C= - Log (1 - z) . n

n=l

Then Sf = Sg = C - { z : R e z 2 l , I m z = 01, and,

for z E B(0,l). However, f * g can be analytically continued around the point 1, and as the integral plainly shows, for a resulting branch, 0 may be a singular point. This example is due to Borel. The phenomenon that 0 may be a singular point for branches other than the "principal one" of the complete analytic function arising from all possible continuations of f * g, occurs because 0 is a singularity of the integrand in the above. As an example in the next section will show (Examples 4.2, 4.3) one can even have f and g analytic in B ( 0 , l ) with Sf = Sg = B(0, I ) , and the complete analytic function generated by analytically continuing f * g in all possible ways have a non-planar Riemann surface.

Note 3.3. Theorem 3.2 does not say that every singular point o f f *g has the form ap where a is a singular point of f and P is a singular point of g. Actually, taking sight of Example 3.6, clearly such a statement can not be true unless, say, we limit ourselves to the "principal branch" of f * g. The following is an accurate statement of the situation, which includes the preceding examples. anzn and g(z) = Cr=obnzn are analytic at 0 and suppose Suppose f (z) = Cr=o y is a (finite) singular point of f * g, then either

(i) y = 0 (possible only if f which is multiple-valued) , or

* g is a branch of a complete analytic function

(ii) There is a singular point a of f and a singular point P of g which are limit points of sequences of regular points for f and g, respectively, such that 7 = ap, or (iii) Every curve which is of finite length and has end points 0 and y contains a point of the form ap where a and p are as in (ii).

220

6. Natural Boundaries

Example 3.5 above is an example of (iii). If f and g have each only one finite singular point, say a and ,B respectively, then case (iii) is not a possibility and the argument of Theorem 3.2 establishes the result just stated. However, in the general case, this argument is not good enough. Hadamard, (and following him, Bieberbach in Analytische Fortsetzung [22], 2325) proves the theorem by replacing rays in Definitions 3.2 and 3.3 by families of similar logarithmic spirals with a fixed incremental angle, thus obtaining "spiral stars". One can then apply the argument of Theorem 3.2, and further conclude that the above stated result holds. Somewhat different approaches to the full result are provided by Pringsheim [207l and Schottlaender [219]. Schottlaender's paper contains a comprehensive survey and criticism of proofs of various variants of the Hadamard Multiplication Theorem and related theorems. Finally, it should be noted that the general theorem can also be stated in the following interesting way: Given a point z # 0; if there is a curve C of finite length, with endpoints 0 and z such that C (including the endpoints) contains no point of the form ap where a is a singular point of f and P a singular point of g , then z is a regular point for

f

*9.

Note 3.4. Mandelbrojt [I531 proves a theorem analogous to Theorem 3.2 for normal families of functions, where here the role of singular points is played by points where a family of analytic functions is not normal. He is then able to derive Theorem 3.2 from this result. Note 3.5. As Example 3.4 already indicates, the question of when does f * g have a singular point of the form ap may be difficult. In fact, as will be seen later anzn is any function analytic at 0 with radius of (Example 4.3), if h(z) = Cr=o convergence 1, say, then h can be expressed in the form f * g where f and g both have C(0,l) as a natural boundary. The attempt to give conditions when f * g necessarily has a singular point has to take into account the nature of the singularities of f and g. Such a discussion began with Bore1 who considered the case when f and g each have a single pole, and was most positively advanced by P6lya [194]. In addition to the usual isolated singularities (which P6lya divides into critical isolated singularities i.e., branch points, and non-critical isolated singularities i.e., poles or isolated essential singularities), P6lya defines "almost-isolated", "isolable" and "easily-approachable" singularities for a function defined by a power series with a finite radius of convergence. (Isolated singularities are always isolable.) Pdlya proves that if f (z) = Cz=o anzn has a single singularity a which is almost-isolated on its circle of convergence, and g(z) = C:=o bnzn a single singularity P which is isolable on its circle of convergence, then ap is a singularity of f * g. He further gives an example which shows that if one only assumes that /3 satisfies the (weaker) condition of being almostisolated, then ap is no longer necessarily a singularity of f * g. In the course of the difficult proof, P6lya obtains results relating the density of the non-zero coefficients of the power series to the nature of singularities on the circle of convergence.

221

6.4. The Fabry Gap Theorem

The reader interested in these and related results, should consult Chapter V of Bieberbach's Analytische Fortsetzung [22], Pblya's original paper which has been reprinted in Volume 1of George Pdlya, Collected Papers [193], 543-589 and two papers by R. Wilson in Mathematical Essays dedicated to A. J. Maclntyre [249]. It is perhaps worth noting that some of this work uses results involving the phenomenon of overconvergence which is discussed in Section 2.

6.4

The Fabry Gap Theorem

In this section we discuss what is presently, essentially the most general gap condition ensuring a natural boundary which is known, and which, in fact, includes those of Section 2. As a preliminary to the proof, however, we need a result interesting in its own right, which connects regions of analyticity of functions initially defined by power- series with the representation of the coefficients as the values of an entire function of a prescribed sort. We will also make use of the material in Sections 2 and 3 of this chapter. We begin with the following general and simple theorem which might well have been in Chapter 3. Theorem 4.1. Let f (z) = C = :o anzn have a positive radius of convergence. Then there is an entire function A(z) of growth ( ~ , T ) ( T< co) such that A(n) = a, for

each non- negative integer n. Proof. Consider the circle C(0, p) oriented positively where 0

Then

1

IanI;

.

Jc(o,r)sdZ

1 a, = 2ai

f (2)

Let z = e-"; then C(0, p) becomes the line segment 8 5 27r). Letting F ( w ) = f (e-"), we have then

.

I? : {w : w = - logp - i8,O <

2

Let A(z) = Jr F(w)eZwdw. Then A(z) = - CEO zk& Jr F(w)wkdw = CEOb k z k , say. One easily sees that Kk,,klbk 1 i < oo and it follows from Theorem 3.2.2 with p = 1 that A is entire of order 1 and finite type. (Note that if limk,, klbkli is finite, then necessarily Gk,, 1bkli = 0.) Note 4.1. Theorem 4.1 is really a theorem about Laplace transforms. It is closely ) to in Note 3.3.13. related to Pblya's theory for functions of growth ( 1 , ~ alluded Although Theorem 4.1 is "obvious", and had been known for a long time previous, it seems first to have been explicitly stated by Dufresnoy and Pisot [59]. It should be noted that there exist other functions with the same growth properties whose values a t the positive integers are the a,.

6. Natural Boundaries

222

A natural question is to what extent is further specification of the entire function A related to properties of the function f . The theorem which we shall use goes back to Wigert in 1899. Theorem 4.2. Let f (z) be analytic in a neighborhood of 0, where it is represented by C,"=oa,zn. A necessary and suficient condition that there be an entire function g such that f ( z ) = g , is that there be an entire function A(z) of growth (1,O) such that A(n) = a, for all integers n 2 1.

(&)

Proof. (Necessity.) Suppose f (z) = Cr=o anzn in a neighborhood of 0 and f (z) =

( )

g where g is entire. Then the series has radius of convergence 1. Let 0 S < 1, then

On the other hand, let 0 < E Theorem (see Diagram 6.3)'

< min(6,l - 6) and p > 1 + S, then

Diagram 6.3

<

from Cauchy's

6.4. The Fabry Gap Theorem

223

+

Diagram 6.3: Starting at the point 1 E, say, traverse (in the positive direction) C(1,c) until the point 1-6, then L1, then C(0, S) in the positive direction until 1-6, then L2 then all of C(0, p) in the positive direction, then L2 in the reverse direction, and the remainder of C(0, S), L1 in the reverse direction and the remainder of C(1, €1. For n 2 1,

Now, since g is entire and p

Hence, letting p have for n 1,

>

> 1,

+ w, the left side of

(4.2) approaches 0, and so using (4.1) we

Define

Then A(n) = a,, and it remains to show that A is entire of growth (1,O). The value of the integral is independent of the choice of 6 , provided only 0 < 6 < min(S, 1 - S), and one sees directly that A is analytic at every finite point and so entire. ) have Finally, letting M(c) = m a x ~ ~ c If( ~((I,, ~we

1 2IT

5 -( 2 7 ~M~ )(c) < cM(c) rnax -

max

CEC(1,c)

"

1

e-('+')

I(z + l)lVl1%

"

1.2

1

elv

v!

C E C ( l ? f )u=O

<

l0gC

+ llV

C , I -max v=o 0

~

~

I log(1 + cei@)1" ~ 2 n

6. Natural Boundaries

224

Hence Log max I Z I = T IA(z)I and so

-

lim

T+"

5 ( r + 1) log

(A) +

0 ( 1 ) , as T -+ m

Log Log maxlzl=r l log T

l

51,

and also lim

for every

E

1-E

r

r+"

E ( 0 ,min(6,l - 6 ) ) . Letting

E

-+

0 , it follows that A is of growth (1,O).

(Sufficiency.) Suppose now A ( z ) = Egoc k z k is of growth ( 1 ,0 ) ,A ( n ) = a, for n 1, and f ( z ) = a0 C:=l anzn. Then a, = CEOcknk, and for z E B ( 0 , I ) ,

>

+

>

For z E B(O, I ) , and k an integer 0 , let $ k ( z ) = C r = l n k z n . Then ~ ! J ~ ( = Z) z- 2 -1 and $ ~ + I ( z=) z $ ( $ ~ ( z ) )= (1 - z ) ( & - l ) & ( $ k ( z ) ) ,and so all the functions $k are polynomials in &,thus functions analytic in @ , - ( 1 ) . Then

,-, ,-,

and it suffices to show that this last series converges uniformly in a neighborhood of every point z # 1. It proves convenient to let z = e W ,for then since $k+l ( z ) = z $ $ ~ ( z ) ,

d dw

q5k+l(ew)= -$k(ew),

and $o(ew)= -1

1 +. 1 - ew

Hence

&

Let a be a finite point such that 1-ea # 0 , and choose R ( a ) > 0 such that is analytic in B ( a , R ( a ) ) . Then, if w E B ( a , $ R ( a ) )= D say, since the coefficients of the power series expansion of about a are

6.4. The Fabry Gap Theorem we have by Cauchy's inequality

say. Furthermore, by Theorem 3.2.2, with p = 1, since A is of growth (1,0), limk,, k ~ ~ k= l *0, and so given E > 0, for all k ko = k0(e),

>

Thus we get from (4.4), on using (4.5) and (4.6),

Since k! ( ! ) k m as k + co by Stirling's Formula, choosing E < R ( a ) , we , see that the series (4.4) converges absolutely and uniformly in B ( a , i ~ ( a ) ) for every finite a such that e" # 1. It follows that (4.3) converges uniformly in some neighborhood of every finite point z # 1. the functions 4k which are polynomials in As to oo, putting = become polynomials in <,and arguing as above, one sees that f is analytic in every neighborhood of = 0. Hence f is analytic in a neighborhood of every point of C, - (1) and so an entire function of

A,

< &,

<

&.

Note 4.2. The sufficiency proof given above has been attributed to Erhard Schmidt (unpublished). If, in this proof, we assume A(z) is of order 1 and type T,0 < T < oo, we get from the same proof (with $ ~ ( a replaced ) by R ( a ) - 6) that if R(a) > T, then the series (4.4) converges absolutely and uniformly in a neighborhood of every finite point such that e" # 1. Consequently, if A(z) is entire of order 1 and type T, A, = a, for n 1, and f (z) = a0 Cr'l anzn, then f is analytic except possibly at points z = eQ such that [a- 2knil 5 T, for some integer k, positive, negative, or zero (since it is only for such points that we may not be able to choose R ( a ) > T). In fact, Hardy proved in 1917 1951, that the necessary and sufficient condition that f (z) = a0 + CrZ1 anzn should be analytic in the region (including m) exterior to the curve given by I Log ($)I = T, where 0 < T < n, and not in any larger region of the same sort, is that there should be an entire function A(z) of order 1 and type T such that A(n) = a, for all integers n 1. If T T ,the curve I Log ($)I = T no longer describes a simple closed Jordan curve (see the following diagrams).

>

+

>

>

6. Natural Boundaries

Diagrams 6.4

Theorem 4.2 also has close connections with results on the Phragmhn-Lindelof indicator function h(8) for the entire function A of order 1 and type T < oo. As remarked in Note 3.3.13, h(8) is the supporting function for a convex set K (the mirror image in the real axis of the set D of Note 3.3.13). Carlson, in his thesis of 1914, gave extensions of Wigert's Theorem 4.2 (published in 1900), which is essentially the case in which K is single point. It should be pointed out, however, that this geometric language is not Carlson's, as the exploitation of the convex set which h(8) supports, originates in Pblya's already oft-cited memoir of 1929 [192]. For further information and proof of a still more general result implicit in a paper of Dufresnoy and Pisot [59], see Bieberbach, Analytische Fortsetzung [22].

6.4. The Fabry Gap Theorem

227

Theorems of this sort are also related to Carlson's Theorem 3.3.12. For example, Dufresnoy and Pisot [59] show that if A(z) is entire of growth ( l , ~ )T , < oo, h its Phragmhn- Lindelof indicator function, and h ( - ~ / 2 ) + h(n/2) < n, then h(0) = lim 1% IA(n) l n+m n This includes Carlson's Theorem 3.3.12 as a special case since if A(n) = 0 for all integers n 2 1, we have h(0) = -m. That h(B) = -oo for B E ( - n / 2 , ~ / 2 ) and hence A(z) E 0 in the right half plane now follows from Theorem 3.3.5 (note especially the last sentence of the proof.) A proof of the sufficiency part of Theorem 4.2 using the residue theorem applied to expressions of the form $$& can be found in Lindeltjf [144], Le CalcuE des Residus. The proof given above follows Bieberbach's Vorlesungen [23]; Hardy's proof (of necessity and sufficiency) follows ideas of LeRoy and is of a still different sort. N o t e 4.3. Suppose the function A in Theorem 4.2 is in fact a polynomial, then g is also a polynomial and, in fact, the degree of g is 1+ the degree of A. To prove this, one simply observes that if f ( z ) = C:=l A(n)zn, where A(n) = ELzoCknk, then, as above

where the q5k(~)as already observed, are polynomials in &. Furthermore, since q5k+l(~)= z&dk(z), and #o(z) = &. It follows (by an easy induction argument) that q5k (z) is a polynomial of degree k + 1 in &,and so f a polynomial of degree r+lin&. Further elaborations of the relation between A and f can be found with references in Bieberbach, Analytische Fortsetzung [22], 16-20. Power series exhibiting the Hadamard gap condition have a very obvious "peculiarity". What is remarkable is that the gap condition can be significantly ameliorated with the same conclusion holding.

EE1

T h e o r e m 4.3 ( T h e F a b r y Gap Theorem). Suppose f (z) = Cnkznk has radius of convergence 1, where limk ,, = m, then C(0,l) is a natural boundary for f. Proof. We have Gk+,ICnkI* = 1. Choose an infinite subsequence of the nk, say {nk,) such that nk,,, > (1 a)nk,, where a > 0, and

+

-

lim IC,,,

V+,

I

1

=1

For notational convenience, we set nk, = p,, for all positive integers v , and let the subsequence of i n k ) complementary to {p,) be denoted by {q,) (in increasing

6. Natural Boundaries

228

order), that is {nk) = {p,) U {q,}. Clearly q, 2 n,, and hence Let

-&

b,,,

= W.

a).

We now prove that, in fact: Clearly &(z) is entire (since Eml < KC:=, (A) 4 is of growth (1,O) and (B) Given E > 0, for all u 2 YO = vo (E),14(pv)l > e-'pv. The proofs of (A) and (B) require somewhat involved calculations, but once they are established, the proof concludes in a few lines on using Theorem 4.2 and material in Sections 3 and 4. = oo, given E > 0, there is an N = N(c) such Proof of (A): Since lim, ,, that % > for Y N = N(E). Hence, since the q, are ordered in increasing order, given E > 0,

>

>

&

q,

Now, given E > 0, for all r sufficiently large, say r ro = ro(r), > and for all r 2 r = r l ( ~ )r, > hence, given E > 0, for all r 7-2 = M a ( r o ,r l ) 1, we get from (4.9),

A;

>

>

for all r r g where c' can be taken as an arbitrary positive number. Hence, q5 is of growth (1,O).

>

6.4. The Fabry Gap Theorem

229

Proof of (B):

For each u, determine the integer n = n ( u ) by

and the integer m = m(v) by

Consider first the terms in (4.11) which are increasing order,

5 n . We have, since the q, are in

Now, each factor in each of the last two products is a positive integer distinct from all other factors in the product in which it appears; hence (4.14) gives us if n even

By a weak form of Stirling's Formula (or more simply, since obviously en we have

and

>

s),

6. Natural Boundaries since (1+ !)"I2 is a monotone increasing function of n. Substituting in (4.15), and then (4.14) we have,

But by (4.12), for each factor in the product in (4.16), more, 1- x2 = . Hence, from (4.16) we obtain

' l+ 5

%

5

-

& 5 i. Further-

-

Now, given E > 0, since lim, , = co, there exists an integer Nl = N1(e) such that for p Nl (r), > l / r , and an integer N2 = N2(E) such that >

>

%

i,

for p > N2 (E). Hence, given E > 0 for n > m a x ( N ~ ( ~N)~, ( E )(note ) that by (4.12), this will be true, provided v is sufficiently large), we have from (4.17),

as v + co, on using (4.12). Hence, given E* > 0, for all v vo = VO(E*), I+(pV)J> e-e*pu,which proves (B). From (A) and (B), together we have that given E > 0, there is a vl = v ~ ( E such ), that for all v vl, e-Epu I I+(pV)IL ecpY,

>

>

and so lim I + ( ~ , ) = I ~I .

Y+,

Since p, = n k , by definition, we have from (4.8),

Hence

6.4. The Fabry Gap Theorem

231

+

has radius of convergence 1. Also, by construction, p,+l > (1 o)p,, and so by Theorem 2.1, (The Hadamard Gap Theorem) C(0,l) is a natural boundary for the power series (4.18). Note, however, that by definition +(I) = n z l ( l - $), and so +(q,) = 0. It follows from the definition of the sequences {p,) and {q,) that the power series (4.18) is the same as the series

Define the sequence a, as follows: a, =C,,,

i f r E {nk)

a, = 0, if r $ {nk)

.

Thus the power series (4.19) is the same as

Thus (4.20) (which is the same as (4.18)) is analytic in B ( 0 , l ) and has C ( 0 , l ) as a natural boundary, and so the chief star of the function defined by (4.20) is B(0, 1). On the other hand, by Theorem 4.2 (Wigert's Theorem) and (A) above, CEO=, +(r)zT is an entire function of &,and SO its only singularity is at z = 1. Thus (cf. Examples 3.2 and 3.3), it follows from Theorem 3.2 (The Hadamard Multiplication Theorem) that the chief star of

must be B ( 0 , l ) as well, that is

x

-x

def

aTrT

Cn, t n k

has C ( 0 , l ) as a natural boundary, which proves the theorem. Although the Hadamard Gap Theorem (Theorem 2.1) was used in the proof of Theorem 4.3, it is still interesting to note also that it is a special case thereof (see also Note 4.8 below). Theorem 4.4. The Hadamard Gap theorem (Theorem 2.1).

6. Natural Boundaries

232

Proof. We need only verify that whenever we have an increasing sequence of integers {nk) such that (i) nk+l - nk > ank, then also we have (ii) hk,, = co. But if (i) holds, then nk+l - nk + co as k + co; hence given a positive number M , for k+r-1 nv+l - n, > r M for all k 2 K ( M ) , nk+l - nk > M , whence nk+, - nk = all positive integers r . It follows that

and so h,,,

>M

for any positive number M , which establishes (ii).

Note 4.4. Although all increasing sequences {nk) satisfying nk+l - nk + co as k + co also satisfy limk , = co, the converse is not true as the example n2k = k2, n2k+l = k2 1demonstrates. Thus the condition of Theorem 4.3 requires less of the series than any gap condition "of Hadamard type".

+

7

Example 4.1. As an example of the uses of Fabry's Theorem 4.3, we have the following result of Carroll and Kemperman [41]. Theorem: Suppose P(k) is a complex-valued function on the integers such that lim IRe P ( k + 1) - R e ~ ( k ) l > i 0

(9

IP(k>l 5 eck

(ii)

k-tm

and

for some constant c > 0and all k > 0, then, if gn is a non- decreasing sequence of real numbers such that limn,, gn = co and limn,, 5 = 0, the function

is analytic in B ( 0 , l ) and has C ( 0 , l ) for a natural boundary (where greatest integer function).

[XI

is the

Proof. As will be apparent, the conditions of the theorem are chosen just so the following proof works. On the one hand, by (ii) and limn,, 2 = 0, limn,, P([~,]) 5 limn,, e = 1, and so the radius of convergence of (4.21) is 2 1. Hence, G(z) = (1 - z) C;=, P([gn])zn also has radius of convergence 2 1. But

*

233

6.4. The Fabry Gap Theorem

+ oo

+ oo,

> [g,_l] for infinitely many n; also IP([g,] .yand so by (i), and En,, $ = ~ ( [ ~ n - l ] ) l>$ IRe P([gn])- Re P([g,-l])l 0, (4.22) has radius of convergence 5 1. Hence (4.22) and (4.21) both have radius Since g,

as n

[g,]

1

of convergence 1. Finally the number of integers n between 0 and m for which P([g,]) # P([gn-l]) is O(g,) = o(m) as m + CQ ([g,] is a non-decreasing sequence of integers). So (4.22) is, in fact, a series with Fabry Gaps and so C ( 0 , l ) is a natural boundary for G, and so also for F.

Note 4.5. Actually, the proof of Theorem 4.3 can be adapted to give results with yet weaker conditions, though the theorems are somewhat more technical. For example, there is the

*

Theorem: Suppose f (z) = Cr==o anzn is analytic in B(0,l) and limn,,lan[ = 1. Suppose there is an a, with 0 < a < 1, and an infinite increasing subsequence {nk) of non-negative integers, and a sequence {yk) of real numbers, such that for each k, if S(k) indicates the number of changes of sign of Re(a,e-7") when n runs through the interval b = [(I - u ) n k ,(1 a)nk], then limk,, = 0. (Here changes of sign means only from + to - or - to +; zeros being struck from the sequence). Suppose also

+

$

-

lim ~ ~ e ( a e, ,- ~ " ~ ) l *= 1 .

k-tw

Then 1 is a singular point of f . A proof is essentially the same as that of Theorem 4.3. In the first place, clearly there is no loss of generality in assuming (1 - a)nk+1 > (1 + a ) n k , so that all the Ik are disjoint, and clearly we can also assume that nk+l > 2(1+a)nk (only necessary if a! < 112). Let those values of R e ( ~ , e - ~ y ~ #)0 which immediately precede a sign change arranged in increasing order, be denoted by r,. Consider instead of the 4(z) of the proof of Theorem 4.3,

Suppose there are infinitely many r,. Let K = K(v) be the index of the interval Ikwhich contains r,; then (since nk-1 < ink)

Hence since, clearlv rv

lim v

v-+w

> lim V-+W

r, ~

f ~ =( k ')~

234

6. Natural Boundaries

and by hypothesis limk,, $ = 0, we get that lim,,, % = co. Now, as in the proof of Theorem 4.3, $(z) is entire of growth (1, O), and breaking the product defining $(nk) into three pieces, the product over all entries in intervals with indices 5 k - 1 is easily seen to be 2 1, while the nk can, without loss of generality, have been originally chosen so that the product over entries in intervals with indices 2 k 1 is

+

The product over entries in Ikis estimated analogously to the proof of Theorem 4.3, and one obtains that given 6 > 0, there is a IC = K(E)such that for all k 2 IC,

Cr=o

Thus, the series an+(n)zn has radius of convergence 1. It can be shown directly that the function represented by this series in B(0,l) has a singular point at z = 1 (using the technique of Note 2.1, together with the fact that Re(ane-7ki)+(n) has no changes of sign on Ik; the technical details can be found in Landau's Darstellung Einige Neuer Ergebnisse der Funktionentheorie, especially p. 77-78). It now follows from Wigert's Theorem, (Theorem 4.2) and the Hadamard Multiplication Theorem (Theorem 3.2) that f (z) = anzn has a singular point at 1. Using this result, one can deduce the so-called

xF=,

CFz0

Fabry Q u o t i e n t Theorem: Suppose anzn = f(z) is analytic in B(0, I ) , = 1, then 1 is a singularity of f . and limn,, The deduction (of a more general result) can be found in Landau ([136], p. 84-86). There are still further refinements of this sort which can be found with proofs and/or references in Chapter I1 of Bieberbach's Analytische Fortsetzung [22]. N o t e 4.6. P6lya has given a somewhat different sort of refinement of Theorem 4.3. Suppose { n k ) is an increasing sequence of non-negative integers and f (z) = ankznb has a finite radius of convergence and, in fact, lh,,, = 6 < oo. Clearly 6 1. Then P6lya proved that every arc of the circle of convergence of length 2 2 ~ / 6contains a singular point of f . Fabry's Theorem 4.3 is clearly the case 6 = oo, 2n/6 = 0, whereas observation that there is always a singular point on the circle of convergence corresponds to the fact that 6 is 1. See, once more, Mathematische Zeitschrift [192]. P6lya extended his theorem to Dirichlet Series in Sitzungsberichte der Preuss. Akademie, Phys.-Math. Klasse, 1923, 45-50. All three of these papers are in Volume I of P6lya's Collected Works [I931 where there is commentary and further references. A proof can also be found in Boas, Entire

CEO

>

>

6.4. The Fabry Gap Theorem

235

Functions [27], Sections 10.3 and 12.6. One should also, of course, consult Chapter 2 of Bieberbach's Analytische Fortsetzung [22]. A refinement of Theorem 4.3, which slightly generalizes the conditions on nk is due to H. Claus [45], and in final form to M. E. Noble [177]. The proofs use results on Tschebyscheff polynomials whose utility in studying such problems was first pointed out by Szego in 1922. The question, of course, arises whether similar conditions weaker than = m will imply that C ( 0 , l ) is a natural boundary for f . P6lya showed lim, , that this was not so: Theorem 4.5. Suppose {nk) is an increasing sequence of non-negative integers such that lim, , y = S < co, then there is a function g(z) = CEO=, an,znk which has radius of convergence 1 and such that C ( 0 , l ) is not a natural boundary for g .

Proof. Note first that if N(t) denotes the number of members of the sequence {nk) which are 5 t , then if N(t) = k, we have nk 5 t < nk+l, and so

Since lim, , y = 6 < co, it follows easily that the number of members of the sequence {nk) in the positive interval (a, b] is N(b) - N(a) > C(b - a ) where C is a positive constant. Now choose two sequences of positive integers ui and vi such that vi = [(I K)ui] and 2 Ui+l > V i , where K is a positive constant and, as usual, [XI is the greatest integer x. Then the number of members of the sequence (for K sufficiently large) i n k ) in each interval (ui, vi) is 2 C(vi - ui) - C - 1 > C K u i - C - 1 > C*ui, say, where C* is a fixed positive constant. Denote the nk in the intervals [ui,vi] by n;. Clearly lim, , < m . Let

+

<

%

where the {a,;) are to be determined. We have to find a sequence of coefficients a,; such that B(0,l) is the disk of convergence of this power series and some point on C(O,1) is a regular point for g. Let r be a positive integer. Then

6. Natural Boundaries

236 where clearly

We need to choose the an; such that

-

1

lim lan;lq = 1

k--tm

and

-

lim 1bmlk< r ,

(ii)

m+w

for some sufficiently large integer r . Then by (i), B ( 0 , l ) will be the disk of convergence for the power series in (4.23), while by (ii) and (4.24), the radius of convergence of the power series around 0 of g(z %) is > and hence 1 is a regular point of g. There are two cases that need to be considered. (A) Given a fixed but small 6 > 0, suppose

+

then n; > E,while if Then, since {nk} c Ui(ui, vi), we have that if m < *, m > *, then n; < Let the {a,:} be any non-zero complex numbers of

z.

modulus

< 1. It follows that if m # Ui (q , F) , then

To estimate Ibml, let

Then

= A,,, and Hence, An is a strictly increasing function of n for n 5 r m - 1, An is a strictly decreasing function for n 2 r m . In particular, from (4.26) we have

6.4. The Fabry Gap Theorem

for 0 < E

< 112 and r > 2 / ~ since , r - 1- E +

ifn>-

rm 1-e

An+1 < , then -

I-;

An

5 r. Similarly, (4.26) also yields em -

1

= 1-

rm+l-c


5e2 2r

5

for 0 < E < 112 and r > 2 / ~ since , (r - 1+ E) + 5 r. Furthermore, since A, = A,,, letting is increasing for n 5 r m - 1, decreasing for n 2 A,,, and a = &,we have for n $! (E, E), on estimating the binomial coefficient by a strong form of Stirling's Formula, for r 2 2 / a

An

5 -

(?)arm J G ( 1 + -)(l1 (arm-m a r m - m d ) 2x(arm - m ) ( F ) m e (arIaTmJ s ( 1 +

1

1 arm-m -i )

( a r - 1)ff'm-m-

(

ar = (arlm ( a r - 1)(2xm) ) + ( I +

(ar-1)m

'

'

&) (-)

Thus, from (4.25), (4.29), (4.27), and (4.28), we have in case (A), for a fixed e , O < e < 1 / 2 , a n d r > 216,

6. Natural Boundaries

SO, in case (A), we have,

=.

where a = 1 (B) For some i, $ ( l - E) < m

< $ ( I + E). In this case, let

say, where b; indicates the sum arising from summing only over those n', such that n', lies in no interval (ui, vi) and bk the sum over the remaining nk. Then (assuming still only that all an; have modulus < 1)

and the above argument for case (A) shows that 1 . r, where a ' = I f€ Finally, we now show that the an; (which so far have only been subjected to the restriction that la,; I 1) can be chosen so as to make all the b; = 0. We need to choose the a,; such that, whenever m E (+(I - E ) , $ ( I + 6)) for some i,

<

But, (4.32) is a system of at most

homogeneous equations in at least C*ui unknowns. Thus, for all r sufficiently large, there are more unknowns than equations, and hence, the system (4.32) always has a solution, furthermore, we clearly can also suppose that the largest value of lan; I is 1. Thus (i) holds. Also, now from (4.30) and (4.31), we have for all m ,

-

lim 1bm

m--to3

< pel-Pr

where p = a or a'. But xel-" has its maximum value 1 at x = 1, and hence, since e was fixed 0 < c < 112, a = and a' = (ii) also holds, and so 1is a regular point of g, which proves the theorem.

&,

&,

6.4. The Fabry Gap Theorem

239

Note 4.7. This completely elementary (but not exactly easy) approach to Theorem 4.5 is due to Erdos [67]. A variant of Erdos' approach was published by Walfisz in Akad. Nauk Gruzinskoi SSR [242]. P6lya's original approach uses his theory of the ) Note 3.3.13) [195]. indicator diagram for entire functions of growth ( 1 , ~ (see Theorem 4.3 has been proved in several different ways. The proof given above is based on ideas of Faber. Compare also, Pringsheim [207]. A version of Fabry's original proof which involves complicated estimates of products of sines, can be found in Bieberbach's Analytische Fortsetzung [22]. TurAn has given a proof of Theorem 4.3 using methods of Diophantine Approximation and "power-sum inequalities" of which the following is typical: k , then for every integer If {zj} and {aj) are non-zero complex numbers 1 j m 2 0,there is an integer n, with m n m k such that

< < +

k

m i... n k IzjI Eajq > j=1

j=1

< <

k

xaj

j=1

(24rnk+ k))'

.

For the proof of Theorem 4.3 using these ideas, see TurAn [231], Eine Neue Methode in der Analysis und Deren Anwendungen (Fabry's Theorem 4.3 itself, in a more general form, is treated on pages 75-81), or Dinghas [56]. Tur&n's method has found a wide range of applications from the theory of the Riemann <-function, to the study of quasianalytic classes of functions, and there is a large literature of applications and refinements. Still another proof of Theorem 4.3, using Diophantine Approximation, was published by Gelfond [84]. Finally, Dufresnoy and Pisot [59] in their paper in Annales Scientifiques de L'Ecole Normale Superieure, prove the Fabry Theorem as a corollary of the following.

'

Theorem: Suppose g(z) = CT=oanzn for z E B(0,r) and En,,la,/$ = T ' then g can be analytically continued into C - B(O,r), if and only if there is an , Phragmkn-Lindelof indicator h satisfies entire function f of growth ( l , ~ )whose h(-n/2) h(n/2) < 2n, such that a, = f (n) b,, where G,,, ~b,l* < $. Dufresnoy and Pisot also apply their theoretical development to a proof of the Fatou-Riesz Theorem 1.2. The reader who penetrates this area further will find still more connections between the various sections of this chapter than those alluded to in these notes. The proof given of Theorem 4.3 (The Fabry Gap Theorem) uses a subsequence of exponents which satisfy the Hadamard Gap Theorem and then reduces the deeper theorem to this one by way of Wigert's Theorem and the Hadamard Multiplication Theorem (Theorem 3.2). Another well-known theorem whose proof includes the selection of a "Hadarnard gap subsequence" to show that the boundary of the disk of convergence is a natural boundary for a power-series, and which at the same time, tends to show that non-continuable power series are rather the rule than the exception is:

+

+

6. Natural Boundaries

240

Theorem 4.6 (Fatou-P6lya-Hurwitz). Let CF=o cnzn be a power series with -

limn,, lcn14 = 1. Then there exists a sequence of numbers {en) such that en = +1 or -1 for all n and C ( 0 , l ) is a natural boundary for CF=o€ncnzn. Proof. Suppose C r = o cnzn = F(z) for z E B ( 0 , l ) . Select a subsequence {nk) of non-negative integers such that nk+l > nk(1 a ) where a is a positive constant and limk,, (cnk = 1 (clearly always possible). Let

+

CO

Then R(z) is analytic in B ( 0 , l ) and by the Hadamard Gap Theorem 2.1 has C ( 0 , l ) as a natural boundary. Since CF=ocnzn converges absolutely in B(0, I ) , we can write F(z) = fo (z) R(z) for z E B ( 0 , l ) .

+

Again, by absolute convergence in B(0, I ) , we can break up R(z) into the sum of countably infinitely many power series fl (z), f2(z), . . . , such that (a) no two of the fj(z) have terms with the same exponent.

(b) each fj(z) contains a countably infinite number of terms. Consider now the collection of all series of the form

where Sj takes on only the values 1 and -1. Clearly there are 2'0 such series, and clearly they all converge for z 6 B(0,l). Suppose now the theorem were false, then all series of the form (4.33) would have regular points on C ( 0 , l ) . But there are 2'0 such series, and the roots of unity are a countable dense subset of C(0,l); hence there must be two such series, Pl(z) = fo(z) CZl 6.j f j(z) and P2(z) = fo(z) Cgl 6; f j(z), where the coefficients cj, c; take on only the values 1 and -1, which have the same root of unity as a regular point. But then this root of unity is a regular point of

+

+

and by construction, this latter series, when rearranged according to increasing exponents, is a power series with radius of convergence 1 which exhibits "Hadamard gaps", and so has C ( 0 , l ) as a natural boundary, which is a contradiction.

6.4. The Fabry Gap Theorem

241

Note 4.8. Clearly the proof also shows that there are 2N0ways of assigning the cn of the theorem so as to produce a series for which C ( 0 , l ) is a natural boundary. This raises the question of whether perhaps there is a "universal sequence" {en) consisting of 1's and -1's such that for every power series CTzocnzn with radius '=oncnznhas C ( 0 , l ) as a natural boundary. The answer t o of convergence 1,CF=o this question is "no" as was shown by Fuchs [79], who actually shows that for any sequence {en) of +l's and -l's, there is a power series CTz0anzn,an > 0 with radius of convergence 1 such that CFzoenanzn is analytically continuable over an (open) semi-circle on C(0,l). Fuchs also observes that the sequence {en : '=on = 1,n 0 , l (mod 4);cn = -1,n r 2,3 (mod 4)) has the property that for every power series f (z) = Czz0anzn, a n > 0, with radius of convergence 1, F ( z ) = C:=, cnanzn has the property that every closed semi-circle on C ( 0 , l ) contains a singularity of F. That there is no "universal sequence" of the type desired if we take all '=on to be complex and only insist on I'=onj = 1 was shown by R. L. Perry [186]. Note 4.9. The proof given of Theorem 4.6 is Hurwitz'. Fatou proved the theorem an = 0 and Cr=olan] is divergent. (This case follows in the case in which limn,, from Theorem 1.2). The first proof of the complete theorem was given by Pblya, and immediately thereafter, Hurwitz gave the above proof in an exchange of letters [116]. Pblya's proof uses an infinite sequence of the sort used in the proof of Theorem 3.2.5 and a lemma of Fabry. P6lya later gave another similar proof in Mathematische Zeitschrift [192], except this time using a quite different lemma originating with Borel.

Theorem 4.6 allows us to give several examples related to the Hadamard product (Section 3). Example 4.2. For z E B(0, I ) , let w

f(z) = Log (1

+ 2) = C ( - l ) k f l k=l

and

03

Then, by Theorem 4.6, there is a sequence {'=on) is a natural boundary for

Let

z

k

of +l's and -1's such that C ( 0 , l )

6. Natural Boundaries

242

Then, since G(z) = zF1(z), G(z) also has C(0,l) as a natural boundary. $ = - Log (1 - z) which when analytically However, F * G = f * g = continued in all possible ways, has a Riemann surface with infinitely many sheets.

CE1

In fact, the same sort of construction gives:

Example 4.3. Let h(z) be any function which (has a branch that) is analytic in B ( 0 , l ) and has a singular point on C(0,l). For z E B(0, l ) , let

--

-

anzn, lim [anl* = 1 .

h(z) =

n+m

1

Then the series C z = o a:zn also has radius of convergence 1, and so by Theorem 4.6, there is a sequence (6,) consisting of 1's and -1's such that

has C ( 0 , l ) as a natural boundary. Nevertheless, clearly, Ic * Ic = h.

Example 4.4. Let

x 00

1 22 F ( z ) = - -= a n r n , for z E B ( 0 , l ) , 1-z2 4-z2 n=O

+

where 1 a n = { 2-n

if n is even if n is odd.

By Theorem 4.6, there is a sequence {en) of 1's and -1's such that

x 00

g(z) =

~~z~~has C ( 0 , l )

n=O as a natural boundary. Let G(z) = g(z) &.

+

Then G * G =

1

+

42 9

and the singularities f4 of G*G are not representable as products of the singularities of G. This is another example of the phenomenon discussed in Example 3.5.

Note 4.10. The analogy between Theorem 4.6 and the result containing lines of Julia, described a t the end of Note 5.1.11, is not accidental. In fact, there is an

6.5. The Po'lya-Carlson Theorem

243

extensive analogy between results on lines of Julia for entire functions and results on singular points of functions with a finite radius of convergence. This was first suggested by Andr6 Bloch who noted the analogy between the existence of a line of Julia for every entire function and the existence of a singular point on the circle of convergence of a power series with a finite radius of convergence, and continued "L'analogie avec les points singuliers des dries entihres peut servir de guide dans la th6orie des directions singulihres ..." Pblya, in Sections 5058 of Mathematische Zeitschrift [192], proved a number of such theorems from a unitary point of view, in which a general argument is made to correspond via two different lemmas either to a result on singular points of power series with finite radius of convergence, or a result on lines of Julia of entire functions of infinite order. For example, the analogue of Fabry's Theorem 4.3 is: If f (z) = C2=oankznk is entire of infinite order and lim, ,, $ = m, then all rays emanating from the origin are lines of Julia for f . And the analogue of the result of Pblya mentioned in Note 4.6 concerning density of non-zero coefficients is: 00 an, znk is entire of infinite order and hk,, = D < m, then Iff (z) = every closed wedge-shaped region with vertex a t the origin and angular opening a t least 2nD contains a t least one line of Julia of f .

6.5

The P6lya-Carlson Theorem

In Sections 2 and 4 we discussed results which say that if a power series with a finite radius of convergence has "too many" coefficients zero, then the circle of convergence is a natural boundary for the function represented by the series. However, in Section 1, we gave examples also of power series (e.g. Example 1.3) for which no coefficients are zero, and yet there was a natural boundary. In this section we attempt to elucidate these examples. The basic problem is somehow to distinguish functions such as C:=)=, 7(n)zn,( ~ ( n= ) Cdln I), which has C ( 0 , l ) as a natural boundary from 03

z c n z n = -which is rational. (1 2)2 n= 1 Clearly the answer does not lie in growth of the coefficients. The question is: where does it lie? A useful first step would be to distinguish power series with finite radius of convergence which define rational functions from others. In this direction there is a famous result of Kronecker.

244

6. Natural Boundaries

Theorem 5.1 (Kronecker). Suppose f nant CoCl Am =

c 1 c 2

(2) =

Crz0Cnzn. Consider the determi-

. . . Cm . - . cm+1

CmCm+l

Then f is a rational function, that is, f ( r )= if and only if there is a p such that for all m

C2m

8where P and Q are polynomials,

> p, A,

= 0.

Proof. Suppose first that for all m 2 p, Am = 0. Clearly, we may assume with no loss of generality that p 2 1 and that A,-I # 0. (If Am = 0 for all all m 2 0, then f G 0). Then the last column of A, is a linear combination of the first p columns, and hence there exist complex numbers a k , 0 k 5 p - 1, such that

<

or, in brief, if we define

then Lj+, = 0 for j = 0,1,. . . ,p. We now prove by induction that Lj+, = 0 for all values of j . Suppose then Lj+, = 0 for j = 0,1,. . . , m - 1, where m 2 p + 1 , and we need to prove that Lm+, = 0. Then

(5.1)

6.5. The Pdlya-Carlson Theorem

Now, to every column from the pth one on in A,, add the linear combination with coefficients ao,a l , . . . ,a,-1 of the p immediately preceding columns. This does not change the value of the determinant and so by definition of Lj+,, and the induction hypothesis (5.1)

However, it is easy to see (for example, by successively expanding by minors of

6. Natural Boundaries

the p

+ l'st column) that this last determinant

So we have, for any m 2 p

+ 1,

By hypothesis, # 0, and Am = 0, hence LP+, = 0, as was to be proved. But this means by induction that Lj+, = 0 for all integers j 1 0. That is P- 1

akCj+r k=O

+ cj+,

= 0 for all j 2 0 ,

or setting j = n - p,

for all n 2 p. Hence, for all n

> p,

and so, iterating,

for all n 2 p, where K is a constant which does not depend on n. Hence, Cz=o Cnzn converges in some open disk centered at 0. Also, clearly all a0zP)f (z) which have index p vanish by coefficients of (1 aP-lz ap-nz2 . ( 5 . 2 ) , and so (1+aP-lt+ a p - 2 ~ +.2. . aozP)f (2) = P ( z ) where P is a polynomial of degree at most p, and so f is rational. Conversely, if f (r)= = = CUzu,then po +plz . phzh = (qO qlz . . . qnzn)(Co C ~ Z . . .). Hence, all coefficients which have index 2 h 1 on the right must vanish; that is C;=-, qkCr-k = 0 for a11 r 2 h 1 (qk = 0 for k n + 1). Hence the columns in

+

+ +

+

+

+

>

+ + + +

:::::::: +

>

+

+

xr.o

+ + +

6.5. The Po'lya-Carlson Theorem are linearly dependent for m. 2 h N o t e 5.1. For a power series minant

Hp)=

247

+ 1, and so A,

= 0 for m >_ h

+ 1.

Cr=,anzn, and k an integer > 0, define the deter-

HP)by HA^) = 1 ; an :

an+k-1

...

an+k-1

k

>1.

an+2k-2

"'

Thus, (0) A, = H,+,,

m20.

HP)

The are called Hankel determinants and play an important role in various aspects of analytic function theory. For further information, see e.g. Henrici [112], and two papers by Pommerenke's Mathematika [204]. Before stating the next result, we need a definition. By the Riemann Mapping Theorem (Theorem 1.5.1), any simply-connected region D of the plane with two boundary points can be mapped by a univalent function F onto B(0, I ) , and furthermore, given zo € D , we can require that F(zo) = 0, and Ff(zo) > 0. Clearly, by a dilation, if we consider instead univalent functions mapping D onto B(0, p ) for some p, then we can require F(zo) = 0 and F1(zo)= 1. Also, by the Riemann Mapping Theorem, given zo, the function, and so p, is uniquely defined by this prescription. Thus we have Definition 5.1. Let D be a simply-connected region in the plane with at least two boundary points. Suppose 0 E D . Let p be the radius of the disk B(0, p ) such that D can be mapped by a univalent function F onto B(0, p) with F(0) = 0 and F1(0) = 1. p is called the mapping radius of the region D . T h e o r e m 5.2. Suppose 00

n=O is analytic in a region D (where 0 E D) whose mapping radius is are all integers. Then f is rational.

> 1, and the Cn

Proof. Suppose $ maps D univalently onto B(O,p), $(O) = 0, $'(0) = 1. Then has a simple pole a t 0, with residue 1 and so

in some open neighborhood of 0. Let

$

6. Natural Boundaries

248 where P,($) is the principal part of (&)m By the residue theorem,

and Rm(0) = 0.

where I? is the image of C(0,r) under the inverse of the Jordan interior of I? (so l+(z)l < r ) . However, (5.3) can be written as

1

1

4 with 0 < r < p, and

m

du

Now, since 4 is univalent in D and +(0) = 0, z E Jordan interior of I'),

z is in

zdu

& is analytic in D , and so (since

Hence, since for u E r , I+(u)I = r,

and so, since lq5(z)l < r , we have, 1 lim ( ~ ( Z ) ) ~ P ~=( 1- ) m--too z

uniformly in every closed subregion of D (since for z in any such region, we can choose an appropriate curve J? C D with z E Jordan interior of I?). Now, if p > 1, then we can always do the above with T > 1, which we assume from now on. Then (since 0 E Jordan interior of I?), for any two integers a and b, by (5.4)

1 I -(length 2n

1 1 1 of I') (const.) max -(const.) max -max% E r 14(z)la L E ~ 14(z)lb Z E ~IzI

K where K is a constant independent of a and b; since for z €

I?,

l$(z)l = r.

(5.5)

6.5. The Pdlya-Carlson Theorem On the other hand, letting Q,(z) = zmpm($),we have

= coefficient of the term za+b in the power series expansion of f (z)Qa(z)Qb(z). This coefficient is easily computed. q?)zu (and q?) = 0, v > n) then Let Qn(z) = CzE0

where dk = ~

k

u E ~ Ok

- ~ q ? )and , so

~1::

(a)

qf) Cj-p-uqv . We may note that q r ) = 1 for all n. A better notation is the following: for any sequence {C,} C-, = O for m > 0), let for n > m 1

where y =

>

and

coc,

= C,

of integers (with

.

In this notation, for j 2 max(a, b ) , e j = CaCbCj = LbLaCj, and, in particular, then

and so by (5.5), ICacbCa+bl

K

5 3 , where l$(z)l < r, and r > 1 .

Consider now the determinant

250

6. Natural Boundaries

Operating first on the columns and then on the rows with the operators Ln, we clearly have

+

Thus, A, equals the m + 1 x m + 1 determinant whose entry in the a l'st row and b 4- l'st column is La.CbCa+b. n o m the definition of the determinant as a sum over permutations, we get, by

Since r > 1, for all sufficiently large m, the right hand side of (5.7) is < 1, but A, is a determinant with integer entries by hypothesis, and so is an integer, whence A, = 0 for all sufficiently large m. By Theorem 5.1 it follows that f is rational. Theorem 5.2 is sometimes called the P6lya-Carlson Theorem, though this name is usually reserved for the following immediate corollary:

Theorem 5.3 (P6lya-Carlson). If

has integer coefficients and is analytic in B(0, I), then either C ( 0 , l ) is a natural boundary for f, or f is rational.

6.5. The Pdya-Carlson Theorem

251

Proof. If f is at all analytically continuable over C(0, I ) , then there exists a region D with B ( 0 , l ) D in which f is analytic. Let be the function F of Definition 5.1 for D. Clearly the mapping radius of D is > 1, for otherwise B ( 0 , l ) would be mapped by C#J onto a subset of itself, and hence by Schwarz' Lemma, since clearly C#J is not a rotation, IC#J1(0)I< 1, contradicting the definition of 4. And so, if C(0,l) is not a natural boundary for f , by Theorem 5.2, f must be rational.

5

Note 5.2. The proof of Theorem 5.2 is essentially Pblya's 11961. P6lya actually proves that the theorem is still true if f is allowed to have finitely many isolated singularities in D . Earlier P6lya [I971 had proved that if f is analytic in B(0, R), R > 1, and has integral coefficients, then f is rational; his proof using Wigert's Theorem 4.2. In this paper he conjectured Theorem 5.3 (p.510) and in 1921 Carlson [40] proved this conjecture for the first time. Later P6lya 11981 proved a much more general result about "determinantal criteria for analytic continuation'' involving as well as the concept of mapping radius, the smallest disk centered at 0 containing all points which are members of the derived set of some (including transfinite) ordinal order of D , and the transfinite diameter of D (see Note 4.1.9). For power series with integral coefficients, this last theorem has a special case: A power series with integral coefficients represents either (a) a rational function, (b) a function whose complete analytic continuation has a non-planar Riemann surface, (c) a function with uncountably many singular points. For an instance of (b), see Examples 5.1, 5.2 below. In the latter paper, P6lya also shows how his results relate to results on overconvergence (cf. Section 2). Note 5.3. It is worth examining the proof of Theorems 5.2 and 5.3 more closely. The statement of Theorem 5.3 is remarkable in that it juxtaposes integers, rational functions, and natural boundaries. However, what is really shown in Theorem 5.2, anzn is analytic is that if p is the mapping radius of a region D , where f (z) = Cr=o in D, and

then

-

1 lim lAmlX

m+m

1

5P

(compare formula (5.7) of the proof). Now, if p > 1 and A, is an integer, we satisfy the criterion of Theorem 5.1 and so get Theorems 5.2 and 5.3.

6. Natural Boundaries

252

Example 5.1. By the binomial theorem, we have for z E B(0, I ) ,

00

(2k-1)(2k-3)...3.1

=C

k!

k=O

k=O

Hence

has integral coefficients, C ( 0 , l ) is not a natural boundary, but there is a branch point singularity at Replacing z by zm we have

a.

which is analytic in B(O, 1) except for a branch point at z = made arbitrarily near the unit circle.

(a)+,

which may be

Example 5.2. Example 5.1 is an algebraic function; however, suppose we continue from it as follows: From formula (5.8) of Example 5.1, formally,

where B is real and w is complex. Using the integration by parts formula /(sin

-(sin 8)"-l cos 6

n-1

n

n

B ) ~= ~ s

(sin 19)"-~d0

repeatedly gives

The right side of this equation is convergent for z E B(0, a); the left side is an elliptic integral of the first kind (viz. Chapter 8) and represents a function which except for branch points at 0, oo, f , - f , each of infinite order. is analytic in C(, Hence, it is not an algebraic function.

6.5. The Po'lya-Carlson Theorem

253

Note 5.4. The study of functions q5 univalent in B ( 0 , l ) with $(O) = 0, #(0) = 1is a way of studying simply-connected regions of the plane analytically. One famous theorem about such functions is essentially due to Koebe:

If R is a simply-connected region containing 0 and with mapping radius 1 (so, in particular, if R = B(0, I)), then the image of R under any univalent map g with g(0) = 0; gl(0) = 1 contains the open disk B(0, (cf. Theorem 7.1.5.) Curiously enough, this theorem also follows from Theorem 5.2 and Example 5.1. For, suppose the theorem were false, and a0 is a boundary point of the image of R under g which is nearest 0, and lzol < Then through rotation and dilation of > 1, and for which f is a R , one can obtain a region R* with mapping radius boundary point. But then

a).

a.

&

&

is analytic in R* which contains 0 and has mapping radius > 1 but is not a rational function, contradicting Theorem 5.2. This proof is due to Szego; another proof follows from a result proved by P6lya in the later paper mentioned in.Note 5.2.

Example 5.3. Let {b,) be a sequence of integers such that 0 5 bn that there are infinitely many non-zero terms in the sequence. Then, arguing as in Example 1.3, Let L(z) = C z = l bn&.

< B , and such

and so is analytic in B ( 0 , l ) and grows most rapidly along the positive real axis. We now show that L(z) is never a rational function. If L(z) were rational, then the only singularities it could have on C ( 0 , l ) would be a finite number of poles. Writing x = reis where 0 5 r < 1, we have 60

lim ,-tea' radially

[(z - eisl2L ( Z ) ~5 lim (1 - rl2 r+~-

Ibnlrn 1 1 rneineI n=l

6. Natural Boundaries Putting r = e-Y, this last limit is 1 <

B lim (1 - e-Y)2 y+o+

n=l

eYn - 1

- e-Y)2

lim Y+O+

C y3/2n3/2 1

n=l

(1 - e-Y)2 = K lim =0 y+o+ y3/2

+ +

+

+

(since for x > 0 we have ex > 1 x $ = 1 x ( 1 + ): > 1 x3l2). So any poles of L on C(0,l) must be of first order, say at the points eisj, 1 5 j 5 r. Hence L(z) has for z E B(0,l) the representation

where the radius of convergence of the power series is

> 1, and so

-

lim lan[+ < 1 .

n+CO

But, for

121

< 1,

So, the mth coefficient of the power series representation around 0 for L(z) is, by (5.10) T

where by (5.11) la,[ is bounded as m + co. But by (5.9), this coefficient is just Cdlm bd. Hence Cdlm bd is bounded as m + oo. But this is impossible, since if {b,,) is a sequence of infinitely many of the integers bn all of which are non-zero, then the . . - b,, 2 e. Hence L is not rational, and so by coefficient of znl...nt is b,, Theorem 5.3, C ( 0 , l ) must be a natural boundary for L(z).

>

+ +

Note 5.5. Example 5.3 (which "explains" Example 1.3) is due to Pblya, who also bn& ; both in the London Journal paper proved the analogous result for CrZo B can be relaxed to of Note 5.2. From the proof, clearly the condition bn Cdln bd = o(n), since from this it will still follow that L(z) can have at most simple poles on C(0,l). Also, the restriction of the bn to integers is used only to apply Theorem 5.3. If the bn are real, non-negative, and infinitely many of them are 2 6 > 0, and Cdln bd = o(n), then the above argument still shows that L cannot be rational.

<

6.5. The Pdlya-Carlson Theorem

255

However, even for integer b,, further relaxation does not seem possible. For example, suppose b, = ,u(n), the Mijbius function. p(n) takes only the values -1,1, and 0, and 1, n = l 0, otherwise. din Hence C r = l totient function. Then

=

Z.

Again, suppose bn = +(n) =

Cdln +(d) = n, and so,

C

ksn

(k,n)=l

1, Euler's

which is rational and has a double pole a t z = 1. N o t e 5.6. It is not necessarily easy to decide whether a power series C r = o anzn, limn,, lanl i= 1 represents a rational function or has C(0,l) as a natural boundary. Consider for example, the function g(z) = C;==,[[an]zn where a is real and irrational, and [XI as usual, indicates the largest integer 5 x. Writing {x} = x - [XI for the fractional part, we see that

thus, if g(z) were rational (and so would have only poles as singularities on C(0, I)), then so would C r = l {an)zn be rational, and arguing as in Example 5.3, this last series (if it represented a rational function) can have as poles on C ( 0 , l ) only those of first order. Nevertheless. there does not seem an obvious wav to rule out this possibility, and the first proof that C:=l[an]zn and C:==,{an}zn have C(0,l) as a natural boundary was given by Hecke [I081 who used the uniform distribution of {no), a irrational to prove the result. Hecke's Theorem stimulated a great deal of further work. Comprehensive results along these lines and bibliographical references can be found in Carroll and Kemperman [41]. For example, they prove: lan[i= 1, an # 0 for n 2 no, and the only singularity of Cmno$zn If limn,, on C ( 0 , l ) is at z = 1, and, furthermore, a is irrational, is a complex-valued Riemann integrable function on [0, I ) , extended periodically to the whole real line and such that J,' + ( ~ ) e ~ " '#~ "0 for all sufficiently large integers rn, and bn = o(an) as n + m, then (an+(an)+bn)zn has C(0,l) as a natural boundary. (Hecke's Theorem is an 1, +(x) = {x}, bn 0.) Carroll and Kemperman prove even more general results, as well as results for Lebesgue integrable (on [0, 1)) and almost all real a.

+

=

N o t e 5.7. An immediate consequence of Theorem 5.3 (P6lya-Carlson), is that there are only countably many power series with integer coefficients which are analytic in

256

6. Natural Boundaries

B(0,l) and analytically continuable over some arc of C(0,l). Taken in conjunction with Theorem 4.6 (see also Note 4.9), one might think that in some sense or other, there are "more" power series with radius of convergence 1 which have C(0,l) as a natural boundary than are continuable over some arc of C(0,l). This question has had considerable study from a number of points of view: topological (beginning with Pdlya in 1917), probabilistic (beginning with Steinhaus in 1929), and set-theoretic (beginning with Hausdorff in 1919). In particular, the corollary of Theorem 5.3 cited a t the beginning of this note was found independently by Hausdorff. For a comprehensive survey of such questions, see the published Stuttgart dissertation of K. Hinderer [113], and two papers by Hinderer and Walk [114]. Finally, it should again be noted that there are many questions, theorems, and relations to other mathematics (e.g. Fourier transforms) which anyone who begins t o look a t the literature will discover have not even found room for mention here.

Chapter 7

The Bieberbach Conjecture By the Riemann Mapping Theorem (Chapter 1, Section 5), given a simply-connected region D in the plane which has at least two boundary points, there is an univalent function G mapping D onto B(0, I ) , and further, given zo E D , we can require G(zo) = 0 and Gi(zo) > 0. Under these conditions G is unique. Alternatively, if instead we are willing to map D univalently by a function F onto a disk B(0, p), and require F1(zo)= 1, as well as F(zo) = 0, then p is uniquely determined, and is the mapping radius of D , which has already been used (Theorem 6.5.2). Suppose now we consider the function 4 inverse to F on B(0,p); furthermore, suppose we consider D such that 0 E D , and zo = 0 (this involves no loss of generality). Then 4 is univalent on B(0, p); 4 : B(0, p) -+ D , 4(0) = 0 and 4'(0) = -= 1,and 4 is the unique such function, for a given D . Finally, clearly the map f (z) = $4(zp) is univalent on B(0, I ) , f (0) = 0, and fl(0) = #(0) = 1. Thus, after normalizing, we are led to study the class of functions f univalent on B ( 0 , l ) with f(0) = 0, fl(0) = 1. Implicitly study of such functions is, by the Riemann mapping theorem, the study of simply connected regions in the plane. On the other hand, one way to study functions is to study their coefficients. The Bieberbach conjecture is a conjecture about the coefficients of normalized univalent functions on B ( 0 , l ) . A tremendous amount of work was done on it and related problems before it was proved by Louis de Branges in 1988. Again, this chapter can only attempt to be an introduction to the subject. Fortunately for the interested, there are books covering the subject in depth; in particular, Pommerenke, Univalent Functions 12031, and Milin, Univalent Functions and Orthonormal Systems [157], as well as useful survey articles by Duren [62], and earlier, by Hayman [106], though this is now a little outdated. In Section 1 below, several distortion theorems other than those immediately relevant to the Bieberbach conjecture are discussed largely because of their intrinsic interest. The reader may wish to look up or try to discover other results which may be obtained by these essentially elementary methods. The Bieberbach conjecture itself is stated in Section 2, and a proof is given following

7. Bieberbach Conjecture the ideas of de Branges and others.

7.1 Elementary Area and Distortion Theorems Definition 1.1. S denotes the class of functions f univalent in B ( 0 , l ) such that f(0) = 0, ff(0) = 1. It is often useful also to consider simply-connected regions G in C,, where co E G. We assume G has at least two boundary points. Picking a point E C, - G, the map transforms G into a simply-connected region properly contained in C. Thus the Riemann Mapping Theorem shows that there is a unique function +(z) = bz + Cr=ob k ~ - kunivalent in G and mapping G onto C, - B(0,l) (with co going onto 00). This leads to the following normalized definition.

<

&

Definition 1.2. C denotes the class of all functions g(z) = a valent in C, - B(0, I), except for the simple pole at co.

&J

+ CEObkz-k

uni-

-

1

Note 1.1. If f E S7 then g(') = = z - l + x r = 2 a k z - k - l+x;=lhn+lz-n =z+. . . , and so g(z) E C. Furthermore, since f E S , g(z) # 0 for z E C, - B ( 0 , l ) . Conversely, if g E C, and is in the complement of the image of C, - B ( 0 , l ) 1 under g, then f ( l ) = g($-i = t+bo-c+Z:, b k z k - L + ( ~ O - C ( ) + ~ ~h=- 1~2 ' ' = a + (C - bo)z2 + . . . , and so f E S. (The choice of a C is, of course, necessary for f to be analytic in B ( 0 , l ) ) . Note, however, that every function in S omits the value co,whereas functions in C need not omit any value.

<

Theorem 1.1. Suppose f (z) = z + CrZ2 avz" E S, then the image of B ( 0 , l ) under f has area n(1-k CrZ2 vlav12). Proof, This is just the well-known "Area Theorem", which we have already used before (e.g. Theorem 1.5.3) specialized to B ( 0 , l ) and a1 = 1 and which is proved in the Appendix.

Theorem 1.2. Suppose g(z) = z + C r = o byz-" E C, then the area of the complement of the image of C, -B(0,l) under g is

Proof. Consider the circle C(0, r ) , r > 1 under g, C(0, r) is mapped onto a simple closed analytic Jordan curve C with the equation w = w(B) = g(reie) ( r is fixed). The area J of the Jordan interior of C is (with w = u + iv) given by u(d)ut(B)dB =

62. (

~(e)+zo(e>

W~(~)-W~(B) dB.

)(

7.1. Elementa y Area and Distortion Theorems Now w1(8) = &(reis get

+ Cr=obvr-ue-vie)

259

= ireis - i C z l ~ b , r - " e - " ~ ~and , so we

On multiplying and integrating termwise, all terms which contain a term ekis,k 0 will vanish, which gives

#

But J >_ 0, and so, C r = l ~ l b , ( ~ r -5~r"2 ; thus

and so, letting first r + 1, and then m + oo, we see that whence, letting r + 1 in (1.1), we get, as claimed,

CzO=, vl bvI2 converges;

As an immediate corollary of J 2 0, we also get

Theorem 1.3. If g(z) = z + C r = o bvzdv E C, then C= :l Ibll 5 1.

vlbv12 5 1; in particular,

Note 1.2. Theorems 1.2 and 1.3 are due to Gronwall (891. Note further that if g(z) = z C r = o bvz-" E C, then not only is lbll 5 1, but if lbll = 1, then g(z) = z bo + e"zv1, for some real 8. It is remarkable that these non-obvious analytic facts follow from the simple geometric proposition that area is non-negative. One may also note that unless g(z) = z + bo, the area in question in Theorem 1.2 is always < n.

+

+

. nzn = Then F is analytic in B(0,l). An easy computation shows that if F ( z ) = F(w), and z # w, then zw = 1, hence F is univalent in B(0, l ) , and so F E S. F maps Example 1.1. Let F ( z ) = C:=,

260

7. Bieberbach Conjecture

< < -a)

C ( 0 , l ) onto the straight line segment L = { z : z real, - co z described twice, and the image of B ( 0 , l ) under F is C - L, which has infinite area. The corresponding function in C, G ( z ) = = z - 2 $ maps C(, - B ( 0 , l ) F(;) onto C, - {z : z real, - 4 z 0 ) .

+

< <

Example 1.2. Suppose F ( z ) = z - $. F is clearly analytic in B(O, 1); and an easy computation shows that if F ( z ) = F ( w ) and z # w , then z2 zw w2 = 4, whence F is univalent in B ( 0 , I ) , and so F E S. The image of B ( 0 , l ) under F is as in Diagram 7.1.

+

Image of C ( 0 , l ) under r

+

-f

Diagram 7.1

%.

and has area ~ ( 1 3(!j)2) + = A corresponding function in C is

So the area of the complement of the image of C - B ( 0 , l ) under G is

7.1. Elementary Area and Distortion Theorems See Diagram 7.2. Image of C(0,l) under

&

Diagram 7.2

+

Example 1.3. Let f (z) = ez2 e > 0. f is clearly analytic in B ( 0 , l ) . However, f is not univalent in B(0,l). For, suppose it were in S, then the function

would be in C; however,

contradicting Theorem 1.3. A direct proof that f is not univalent seems considerably more involved.

+

Example 1.4. Let f ( z ) = a z k (1_ZZ)2, k an integer 2 3, a any non-zero real number, then f is clearly analytic in B ( 0 , l ) ; however, f is not univalent in B ( 0 , l ) . For, suppose f E S ; then

+

+

would be in C. But g(z) = z - 2 - azk-' . . ., which contradicts Theorem 1.3. Again, a direct proof that f is not univalent seems considerably more complicated. Theorem 1.3 already indicates that Theorems 1.1 and 1.2 may have non-obvious analytic consequences; another important instance, which explains Example 1.3, is Theorem 1.4. Suppose f (z) = z

+ zZ2a,zV E S , then la21 5 2.

7. Bieberbach Conjecture

262

Proof. The idea of the proof is to make a suitable transformation of f , pass to the corresponding function in C, and use Theorem 1.3 to obtain information about f . Thus we note that

and let

F ( z ) has the following properties:

(i) All powers appearing in the expansion (1.2) are odd. (If an even power appeared, on squaring, we should get the contradiction that f (z2) contained an odd power of z in its power series expansion.) Hence F is an odd function. (ii) F is analytic in B ( 0 , l ) . For f (0) = 0 and f is univalent in B(0, I), so f (z2) takes the value 0 in B ( 0 , l ) only at z = 0, where it has a double zero, and it follows that (f ( z 2 ) ) i = F(z) is analytic (and F(0) = 0). (iii) F is, in fact, univalent in B ( 0 , l ) . For if

then f (z;) = f (zi), and so (since f is univalent in B ( 0 , l ) ) zl = fz2. If zl = -z2, then since by (i) F is odd, F ( z l ) = -F(zg), and so by (1.2), F ( z l ) = 0, whence f (z:) = 0, and so zl = 0 = 22. By (i)-(iii) above, F is an odd function E S. Let $(z) = 1 be the corre(;I sponding function in C. Then

It now follows from Theorem 1.3, that I $az1 I:1, and so la21

< 2,

Note 1.3. If, for example, instead of F we use H(z) = (f (2')) in B ( 0 , l ) and

4,then H is univalent

and Theorem 1.3 (which shows that lbl l2 result than Theorem 1.4.

+2jb2l2 < 1) now gives la21 < 3, a poorer

7.1. Elementary Area and Distortion Theorems

263

+

Similarly, f (;I = z - ;a2 . . . and we get no further information from Theorem 1.3, if we try t o use this. Indeed, Example 1.1 shows that Theorem 1.4 cannot be improved. An immediate consequence of Theorem 1.4, that was already proved in a quite different way (Note 6.5.4) is

Theorem 1.5. Suppose f E S. If I3 is the boundary of the image of B ( 0 , l ) under f , then every point of B has a distance of at least from the origin, (i.e. the image and is the best possible such constant. contains the disk B(0,

i

i));

Proof. Suppose for z E B(0, I ) , f (z) # C (there is some such C by Liouville's Theorem). Then C # 0 (f (0) = 0) and so fl(z) = is univalent in B ( 0 , l ) . But letting f (z) = z C= :2 a,zV,

+

+

+

Cf (z) - C z Ca2z2 - . . C - f(z) C-z-a2z2-... and so fl E S . It now follows from Theorem 1.4 that la2 &I 5 2. Hence I 1 5 2 la21, and so again by Theorem 1.4 (since f E S ) , 5 4, or ICI 2 $, which proves the theorem. Example 1.1again shows the constant is best possible.

+

&

+

i

Theorem 1.4, taken with Note 1.3, leads to the question: for what functions f E S does la21 = 2. This is answered by the following

Theorem 1.6. Iff6s; f (z) = z + C z 2a,zY, and la21 = 2, then f (z) = for some real 8. Proof. If f E S, F ( r ) = (f(z2))'l2 and )(z) =

(l-efs,)~

7

&,then the proof of Theorem

%+

1.4 shows that ) 6 C where )(z) = z C,, b,z-". By Theorem 1.3, ila2I2 Cr=2vlbv12 5 1. SO if la21 = 2, all the b, = 0 for v 2 2, and hence

+

i8

)(z) = z - %, whence F ( z ) =

l-:ezz.

and so f ( z ) =

(,-:BZ)2

.

Note 1.4. Theorem 1.6 explains Example 1.4 above. Theorem 1.4 was first proved by Bieberbach who also discovered Theorems 1.2 and 1.3 independently of Gronwall [go]. The existence of a constant M such that B(0, M ) is taken on by every f E S was first noted by Koebe [134], and the exact value M = found by Bieberbach (implicitly) op. cit. and Faber [69]. Because of theorems like Theorems 1.41.6, the function (,_ZZ), is often called the "Koebe function" and the functions " 5 are called "rotations of the Koebe function". Another proof of Theorem (1-eiez) 1.5 (different from the ones above and in Note 6.5.4) can be found in Ahlfors [3] Conformal Invariants, p. 29. An important result further justifying a special denotation for these functions is the following "distortion theorem".

i

7. Bieberbach Conjecture Theorem 1.7. I f f E S , then for all z E B(0, l),

and equality on any side only holds for some rotation of the Koebe function, i.e. for 0 real. a function of the f o m (l-e:Q,2, Proof. The proof again depends on Theorem 1.3, except that this time we first use a Mobius transformation in order to obtain nontrivial information. is a composition Given a fixed point z0 E B(O, I), the function g(C) = f of univalent functions and so univalent in B(0,l). Expanding in a power series around 0, we have

(&+

1

and so

Writing

h(C) = C + p2c2 + . - . ,

we have from (1.4) and (1.5) that

and from Theorem 1.4 that 20 E B(O,l),

lP21 5 2; hence, multiplying by

Since zo was an arbitrary point of B(O,1) and IRewl triangle inequality that for all C E B(0,l)

*2

we have for any

< Iwl, it follows from the

7.1. Elementary Area and Distortion Theorems However, writing

C = reie, d

Re Log f'(reie) (1.8)

Hence, (1.7) becomes

Integrating throughout from 0 to 121, we get (since fl(0) = 1 by hypothesis)

or, by exponentiating,

1- IzI < If1(4l (1 + 1 ~ 1 )~

5

l +1' for z E B(O,1) , - ,4)3 ,

which proves (b). Now, for z E B(0, I), integrating fl(C) along the straight line from 0 to z gives (since f (0) = 0) by (1.9),

On the other hand, let I? be the curve with endpoints 0 and z such that f (I?) is the straight line from 0 to f (2). Since f is univalent I? does not cross itself and is rectifiable. Also, if (' = reie, then an easy computation shows that ldCl dl
>

(1.lo) and (1.11) together prove (a). If equality holds in one side of (b), say for zl # 0, then the proof shows that equality must hold in the corresponding side of (1.7) for all C with 0 < ICI 5 lzll. Thus, taking (' = rzl,O < r 5 1, and dividing both sides of (1.7) by r, and then letting r -+ 0, we get (since fl(0) = I ) , that equality must hold on one

7. Bieberbach Conjecture

266

side of -41z11 5 Re(zl fl'(0)) 5 41z11. But if Re(zl fl'(0)) = f41zll, then 41z11 5 lzll If"(0)l and so I f1'(0)l 2 4. But then by Theorem 1.4, lfl'(0)l = 4, and so by Theorem 1.6, f (z) = (l+e:,z), for some real 8. Since (a) follows from (b) by integration, similar remarks hold for it.

Note 1.5. Theorem 1.7 also shows immediately that for f E S , and z E B(0, I ) ,

1 5 (%) however, one can do better by noting that if f E S, / then also for the function h defined by equation (1.5) of the above proof, h E S , so,

(*13

for all

';

5

t E B(0, . . I),,.

ICI

IS1

< lh(S)l 5

- lSl)2 . (1 + IC1)2 Given any point zo E B(0, I), let C = -20, then by equation (1.5) above,

since g(-zo) = f (0) = 0.

or (replacing zo by z)

I

-1

I

1< f

I

5-

+

1" , for all z E B(O, 1)

1-121

Here again equality can only hold for a rotation of the Koebe Function. One may also note that taking z = reis , in the left side of Theorem 1.7 (a), and letting r + I-, one obtains another proof of Theorem 1.5. Theorem 1.7 is also due to Bieberbach in Gottingen Nachrichten [24], as well as Pick and Plemelj (see p. 946, footnote 2 of Bieberbach's paper) who earlier assumed la21 5 2, first proved by Bieberbach. As another example of the same sort of arguments we have been using, we have the following somewhat surprising Theorem 1.8. Suppose $ is univalent in S = C, - {z : z 5 0); that is the plane with origin and the negative real axis deleted. Then lqhl(x)l is non-increasing on the positive real axis. Furthermore, if l$'(x)l takes on the same value at two digPerent points of the positive real axis, then $(z) = Az B , where A, B are constants.

+

Proof. The theorem can be proved by mapping B ( 0 , l ) onto the slit region S and applying Theorem 1.4.

(e) 2

Precisely, any function of the form a where a is a positive real number, maps B ( 0 , l ) onto S , and is univalent in B ( 0 , l ) .

7.1. Elementary Area and Distortion Theorems Hence F(z) = ) (a

267

(E)~ is univalent ) in B ( 0 , l ) . Furthermore, expanding

F in a power series about 0,

and so if

Ifz 2

f (z) =

)(a(=) 14a4'(a)

,

then f E S, and by Theorem 1.4,

I

am"(a)

I

+ 1 5 1, for all positive real a .

Hence.

and so, since a

> 0,

whence [)'(a) 1 is non-increasing on the positive real axis. Furthermore, if I)'(x)l takes the same value at two different points of the positive real axis, then, for some = 0, and so we must actually have real a , we must have

but this means, by Theorem 1.6, that f (z) = i +(the Koebe function itself is the only rotation of the Koebe function for which the second coefficient has argument 0). Hence z ) ('11 1) = labf(a) -2 (1 - z)2 + $(a)

(.

where a and b are constants, which proves the result.

Note 1.6. Trivial examples of Theorem 1.8 are )(z) = Log z and )(z) = zllm, m a positive integer. By applying a preliminary rotation first, Theorem 1.8 can be stated for the plane slit along any ray. Theorem 1.8 is due to Loewner [147], where a more detailed study of univalent maps of slit regions appears. The following paper in the same journal (by Frank and Loewner) contains an application of these results to a hydrodynamical problem.

7' . Bieberbach Conjecture

268

Another theorem on slit regions which may be proved similarly is: Theorem 1.9. If h(z) = z + Cr=o Cnz-" is univalent in C - {Z : -p 0), then Re C1 5 0.

maps C - B(O, 1) onto C - {z : -p Proof. The map z + f univalent in C - B(0,l). Hence

5 z 5 0,p >

< z < 0) and is

is univalent in C - B(0, I), and so H E C. Clearly the terms involving C,, v 2 2 as coefficients do not contribute to the coefficient of $ in the Laurent series expansion of H around 0, and so we get from Theorem 1.3 and (1.13),

Hence Re

(:c1) 5 0, or since 4/p is real, Re Cl < 0.

Parallel to Theorem 1.7 we may also inquire about inequalities for arg fl(z). Theorem 1.10. If f E S, then

Proof. If arguing from inequality (1.6) of the proof of Theorem 1.7, we use IIm wl IwI instead of lRe wl 5 Iwl, and replace z0 by 5 we are led to

-41'1

1 - ICI2

(rn)

4111 < I m cfU(1) 5 1- ICI2 '

But, writing C = reie,

and so (1.14) becomes

-4 1- r 2

d

5 ;i;arg fl(re")

4

1-1-2

5

7.1. Elementary Area and Distortion Theorems Integrating from 0 to

121,

-2 Log

gives

( a 5 )5 1

IzI

argfl(z)

(i T izi)

2 ~ o g-

Note 1.7. Theorem 1.10 is a result of Bieberbach [25]. For the rotations of the Koebe function (l+e?s,)z , 1 - eiez f'(2) = (1 + ,ieZ)3 . Also, since for z

and so for z

# 0 the points 1f eisz belong to the circle C(1, lzl), for z # 0,

# 0,

arg f '(2) = arg(1 - eiez) - 3 arg(1

+ eisz) 5 4 arcsin lzl 1- r 2

dr = 2 Log

(-)1+- IzIIzI 1

Thus (for z # 0), Theorem 1.10 is not sharp for the rotations of the Koebe function. In fact, it is not sharp at all. Sharp estimates for I arg f1(z)l were found by Golusin [86]. These are

I arg ft(z)l 5 4arcsin It/ if lzl 5

fi

For further sharp bounds on arguments associated with univalent functions, see, for example, Chapter IV of Golusin, Geometric Theory of Functions of a Complex Variable [85]. It is of some interest to ask what results are obtained if suitable restrictions are made on the image domain of B(0,l) under an f E S. For example, since the Koebe function does not map B(O,1) onto a convex region, we might ask after Theorems 1.5 and 1.6, what happens if in Theorem 1.5, we required the image to be convex.

Theorem 1.11. Suppose f E S and the image 7 of B(0,l) under f is convex, then the disk B(O,1/2) E 7, and in general, this is the largest such disk centered at 0.

7. Bieberbach Conjecture

270

Proof. The proof depends upon Theorem 1.5. Suppose f ( z ) # C for all z E B ( 0 , l ) . We claim that if g ( z ) = ( f ( z ) - CI2 , then g is univalent. For g is clearly analytic in B ( 0 , l ) and if for some z and w in B ( 0 , I ) , g ( z ) = g ( w ) , then f ( z ) - C = f( f ( w ) - C ) , and so since f is univalent, if z # w, then f ( z ) # f ( w ) , whence we have f ( z ) f (w) = 2C. But then since 7 is = C E 7; contradicting the definition of C , since by hypothesis convex, 7 is the image of B ( 0 , l ) under f . If f(z) =z+a2z2+... ,

+

then

g ( z ) = ( f ( z )-

c )=~c2- 2Cz + (-2Ca2 + 1)z2+ . . .

and so is univalent and h E S. Since by definition of C , g ( z ) # 0 for any z E B ( 0 , I ) , h ( z ) # C / 2 for any z E B ( 0 , l ) . Since h E S, by Theorem 1.5, 1C/21 2 $, and so ICI 112, which proves the theorem. The function & shows the result is sharp since it maps B ( 0 , l ) onto the halfplane { z : Rez > -112).

>

As a parallel to Theorem 1.10, we have

Theorem 1.12. Suppose f E S and the image 7 of B ( 0 , l ) under f is convex, then / a r g fl(z)l 2arcsinlzl ,

<

and there is a function f E S with convex image 7 so that equality holds for a point zo of B ( 0 , 1 ) . Proof. We use Theorem 1.5.5 (The Schwarz-Christoffel Formula for B ( 0 , l ) ) . Suppose f maps B ( 0 , l ) onto the Jordan interior of a convex polygon ??3, with n sides. then, by Theorem 1.5.5,

Since f ( 0 ) = 0 and f l ( 0 ) = 1, we get C2 = 0 and C1 = n ; = l ( - B k ) l - a k . Since the points B k which go onto the vertices of the polygon all lie on C ( 0 ,I ) , we can write B k = e - i e k , Ok real. Finally, recalling that the quantities T a k are the interior convexity requires ar, _< 1, and the formula for the sum angles of the polygon of angles of a polygon gives

vn,

7.I . Elementary Area and Distortion Theorems Taking these considerations into account (1.15) becomes f (z) =

1' fi O

(1 - eiskw ) ' ~dw

,

k=l

ELl

where p k = a k - 1 5 0, and p k = -2. Since I arg(1 - eiekz)l the points 1 - eiskz lie on C(1, lzl)), we have from (2)

I arg f '(2) 1 =

1

< arcsin lzl (all

n

pk

arg(1- eiskz)

5 2 arcsin lzl .

k=l

Since this last inequality does not involve the number of sides of the polygon, and since any convex set may be arbitrarily approximated by a polygonal convex set, it holds for all convex 7. The function f (z) = & again shows that the result is sharp, as fl(z) = &I and so arg fl(z) = -2arg(l - z) which takes on the value 2 arcsin lzl for suitably chosen z (which in fact can be explicitly computed). Note 1.8. The elegant proof of Theorem 1.11 given is due to MacGregor [150], although the theorem had been long known. Theorem 1.12 is a result of Bieberbach in his already cited paper of 1919. Many results similar to those in Theorems 1.51.12 can be found by similar methods; for some further examples, in addition to the literature already cited, see Chapter V, Section 8 of Nehari, Conformal Mapping 11681. Theorem 1.13. If f E S, and 2.1,z2 E B(0, r ) where r

< 1, then

Proof. Since f is univalent in B(0, l ) , f ' does not vanish in B(O,r), and hence by the maximum and minimum modulus principles, and Theorem 1.7 for all z E

and so the result follows. Note 1.9. The function again shows the result is sharp. Koebe proved a more general distortion theorem of the same sort as Theorem 1.13, but with less precise bounding functions: If R is a region in the plane and D another region such that D C R, then there is a positive constant M , depending only on R and D, such that if f is univalent in R, then for any zl, z2 E D ,

7. Bieberbach Conjecture

272

This result is of importance in proving Koebe's famous results on uniformization (which were established in 1907-1909 in a series of papers in Gottingen Nachrichten; about which there is copious literature, but which cannot find place here. The above result is stated in [135].

Note 1.10. The theorems of this section dealing with limits on distortion of B ( 0 , l ) under univalent maps f , or the size of disks in the image centered at f (0) are clearly different from those of Chapter 2, where we were examining the largest disks, necessarily contained in the image Z of B ( 0 , l ) under an analytic or univalent map f , which were centered anywhere in 1.The difference between disks centered anywhere and disks centered a t f(0) is considerable (cf. for example the image Nevertheless, as Landau first noted in 1922, there is a of B ( 0 , l ) under connection which can be used to establish distortion theorems for analytic maps which are not necessarily univalent. As an example of Note 1.10, we have the following result.

Theorem 1.14. Iff is analytic in B(0, I ) , f (0) = 0, fl(0) = 1, then the image of B ( 0 , l ) under f contains a disk B(O,p), where p is a positive constant. Proof. Suppose f is analytic in B(0, I), f (0) = 0, f'(0) = 1, and f omits both the points eie and aei@for some real 8, $, and a # 1. Let n be an integer 2 2 and let

('1'

=

(

f (z) - aei* l/n f ( 2 ) - .ie

)

+ + ... (-aei.li' -ei0 + + . . . 2

=

2

- (aei(@-e) - (e-ie - aei(@-2@)z+ . . .)I/" =

Then, F is analytic in B(0, l ) , and F does not take either 0 or 1 for z E B ( 0 , l ) . i(+-e)

Hence, letting a = a1Ine-, and b = Theorem 2.2.3, for example, that

i(J.-s)

al'nen

(evie -

'e - ' 9 , a

we have by

where v is a fixed branch of the inverse of the elliptic modular function p. Taking for simplicity n = 2-", and $ = 8, and setting L = = we get from (1.17), 1 L 2 lbl = - ( 2 " - I ) , 2n

w,

> L we have a contradiction. and thus if n is so large that Hence, we get that if n is the least positive integer such that the image of B ( 0 , l ) under f contains the disk B(O,2-").

> L, then

7.2. Some Coeficient Theorems

273

Note 1.11. The above result is due to Landau [138]. For yet another covering theorem of the same sort, under the auxiliary condition that for z E B(0, l ) ,f (z) # 0 for z # 0, see Theorem 8.6.11. Actually, many results on functions univalent in B ( 0 , l ) can be made t o yield results on functions analytic in B(0, I ) , by what is known as the principle of subordination introduced by Littlewood in 1925. This principle is dealt with in almost any discussion or book on univalent functions. See, for example, Chapter VIII, section 8 of Golusin's cited book, or Chapter 2, section 2.1 of Pommerenke's. Both of these contain references t o further literature. There are also generalizations of distortion theorems to multiply-connected regions, and to functions of several complex variables.

7.2

Some Coefficient Theorems

Let us note first Theorem 2.1. The family S is compact; that is, if {f,) is any sequence of functions in S , then {f,) contains a subsequence converging for all z E B ( 0 , l ) to a function of S .

Proof. By Theorem 1.7,

Hence the family S is locally uniformly bounded for all z E B(O, I), and so by Theorem 1.4.2, S is a normal family, and {f,} has a convergent subsequence. By Theorem 1.2.6, a convergent sequence of univalent functions converges t o either a univalent function or a constant. Since for all f E S, fl(0) = 1, the limit function cannot be a constant, so it must be a function f univalent in B(0, I ) , and, as is easily seen, satisfying f (0) = 0, fl(0) = 1. Note 2.1. It is worth noting that since B ( 0 , l ) can be carried onto any other disk by a Mobius transformation, and since a family of functions is normal in a region R if and only if it is normal in all disks C R, we get by the above argument that any family of functions univalent in a region R is a normal family. Furthermore, given a point C E R, the subfamily of this family consisting of all functions such that I fl(C)I C > 0, is compact by the above argument. In fact, any condition ruling out constant limits will suffice to demonstrate compactness.

>

As an immediate consequence of Theorem 2.1, we have Theorem 2.2. There exists a function F, E S , such that A, = supfEs lnth coefficient o f f ( = lnth coeficient of F,I.

7. Bieberbach Conjecture

274

Proof. Let J ( f ) = [nth coefficient of fl. Clearly there is a sequence of functions S such that limk+, J(fk,,) = A,. Since S is a compact family, there is a of {fk,,) converging to a function Fn E S , and, as is easily subsequence {fk,,,) verified, J ( F n ) = lim J(fk,,,) = An . fk,n E

V+OO

After the results of Section 1, in which the function (,_Zz), frequently played the role of extremal function, it is reasonable to conjecture that it might well be extremal for the problem of the maximum modulus of the nth coefficient (as it is for la21). This is the Bieberbach conjecture. Precisely

Definition 2.1. (The Bieberbach Conjecture) is that iff (z) = Z+C;.~ anzn E S , then la,/

5 n.

The Bieberbach conjecture stimulated much research on univalent functions throughout the 20th century and was attacked by a great variety of methods since Bieberbach first posed it in 1916. In 1984 a proof was announced by Louis de Branges. The proof given here basically follows the version of his proof given by Fitzgerald and Pommerenke in 1985 (see also Conway [51], vol. 2). A basic tool of this proof goes back to 1923 when it was proved by Karl Lowner (who later immigrated to the U.S. and Anglicized his name to Charles Loewner). Lowner used his approach to prove the Bieberbach conjecture for a3. We begin thus with the definition of Loewner chains.

Definition 2.2. A Loewner chain is a continuous function f : B ( 0 , l ) x [0, oo) + C such that

(i) for all t E [0, m),f (z, t) is analytic and univalent (ii) f (0, t) = 0 and (iii) if 0 5 s < t

(0, t ) = et

< oo, f (B(0, I),s) c f (B(0, I),t)

Example 2.1. f (2, t) =

is a Loewner chain. Note that f (B(0, l ) , t ) is the

slit region C-{z:zreal,

et - o o < z I : --) 4

(see Example 1.1).

Note 2.2. For any function g E S, f (z, t) = etg(z) automatically satisfies properties (i) and (ii) of Definition 2.2, but property (iii) is not automatically satisfied. This is corrected by the following result which depends on the Riemann Mapping Theorem (Theorem 1.5.1).

7.2, Some Coeficient Theorems

275

Theorem 2.3. Suppose {R(t) : 0 5 t < co) is a family of nested simply-connected regions (i.e. for 0 5 s < t 5 co, R(s) R(t)) such that as n -+ co, R(t,) -+ @. For each t > 0, let ht be the inverse of a Riemann mapping between B ( 0 , l ) and R(t) . So ht : B ( 0 , l ) + R(t) and we can prescribe ht (0) = 0, h: (0) = P(t) > 0. Let ht (z) = h(z, t) and P(0) = Po, then (a) p is a continuous strictly increasing function and P(t) -+ co as t -+ co.

(9)

(b) Let X(t) = log and f ( z , t ) = &h(z,X-l(t)); then f defines a Loewner chain with f (B(0, I ) , t) = &R(X-' (t)). Proof. Note that if t, -+ t, then htn -+ ht and so P(t,) -+ P(t); hence p is continuous. Now if s is fixed and < t and R(s) R(t), then there is an analytic function taking B ( 0 , l ) onto itself such that h,(z) = ht($(z)) for all z E B(0, l ) , and $(0) = 0. Then l$(z) 1 IzI by Schwarz' Lemma, and in fact, because R(s) # R(t), l+(z) 1 < 121. Also, here I$I(O)I < 1 (again because of the inequality). Hence P(s) = hl(O, s) = hl(O,t)$'(O) = B(t)$I(O). Now I$'(O)I < 1 and so for s < t , P(s) < P(t) so P is a strictly increasing function from [O, co) t o [O,co). Furthermore, since R(t) + @ as t + co,P(t) + co as t -+ co. SO P : [O, 00) -+ [PO,co) is a continuous strictly increasing, and onto map. So X(t) = log is a strictly increasing function taking [0, m ) onto itself.

5

<

(y)

Then, f (2, t) = & h ( z , A-'(t)) is a continuous function which is clearly analytic and univalent for all t 2 0. Also, clearly f (B(0, 1), s) C f (B(0, 1), t ) . Clearly f (0,t) = 0, (ht(0) = 0). Also, if u = A-'(t), then t = X(u) and so et =

y,so g ( f (z, t))l

= P(t)). Clearly f (B(0, I ) , t) =

= et ((since e h ( i , t)l z=o

-1fl(X-l ( t ) ) .Thus f is a Loewner chain. Po

z=o

&

Note 2.3. Clearly must appear in the result since for an arbitrary region 0 , there is no reason to think that the map from B ( 0 , l ) to fl (inverse of the Riemann map from R to B ( 0 , l ) ) is in class S, while for a Loewner chain f (z, 0) is in class S. An important example is: Example 2.2. Suppose y is a Jordan arc that does not pass through 0 and such that y(t) -+ co as t -+ co. Let y(0) = ao. For 0 t < co, consider yt, the restriction of y to [t, co) and let R(t) = @ - yt. Then, by Theorem 2.3, we can find a Loewner chain f ( z , t) .

<

Note 2.4. A Loewner chain can be thought of as a parametrized family of univalent functions ft(z) = f (z, t) starting a t fo and as t + co, the ranges of the functions expand t o fill the whole plane. Theorem 2.4. For every function fo E S, there is a Loewner chain such that f (z,O) = fo(z) in B(O,l).

7. Bieberbach Conjecture

276

Proof. Assume first that f ( z ,t ) is analytic in a neighborhood of B ( 0 , l ) . then 7 = f ( C ( 0 , l ) )is a closed Jordan curve. Let g : @ , - B ( 0 , l ) +-@ , (closure of the Jordan exterior of y) where g(co) = co and gl(co) > 0. Let, for 0 5 t < co, R ( t ) = the Jordan interior of the curve g ( { z : lzl = e t ) ) , so O(0) = f o ( B ( 0 , l ) ) . Also, clearly for 0 5 s < t < oo,R ( s ) $ R ( t ) . Then by Theorem 2.3, in the notation there, let h ( z ,0 ) = f o ( z ) and then Do = 1 (this is true by the uniqueness of the mapping function). Theorem 2.3 then proves Theorem 2.4 in this case. In general, for an arbitrary function f E S , put for each positive integer n, r, = 1 - and suppose f,(z) = $f (r,, z ) (so f,(z) E S ) . Then clearly each f , is analytic in a neighborhood of B ( 0 , l ) . So there is a Loewner chain F, with F,(z, 0 ) = f,(z). It is easy to prove that the set of all Loewner chains is compact. So some subsequence of {F,) converges to a Loewner chain F and then F ( z ,0 ) = f ( z ) in B ( 0 , l ) .

A

Note 2.5. Suppose f,(z) = f ( z ,0 ) maps B ( 0 , l ) onto the complement of a Jordan arc extending to co, then Theorem 2.4 is just Example 2.2. Definition 2.3. L is the set of all Loewner Chains. Theorem 2.5. If f E L and 0 5 s 5 t < co, then there is a unique analytic function + ( z ,s , t ) defined for z E B ( 0 , l ) such that (i) 4 ( z ,s, t ) E B ( 0 , l ) and f ( z ,s ) = f ( 4 ( z ,s, t ) ,t ) for all z E B ( 0 , l ) (ii) and

4(O,s, t ) = 0, 4 ( z ,s , t ) is univalent, l4(z,s, t)I 5 IzI (for all z E B ( 0 , I ) ) ,

&( $ ( z ,

S,t))

= es-t.

(iii) For all z E B ( 0 , I ) , $ ( z , s , s ) = + ( z ) . (iv) If s 5 t 5 u , then q5(z,s,u) = $ ( $ ( z ,S , t ) ,t , u ) for all z E B ( 0 , l ) . Proof. Since f ( z ,s)(B(O,1 ) ) C f ( z ,t ) ( ( B ( OI, ) ) , there is a unique analytic function defined on B ( 0 , l ) such that (writing f,(z) = f ( z , ~ ) )f t,( d ( z , s ,t ) ) = f s ( z ) and 4(O,s, t ) = 0. Since ft and f s are both univalent on B ( 0 , l ) ,so is 4. By Schwarz' Lemma I ~ ( z , S , t)l IzI for all z E B ( 0 , l ) .

<

Taking the partial derivative with respect to z at both sides of (i) and evaluating a t 0 (denoting the derivative by I )

But f l ( O , s ) = eS and fl($(O,s, t ) ,t ) = et so @(O,s, t ) = eSwt. Since + ( z ,s, s ) must = z for all z E B ( 0 , l ) (since f ( z ,s ) = f ( $ ( z ,s, t ) ,t ) ) .

4 is unique,

7.2. Some Coeficient Theorems Again, the uniqueness of

4 shows that if w = 4(r, s, t), then

and so (iv) follows.

Definition 2.4. The function $(z, s, t) defined for z E B ( 0 , l ) of the preceding theorem is called the transition function for the Loewner chain. Note 2.6. The transition function $(z, s, t) is given by 4(z, s, t) = in fact, establishes its uniqueness).

fcl(f, (z)) (this,

From now on, we consider only the situation in Example 2.2 (see also Note 2.4). So, to restate: y : [O,oo) + (C is a Jordan arc with y(0) = a0 that does not pass through 0 and y(t) + oo as t + oo. For 0 5 t < oo, yt is the restriction of y to [t, oo) and R(t) = C - yt. We assume there is a Loewner chain f such that ft(B(O, 1)) = R(t) for all t 2 0 and Po = 1 (see Theorem 2.3 and Note 2.2 above). Let gt = R(t) + B ( 0 , l ) (so in the notation of Chapter 1, gt is a Riemann mapping function). Define g((, t) = gt((). Let 4 be the transition function for the chain f and for S I t14st (.z) = 4 ( ~S,,t). is unique and = f r l o f,. The functions f, and f t have continuous Recall that extensions to C(0,l) (this, in fact, is an application of the Osgood-Caratheodory Theorem mentioned Chapter 1; but which is not proved in this book). Similarly gt has a continuous extension to R(t) U {y(t)). There is a unique point on the unit circle, say X(t) such that ft(X(t)) = y(t). Let y([s, t]) be y, - yt and let Cat be the closed arc of C(0,l) defined by {. : f s ( ~ E) y([s,t])). Let Jst = gt(y[s,tl). So Jst B ( 0 , l ) U X(t) and is a Jordan arc. So q5,t maps B ( 0 , l ) onto B ( 0 , l ) Jst, has a continuous extension to B ( 0 , l ) which maps Cst onto JStand C ( 0 , l )C,,onto C(0,l) - {X(t)). Note that X(s) is an interior point of C,t and as t + s, Cst shrinks to X(s) while for fixed t and s + t, Jstshrinks to X(t). Let J,*, be the reflection of Jst across the unit circle, then by the Schwarz Reflection Principle (see Appendix) @ - C,t can be conformally mapped onto (C (JSt U J;); this is an analytic continuation of to (C - CSt, which will continue to be denoted by 4,t. With the preceding notation, we have

frl:

c

Theorem 2.6.

(a) F o r r E @ - C s t ,

IeI5

4et-I.

(b) X : [0, oo) + C ( 0 , l ) is continuous. Proof. JstE

(C

- +,t(C(O, 1)). Now, lim 4 ( t , s, t) = lim Z+OO

z

2-0

Z =

(7%

S, t)

1 - et-s 4'(O, S, t)

278

7. Bieberbach Conjecture

(by Theorem 2.5). So, by Theorem 1.5 (or 6.5.4), every point in Jstis contained in the complement of {z : lzl < @(o, s, t)}, i.e. has a distance of at least +$'(o, s, t) So for Jzt, which is from boundary of dSt(B(O,1)). So J s t C - {z : Izl < the reflection of Jstacross C(0, I ) , we have

q).

J:t

2 {z : IzI 5 4et-s) .

This proves (a). For (b), note that (a) shows that for any T 2 s, {$$,t(z) : s t 5 T ) is a normal family (Section 1.4, particularly Theorem 1.4.2). Hence, by the uniform boundedness, every sequence has a convergent subsequence to an analytic function. So if {&,) converges to say $ analytic on C - {X(s)) as tk + s (tk > s). Then $ is bounded on C- {X(s)), so X(s) is a removable singularity, so $ must be constant. + 1 uniformly on compact Furthermore, $(O) = limt,, $'(0, s, t) = 1. So sets of C - {X(s)) as t -+ s(t > s). So dSt(z) + z uniformly on compact sets of C - {X(s)) as t + s(t > s). Fix s 2 0. Given E > 0, choose 6 > 0 so that for s < t < s 6, Cst B(X(s),E). For C = C(X(s),e), let x = $,t(C), so that the interior of x (a Jordan curve) contains both Jstand J,*,and hence also contains X(t). Now for sufficiently small 6, I$st(z) - ZI < E for all z E C. So for z E C ,

<

+

IX(s) - X(t)l 5 IMs) - 1 + lz - $st(t)l + I$st(z) - X(t)l < E + E + I$st(z) - X(t)l But I&(z) - X(t)l is less than the diameter of X, and so less than 36 (for all t sufficiently near s). So, IX(s) - X(t)l < 5 ~ .SO X is right continuous. The proof that X is left continuous is exactly similar. This proves (b). Now $,t(z), for z E B ( 0 , l ) is zero only for a simple zero at 0, so has no zeros in B ( 0 , l ) and so we can define the analytic function

e

by choosing the branch of the logarithm where @(O)= s - t. (Recall that $it(0) = es-t). @(z) is analytic on B ( 0 , l ) and continuous on B(O, 1). Now $st(Cst) = the Jordan arc Jst and if z 6 C ( 0 , l ) - Cst, then $,t(z) E C ( 0 , l ) - {X(t)). (See discussion before Theorem 2.6); so Re@(z) = log = 0, everywhere on C ( 0 , l ) except Cat. Now, if we let eia and eiP be the endpoints of CSt,then, by the Poisson formula,

1 1

But having chosen the branch of the logarithm where @(O)= s - t, so Re @ ( z ) = 0, we get 1 @(z) = g

Re @(e8)-

+

eie z dB. eze- z

7.2. Some Coeficient Theorems Note that then

Now $,t(z) = $(z,s,t) = frl(fs(z)). (Note 2.5); or ft o $st = f,. Since we defined gt = f ~ and ' g(C, t) = gt(C), we get gt = $st o g,, and putting z = gs(C), get that st(<) . (+)0 = log gs(0 '

log $ so, by (2.1)

We break this last integral into real and imaginary parts of the integrand and apply the Mean Value Theorem to each part. Since, as already observed,

this gives

< <

<

where a u @ and a 5 v @. Divide both sides by t - s and letting t converge to X(s); so

+ s(t > s); then

eiu and eiv both

Let x(s) = i ( s ) . Then, since X(s) E C(0, I), we have

(-)

1 gt(C) lim -log S gs(C) t>s

t+s t

- 1+ x(s)gs(C) -1- x(s)gs(C)

'

But the left-hand side of this equation is the derivative (from the right) of loggt(s) at t = s. A similar argument shows that the same formula holds for the derivative from the left. So we get

This last allows a proof of Loewner's differential equation.

7. Bieberbach Conjecture

280

Theorem 2.7. Let f be a Loewner chain such that fo is a mapping onto a "slit region" (as in Example 2.2). (Such a mapping will be called a "slit mapping".) Then there is a continuous function x : [0, oo]+ C(0,l) such that f (z, t ) exists and

Proof. Let F ( z , t ) from C ( 0 , l ) x [0, oo]to C x R be defined by F ( z , t ) = (f (z, t ) ,t ) . The image of F is the open set

and F is a one-to-one mapping whose inverse is

So F-' is continuously differentiable with Jacobian

# 0 (in fact = &f ( 9'( ~ > t ) ?) t.) It follows that and so &(f (z, t ) ) exists and is continuous. Let x

which equals &g(<, t ) which we know

F is continuously differentiable, be as above. Since

c = f (st(c, t ) ) ,

differentiation gives

With z = gt(<) = g(C, t ) (cf. argument before Theorem 2.7); this becomes

Equation (2.3) is called Loewner's differential equation. It is now necessary to consider relations on the coefficients of power series which are exponentials of power series and their relations to the original coefficients. The idea of utilizing this relationship in some manner is quite old as is the use of

7.2. Some Coeficient Theorems

281

Loewner's equation (which Loewner himself used to prove lasl 5 3 in Bieberbach's conjecture). Suppose q!J is an analytic function in some neighborhood of 0 with +(O) = 0, and

its power series. Let $ ( z ) = e@(') = CEO bkzk. With this notation we claim that Theorem 2.8.

Proof.

Replacing k by k - 1 and m by m - 1, this gives

So, by Cauchy-Schwarz,

7. Bieberbach Conjecture

Now by (2.4),

+

But exp(x) 2 1 x, so

Repeating for

b: gives

n-l

n-1 Em=l m21amI2- (n - 1)

k=O

Combining (2.5) and (2.6) gives

Continuing in this way, gives

But, writing

Ak for

k

m21a,12, by partial summation (since

&=

7.2. Some Coeficient Theorems 1 - -I_) we get n+l

Thus in (2.7), the quantity being exponentiated is equal to

Writing this as a fraction with denominator n

+ 1, it equals

Combining this with (2.7) proves Theorem 2.8. Note 2.7. All the above was known many years before 1984-the Loewner differential equation since 1923. Milin and Lebedev investigated the relationship between the coefficients of a series and those of its exponential in the 1960's and versions of the Milin-Lebedev inequalities (though not Theorem 2.8) appear already in Pommerenke's 1975 book on Univalent Functions [203]. Theorem 2.8 appears as the "Second Lebedev-Milin Inequality" in Duren, Univalent Functions [61]. It appears to have been first announced by Milin in 1967 but a proof waited until his book

7. Bieberbach Conjecture

284

of 1971 (that was not translated into English until 1977). There are two other "Milin-Lebedev" inequalities, but for DeBranges' Theorem only the second one is necessary. Readers interested in these should consult either Duren's aforementioned book or Volume I1 of Conway's Functions of a Complex Variable [51]. Milin had made the conjecture that iff 6 S and h is the branch of $ log with h(0) = 0; and

(y)

then for all n 3 2,

2

m=l ' k = l Milin's conjecture, if true, implies Robertson's conjecture that if g 6 S and is odd and has the power series expansion

then

n-I

and this in turn implies the Bieberbach conjecture. Fitzgerald and Pommerenke have shown that DeBranges' ideas cannot be applied directly to a proof of the Bieberbach conjecture. To make use of Loewner's differential equation, let f be a slit mapping in S and F the Loewner chain with Fo = f , e-tFt E S for all t 2 0. Define

h ( z ,t ) =

1

log

(

) = c rk(t)rk, where h(0,t ) = o 00

k=l

Define the function

where the r k ( t )are certain functions introduced by DeBranges. They will be defined later, but the properties that they have that will be used are:

( a ) n ( t ) - ~ k +(l t ) = -

(b) rk(0)= n + l - k ( c ) limt+, rk ( t ) = 0

(d)

&7k

(t)< 0

&~k(t)

&7k+l(t)

k+l

7.2. Some Coeficient Theorems ( e ) ~ , + l(t) = 0

In other words, DeBranges applied "weights" ~ ( t to ) the Milin coefficients, and the 0. heart of his proof is to show

3>

&

6

As above, we shall use to indicate differentiation with respect to t and to indicate differentiation with respect to z even when there is only one variable in the argument. Of course, in Loewner's differential equation, there are two variables to be concerned with.

> 0.

Theorem 2.9. For $(t) as defined above $$(t)

Proof. By Loewner's differential equation (Theorem 2.7) and the definition of yk(t),

But, we know that jx(t)l = 1 and so for lzl

< 1,

k

On the other hand, with h(z,t) =

yk(t)z

,

Substituting this in (2.8) and using (2.9) gives

Thus we get

Comparing coefficients in (2.10) gives d -yk (t) = kyk (t) dt

+~

+

k-1

( t ) 2~ r=l

rx(t)"'y7,(t)

.

7. Bieberbach Conjecture

286

c:=~

Let bk(t)= rx(t)-'-y,(t) for k 2 1 and bo = 0. Then, the above becomes

a

x k

% ~ k ( t= ) x(tlk - kkyk(t) + 2

~ ( x ( t ) ) ~ - ' y , (= t )~ ( t- )k ~ (kt )+ 2 ~ ( t ) ~ b *,( t )

r=l

and we observe that

Now, clearly

-

Now, since x ( t ) is on the unit circle, ( x ( t ) ) - l = x ( t ) , and so by (2.11)

- -k bk ( t ) - bk-1 ( t ) = Icx(t) Tk ( t ) . Using this we get

So, for the function

With

4 defined preceding the statement of Theorem 2.9, we have

7.2. Some Coeficient Theorems we get since bo = 0,

By partial summation, then, the second sum in (2.12) equals

(since rn+l( t )= 0 by (e), and where we have suppressed the dependence on t ) . For the first sum in (2.12) we have

since Ix(t)l = 1. Thus, (2.12) now becomes

For the terms in the first sum here, we use (a) and so get

( R e bk

+ Ibk12)(n(t)- n + l ( t ) )= -(Rebk

Summing from 1 to n gives

since bo = 0 and Tn+l(t) = 0. So (2.14) now is

+ Ibkl)2

k+l

7. Bieberbach Conjecture

288 on writing Ibk - bk-l that this last gives

l2 = (bk - bk-1) (&- bk-l).

It is easy to see in the same way

) this proves Theorem 2.9. Since by (d) & ~ ~ (
+

Since by (c), rk(t) + 0 as t

+ 00,and by Theorem 2.9 g $ ( t ) > 0, we get

which proves Milin's conjecture for a slit mapping f E S (recall that the yn are with h(0) = 0). defined by h(z) = C;=, y,tn where h(z) is the branch of log Two issues for Milin's conjecture remain: (i) what about other f E S?, and (ii) what are the mysterious functions rk(t)? There is also the problem of how to deduce Bieberbach's conjecture from Milin's. Question (i) is easy to answer since the slit functions turn out to be dense in S. That is, precisely,

Theorem 2.10. To each function f E S, there is a sequence of slit mappings E S SO that f n + f uniformly on each compact subset of B ( 0 , l ) .

fn

Proof. Let f be a function in S . All that is necessary is to produce a slit mapping g E S so that If (z) - g(z)l < E

<

for lzl p < 1 where E and p < 1 are given positive numbers. The theorem will follow by choosing sequences { e n } and {p,) where en + 0 and p, + 1. Suppose f E S maps B ( 0 , l ) onto a domain D bounded by an analytic Jordan curve r. Let I?, be a Jordan arc connecting oo to a point wo E r then going part of the way around l? to some point w,. If D, is the complement of I?,, let gn map B ( 0 , l ) conformally onto Dn with gn(0) = 0 and gk(0) > 0. Suppose the points wn are chosen so that F, C r,+l and w, + wo. Then clearly D is the kernel of the sequence of regions {D,), that is D is the largest domain so that 0 E D and every compact subset of D lies in all but a finite number of the domains D,. Clearly every subsequence of {D,) has the same kernel. Then, by the Carathkodory convergence

7.2. Some Coeficient Theorems

289

theorem (see Appendix) gn + f uniformly on compact subsets of B(0,l). But then gk + f' and in particular, gL(0) 4 fl(0) = 1. So, E S, and they are slit mappings converging to f uniformly on compact subsets of B ( 0 , l ) . This proves Theorem 2.10, and justifies the fact that Milin's conjecture need only be proved for slit mappings.

#

There remains the question of the mysterious functions rk(t) satisfying (a)-(e) (see discussion preceding Theorem 2.9). DeBranges' functions are defined (for n 1 and 1 k 5 n ) by

>

<

n-k

(2k

rx(t) = k C ( - 1 ) " u=O and

rn+l( t ) E

+ v + 1)v(2k + 2v + 2)n-k-v (k+v)v!(n - k - v)!

e

-(k+,)t

0; where, as usual,

for any complex number and any non-negative integer v. This seems to make them even more mysterious, until it is realized that 2 r k ( t ) can be expressed in terms of the well-known Jacobi polynomials. That is

Theorem 2.11. n-k

n

where P,$~")(X)is a Jacobi polynomial. Proof. Here the Jacobi polynomials are defined as usual as the orthogonal polyno( l a > -1, ,4 > -1. Here, as mials on [-I, 11 with weight function (1 - ~ ) ~ x)P, usual ( n + a ) ( n + a - 1 ) . . . (a + 1 ) n+a ( 1 )= = n! Explicitly the Jacobi Polynomial

+

( )

c (nia) n

p p p ) ( X ) = 2-n

( n +- m @)(x-l)n-m(x+l)m.

m=O

Askey and Gasper have proved that

+ 1)2,(2k + 2v + 2),-, C pj2"O' (x) = C (2k22V(2k + l),v!(m - v)! rn

m

v=o

v=O

(2(x - 1))"

(see American Jnl. of Math., 1976, pp. 709-737; p.717, where we have used the fact that

7. Bieberbach Conjecture

290

for a = 2k in their formula.) The reader interested in the Jacobi polynomials is referred to the classic text by Gabor Szegij on Orthogonal Polynomials, published by the American Mathematical Society[226]. Now, ekt 3 ---n(t) k at

=

+ 1),(2k + 2v + 2),-k-" C (-1)~(2k+ uv!(n - k - v)!

n-k

e

-vt

(2.17)

v=o

But, by (2.16) we have n-k

C P;'^~~) (x) =

u=o

22"(2k

v=O

x

n-k

+ l),v!(n

-k

- v)!

n-k

pj2k.O)(1 - le-t) = x ( - l ) v (2k + 1)2v(2k + 2v 2)n-k-v V=O V=O 2'"(2k l),v!(n - k - v)!

22ve-tv

+

.

But (2k (2k

'I2'

+ 1)"

+

= (2k + u + 1).. . (2k

+ 2v) = (2k + v + l), .

So we get n-k

C

n-k

pj2k.O)

(1 - 2e -t

v=o

- C ( - l ) " (21: + u + 1),(2k

+ +

2v 2)n-k-v v!(n - k - v)!

v=o

e-vt

by (2.17), which completes the proof. Now, in the discussion preceding Theorem 2.9, (e) holds by definition, and (d) follows from Theorem 2.11 and a theorem of Askey and Gasper that depends on the hypergeometric function, that will not be proved here, but that asserts that

C

P;"'O)

(x)

>0

) (*). To prove (b), observe As to (c), it is immediate from the definition of ~ k ( t in that ~ ; ' ~ ' ~ ) ( - 1=) (-I), (see the explicit definition in (2.15)), so by Theorem 2.11 n-k

n

i

0 = -k

n-k

if n - k is odd ifn-kiseven.

7.2. Some Coeficient Theorems But if (a) holds, then

in every case; so ~k

( 0 ) - Tk+l(O = 1

+

and so summing ~ ~ (=0n ) 1 - k . It remains to prove (a).

Theorem 2.12. With the definitions above

as stated i n (a) i n the discussion before Theorem 2.9. Proof. We need to show that

+ -L1 ata

7k (t)

-7%

1

a

( t ) = Tk+l ( t ) - k at7k+1 ( t ) +

and to do this, given the form of what we need to prove, it is useful to introduce exponential factors and to consider Tk n ( t )e-kt a k ( t )= ekt and bn ( t ) = k k

< k < n) n-k (2k + u + 1),(2k + 2v + 2),-r-, a k ( t ) = C(-l)v

Then from the definition of

~ k ( t ) we ,

(2k + v + I)"(% + 2v + 2),-t-v C(-')' ( k + o)u!(n- I; - v ) ! v=o

n-k

bk(t) =

n-k

a =

at

e-vt

( k + v ) v ! ( n- k - v ) !

v=O

and

have (for 1

(2k

e-vt-,kt

+ u + 1),(2k + 2v + 2),-a-,

C(-l)"+l( k + ,y)(v -

v=1

- k - V)!

e-vt

and

a --bk(t) at

2~ + 2)n-k-v C (-1)v+l (2k + ( k-I-+1)"(2k v ) v ! ( n- k - v ) !

n-k

=

V

$

v=o n-k

=

C(-l)"+l ( k + v ) v ! ( n- k - v ) !

v=o

(2k + v)e-~t-2k+kt

7. Bieberbach Conjecture

292 On using (2k

+ v)(2k + v + I ) , = (2k + v),+l. n-k

= C(-1)" v=l

So we have

(2kv + 1)u(2k + 2~ + 2)n-k-v ( k + v ) ( v - l ) ! ( n -k - v ) !

e-vt-(2k+l)t

+ 1 and then v by v - 1, thus n-k (2k + v + 1)v(2k + 2v + 2),-n-, =C(-I)~+~

On replacing k by k

8 e-kt--ak(t) dt

e-(k+,)t

( k + v ) ( v - l ) ! ( n -k - v ) !

v=l

But a k ( t ) = y e k t and bk+1(t)= w e - ( k + l ) t so we have 1 d

n ( t )+ -k -n ( t )= n + l ( t ) dt

1

d g k + i (t)

which proves Theorem 2.11 and so (a). This completes DeBranges proof of Milin's conjecture, except, of course, for the unproved statement (d) depending on Askey and Gasper's results, and, of course, for the motivation of DeBranges' choice of weight function ~ k ( t ) . It remains to see how Milin's conjecture implies Bieberbach's (especially since DeBranges methods cannot be used directly to prove the Bieberbach conjecture.)

Theorem 2.13. Let f E S. So f ( z ) = a + a2z2 + a3z3 + . . . then the Bieberbach conjecture, lanl 5 n is true. Proof. Let g ( z ) be an odd function in S such that ( g ( ~ )=)f~( z 2 )on B ( 0 , l ) . SO

Suppose h ( z ) = log

*

= Cr=lynzn. If z E B ( 0 , l ) - (-1,0], then

But

and so is analytic in B ( 0 , l ) . Thus h ( z ) = (with cl = 1)

1 log

*

is a branch of log

9. So

7.2. Some Coeficient Theorems So, by Theorem 2.8,

But, by Milin's conjecture proved in Theorems 2.9 and 2.10 and the discussion there, (see discussion following proof of Theorem 2.9), this means that C i = O ' I ~ 2 k 1+' 1 n 1, or replacing k by k - 1, and n by n - 1, Ci=, I c ~ ~2 5 - ~n.~ This is, in fact, a conjecture made by M.S. Robertson in 1936, and it implies the truth of the Bieberbach conjecture. For consider f and g as above. Then

<

+

Expanding and identifying coefficients, this gives for all n

So by Cauchy-Schwarz,

> 1 (with cl = 1)

n

by the above.

Note 2.8. There is a way to prove the inequality on r k (t) independently of Jacobi polynomials. This is brief but still requires knowledge of generalized hypergeometric series. See the paper by Gasper [83]. It is, in fact, interesting that the critical fact in this proof comes down to an inequality on generalized hypergeometric series (and unfortunately omitted above). In the nearly seventy years between the posing of the Bieberbach conjecture and its final solution by DeBranges, much work went into various attempts to prove it, at least for some classes of univalent functions. Thus it was proved for typically real univalent functions and for univalent functions that mapped B ( 0 , l ) onto a "star-shaped" region. These and similar results can be found in the first edition of this book. However, for some univalent functions, a sharper result than lan[ n has been known for a long time prior to DeBranges Theorem. We close this chapter with such a result.

<

Theorem 2.14. I f f (z) = z + x:==, a,zn E S, and the image 7 of B ( 0 , l ) under f is convex, then lan[ 5 1, and this result is sharp. Proof. Let 7, be the image of B(O,r), 0

< r < 1, under

7,, where zl ,z2 E B(0,l) and zl # z2, lzll 5 z E B(0, I), and 0

< t < 1,

f . Suppose f (zl), f (22) E

1 . ~ ~ 1 Then . since 7 is convex, for all

294

7. Bieberbach Conjecture

So writing f - l for the inverse of f ; fP1(F(z,t)) is analytic in B ( 0 , l ) and I f-l(F(z, t)l 5 1, f-l(F(0,t)) = 0; and so by Schwarz's Lemma, I f-'F(z, t)I 5 lzl for all t E (0, I), and so in particular, taking z = 22,

Hence the line joining zl and 2 2 is in 7,; that is, 7, is convex, 0 < r < 1. So the curve {f (z) : lzl = r) is convex for each r,O < r < 1. The tangent at the point reie makes the angle n/2 B (measured counter-clockwise) with the real axis; hence the tangent to the point f (reis) on the image curve makes the angle

+

w

= n/2

+ B + arg f '(reis) .

Now, if s denotes the arc length of the image curve, then,

2

Also, denotes curvature, and since 7, is convex (and 0 E 7,), $ is nonnegative. Hence -dw= - - >dwo . ds dB ds dB That is d -(n/2 B arg f'(reie)) 0 , dB or d 1 -Im( Log f'(reie)) 0 , dB

+ +

>

>

+

with z = reie. And this holds (by the maximum modulus principle for the real part) for all z E B ( 0 , l ) . It follows from this that

from which it easily follows that zfl(z) is univalent on B(0,l). Since clearly I

I

= 0 and $-(zfl(z))l

rf1(z)l z=o

and

SO

lan/ 5 1.

z=o

= 1, it follows that zfl(z) E S . But

7.2. Some Coeficient Theorems

295

Note 2.9. Before DeBranges' proof of the Bieberbach conjecture, this could be deduced from the fact that the image of B(0,l) under z fl(z)was starshaped and R. Nevanlinna's proof in 1917 that for such functions la,l I: n.

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Chapter 8

Elliptic Functions The subject of elliptic functions is a very old one. There are two "theories" associated with the names of Jacobi and Weierstrass respectively. Although, as will be seen, all the formulas of each of these "theories" can be expressed in terms of the other; nevertheless, depending on the sort of problem being treated, the functions and notation of one approach can be vastly simpler to use than those of the other. Elliptic functions have already been alluded to several times in this book; we consider first three different ways in which they arise.

Problem 1: Consider an ellipse centered at the origin in B2 with semi-axes of lengths a and b, 0 < b 5 a in a system of rectangular coordinates. Its equation may be written

Diagram 8.1 Suppose we wish to find the length e of the arc from (0, b) to the point (a,P) on the ellipse in the first quadrant. By a usual calculus formula

8. Elliptic Functions

, we have 2=* a-

f,

Since y = bdl-

Letting x = a sin 0, we have with

4=

Arc sin(a/a)

> 0,

where k = (1 - $) I , and so 0 5 k < 1. 4 is an angle measured clockwise from the y-axis, however, it is not the angle formed by the y-axis and the line from the origin to ( a , P ) , so measured. Indeed, the sine of this angle is -and, in I

J-+

+

general, a2 # a2 P2. We may also note that /3 = bcosd, so that -knowledge of q5 determines the point ( a , p). Also, clearly, the perimeter of of the ellipse is given by the integral with a = a, that is

Readers who remember analytic geometry will recall that k is the eccentricity of the ellipse. If a = b, then k = 0 and we have a circle. Otherwise it turns out that the exact evaluation of the integral in (0.1) cannot be given in terms of elementary functions. Clearly there is no harm in normalizing by letting a = 1. We might then consider the problem: given the eccentricity and arc length of an ellipse to find the angle 4 in (0.1), or equivalently, of inverting the integral.

Problem 2: Consider the Schwarz-Christoffel formula for the conformal mapping G of the upper half-plane { z : I m z > 0) onto the interior of a rectangle. Suppose we let the points -c, -1,1,c, where c > 1, correspond to the vertices of the rectangle, then by Theorem 1.5.4 we have (since a k = 112, k = 1,2,3,4).

If we take K 2 = 0, K1 = c and put c = 115, then

where 0

< k < 1.

8. Elliptic Functions

299

Hence for z real, -1 5 z 5 1, G(z) is real and G(-z) = -G(z) and so one side of the rectangle lies along the real axis and has vertices K and -K where

Then G takes -l/k onto the vertex Here

iK' =

Ilk

-K + iK' say and 1/k onto K + i K 1 .

1

dw and -K = (1 - k2w2)i (1 - ~ 2 ) i

Ilk

1 dw (1-k2w2)i(l-w2)i

.

Making the change of variable (1 - k2w2)4 = U, and letting k' = (1 - k2)4 , we have 0 < k' < 1 and the expression for K' becomes

Diagram 8.2 If in (0.2) we take z real (for simplicity), and make the change of variable w = sine, putting = arcsin z, we get

+

which has a certain "family resemblance" to the integral of Problem 1. Here again it can be shown that the integral is not expressible exactly in terms of elementary functions, and here again we may wish to consider the "inversion problem": in (0.2), given k and a point on the rectangle, find the z which is the pre-image of that point under G.

Problem 3: Non-constant periodic functions meromorphic in C with one fundamental period are well-known, e.g. the rational functions of e C Z ,c a constant. Such functions are called simply periodic, since every period is an integral multiple of the single fundamental period.

300

8. Elliptic Functions

A natural question is, are there non-constant multiply-periodic functions meromorphic in C ? We start with an investigation of this question rather than the ones raised in problems 1 and 2. Although clearly Problem 1 could motivate naming some object an "elliptic function", and the integral of Problem 2 has a certain resemblance to that of Problem 1, while any connection with Problem 3 seems remote a t present; nevertheless, we will see that the three problems are closely related. Since the work of Weierstrass, Problem 3 has often seemed the most appropriate starting point for such a discussion, though the name "elliptic" as will be seen, originates in Problem 1. The connection of Problem 3 with Problems 1 and 2 is brought out in Section 4. Elliptic integrals were first encountered by John Wallis around 1655 in connection with Problem 1 as well as the arc length of other curves, they thus extend back to the beginning of analysis. Yet such terms as complex multiplication, abelian integrals, Eisenstein series, modular functions, elliptic curves, and others, all represent areas of active mathematical interest today, and all originated in the study of elliptic functions. The subject has, in fact, given rise to a wealth of mathematics of ever-expanding interest. Contemporaneously this sometimes has a heavily algebraic guise; nevertheless, the more purely complex analytic theory is still of value and an introduction to it is often missing from a course in complex analysis. This chapter, despite its length, is still only an introduction, many interesting and valuable results being omitted without mention. The analytic theory of elliptic functions received what is perhaps a definitive statement in Robert F'ricke's two-volume work Die Ellzptischen Funktionen und Ihre Anwendungen [74]. The contemporary value of this book is perhaps indicated by its being reprinted (Johnson Reprint, New York) in 1972. On elliptic modular functions, a famous and similarly definitive work is F'ricke-Klein, Theorie der Elliptischen Modulfunktionen [75]. Since we will begin our development with an approach historically posterior to the work of, for example, Legendre and Jacobi, some of which will be discussed in Sections 4 and 5, a few historical remarks may be in order. Jacobi called December 23, 1751, the birthday of elliptic functions, because on that day Euler was asked his opinion of a paper by Fagnano on arcs of lemniscates. Stimulated thereby, Euler discovered the addition theorem for elliptic integrals. See notes 4.3 and 4.5 below. In the 1780's, Legendre took an interest in elliptic integrals which was to last for forty years; what we today call the three kinds of elliptic integrals in Legendre normal form, he called "fonctions elliptiques" in his treatise published in 1825-1828. With Legendre's treatise, elliptic integrals became an independent subject of study. In 1823 Abel and slightly later, Jacobi, began the study of the "inversion problems" mentioned in Problems 1 and 2. They called these "inverse functions" elliptic functions, as we do today, and discovered their double periodicity. Abel's papers were published in the Journal fur die Reine u. Angewandte Mathematik [I]. Jacobi's "Fundamenta Nova" of the theory of elliptic functions appeared in 1829. It seems as though many of these results were discovered by Gauss in the years 1796-1814 but not published by him. In Jacobi's lectures, edited and published in 1838 by Borchardt, he introduces Theta-functions as the foundation of the theory of elliptic

8.1. Elementary properties

301

functions. (Gauss apparently had already come across these as well, without taking them as fundamental.) Although there were earlier anticipations, principally by Eisenstein, beginning with Liouville's lectures of 1847 and Weierstrass' of 1862-63, the double periodicity discovered by Abel and Jacobi was taken as fundamental, as it generally is today in a systematic development, (see Note 1.2). Perhaps an indication of the importance of elliptic functions to complex analysis, in general, even early on is indicated by the fact that the familiar "Liouville's Theorem" and "Weierstrass Product Theorem" of elementary complex analysis both have their origin in the study of elliptic functions. The reader interested in a historical presentation of the results of elliptic integrals and functions until the early twentieth century, cannot do better than read the magisterial article by Fkicke "Elliptische finktionen" Section IIB3 of the Enzyklopadie der Mathematischen Wissenschaften [76] (Volume 11, part 2, pages 177-348). The historical remarks above, and much of those below are drawn from this source. A final remark: Elliptic functions were not least important because of the early realization of their manifold applications to physical investigations; the most classic of these being the theory of the pendulum. A selection of these in English, none of which can find room here, can be found in Greenhill, The Applications of Elliptic Functions [88]. Applications of elliptic functions have continued to be important to the present day. A word is also necessary about notation, especially as in the theory of elliptic functions, many letters and symbols have acquired persistent definitions. However, the notation is only semi-standard in the sense that there are slight variations from author to author which can prove annoying if care is not taken to establish a particular author's conventions. Any notation established by definition in this chapter will persist throughout the chapter. Often, but not always, the statements of theorems will contain reminders of such defined notation.

8.1

Elementary properties

With respect to Problem 3, we have first two easy results which serve for orientation.

Theorem 1.1. For a non-constant periodic meromorphic function f defined in C

(i) ca is an isolated essential singularity. (ii) The set of periods off has no finite limit point.

+

Proof. (i) Suppose f (21) # f (22) and f (z w) = f (z) for all z E C and some fixed complex number w . Then any neighborhood of w contains infinitely many points of the forms zl +nw, 2 2 +nw. Hence f (z) has an isolated essential singularity f ( z ) does not exist). a t ca(lim,,, (ii) Let { w ) be the set of periods of f (z), and suppose wo were a finite limit point of { w ) . Suppose f is analytic at zo, then g(z) = f (2) - f (zo) has the infinite

302

8. Elliptic Functions

set of zeros {zo + w) which have the finite limit point {zo identity theorem for analytic functions, f (z) f (zo).

+ wo} and so by the

Theorem 1.2. The periods of a non-constant periodic meromorphic function f are either (a) integral multiples of one period, or (b) the sums of integral multiples of two periods with non-real ratio.

Proof. (a) Suppose all the periods lie on a line through the origin. Suppose wl is a period of smallest modulus (exists by Theorem 1.1 (ii)). Then all other periods have the form Awl, X real. Let m = [A]. Then Awl = mwl {X)wl is a period, mwl is a period, and 0 {A) < 1. Hence, by definition of wl, {A) = 0 and X is an integer. (b) Suppose all the periods do not lie on a line through the origin. Then there are two periods wl, w2 such 0, wl, w2 are the vertices of a non-degenerate triangle. By Theorem 1.1 (ii), in the Jordan interior and on the boundary of the triangle, there can only be a finite number of periods. Let w3 be such a period. w3 does not lie on a line through the origin and at least one of wl, w2; say wl with no loss of generality. Then 0, wl, w3 are the vertices of a triangle containing fewer periods in the closure of its Jordan interior than the original one. Continuing, if necessary, in this way, after a finite number of steps, one arrives at a triangle whose vertices are periods and the origin, and there are no other periods in the closure of its Jordan interior. Let O,wl, w2 be the vertices of such a triangle T. Consider the parallelogram Q with vertices 0, wl, w2,wl w2. Let w be a period in the closure of the Jordan interior of Q, w # wl,w # w2. Since w is a period, w' = w2 wl - w must also be a period or 0. If w' # 0, then since w w' = wl w2,w' must lie in the closure of the Jordan interior of T (see Diagram 8.3) which is a contradiction. So w' = 0.

+

<

+

+

+

+

Diagram 8.3 Now suppose p were any period of f . Given any complex number, it can be written in a coordinate system with the straight lines determined by (0, wl} and

8 , l . Elementary properties

303

(0, w 2 ) as coordinate axes. That is, we can write p = Xlwl then

+

{ X l ) ~ l {Xz)wz = P

+ X2w2, X I , X2 real. But

- [XlW - [X2lw2

+

is a period or 0, and 0 5 { X I ) < 1,O 5 {X2) < 1. Hence { X l ) w l {X2)w2 lies in the closure of the Jordan Interior of Q. By the preceding argument, it follows that { X l ) w l + {X2)w2 = 0 and p = [Xllwl [X2]w2.

+

Definition 1.1. A meromorphic periodic function with two periods with non- real ratio is called an elliptic function (it proves convenient to include constant functions in the definition). Two periods w l and w2 of an elliptic function such that all other periods are sums of integral multiples of w l and w2 are called fundamental or primitive periods. If w l , w2 are fundamental periods of an elliptic function f , then the region whose boundary is the parallelogram with vertices 0 , w l , w2, w l w2, together with the straight-line portion of the boundary running from 0 to w l and 0 to w2; including 0 but not w l or w2, is called a fundamental parallelogram. Any similar parallelogram with vertices zo, zo w l , zo w2, t o w l w2 is called a period parallelogram. It should be noted that henceforth, the word parallelogram will refer to such open regions, together with a portion of their boundary and not the boundary of the open region. It is clear that the whole plane can be covered by a network of non-overlapping similar copies of a fundamental parallelogram; the corresponding vertices (i.e. the set of points m w l nw2, m, n integers) is called the set of lattice points associated with the periods wl,w2.

+

+

+

+ +

+

N o t e 1.1. A pair of fundamental periods w l , w2 for an elliptic function f is not bw2 and unique since if ad - bc = f1 , then one easily sees that wf = awl w,* = cwl dw2 are also a pair of fundamental periods for f . Clearly the values of an elliptic function are completely determined by the values in a period parallelogram.

+

+

Two nearly trivial observations are Theorem 1.3. The set of all elliptic functions having the same periods forms a field.

Proof. A clear verification. T h e o r e m 1.4. The derivative of an elliptic function is elliptic with the same fundamental periods.

Proof. Clear.

A basic result going back to Liouville's lectures of 1847 is T h e o r e m 1.5. An elliptic function f which has no poles in a period parallelogram is constant.

8. Elliptic Functions

304

Proof. The function f is analytic and bounded in a period parallelogram; hence in the whole plane; hence by "Liouville's Theorem", it is a constant. Since it is clear that there can be only finitely many poles in a period parallelogram, we may make Definition 1.2. The sum of the multiplicities of the poles of an elliptic function f in a fundamental parallelogram is called the &r of the elliptic function. T h e o r e m 1.6. The sum of the residues of the poles of a non-constant elliptic function f which are in a period parallelogram is 0.

Proof. By translating the period parallelogram by a small amount, if necessary, we can assume without loss of generality that there are no poles on its boundary. Call the perimeter, oriented positively, C. But, if w l and wz are the fundamental periods for f corresponding to the period parallelogram, then

which proves the theorem. T h e o r e m 1.7. The order of a non-constant elliptic function is

Proof. By Theorem 1.5, the order must be be 1.

> 0; by Theorem

2 2.

1.6 the order cannot

Theorem 1.8. The sum of the multiplicities of the zeros of a non- constant elliptic function f in a period parallelogram is equal to the order o f f .

Proof. Since the zeros and the poles of f both form isolated sets of points, again by translating the period-parallelogram by a small amount if necessary, we see that there is no loss of generality in assuming that there are no zeros or poles on its boundary. If N denotes the sum of the multiplicities of the zeros of f in the periodparallelogram and P the corresponding sum for the poles = the order of f , then N - P = i?;; dz, where C as before is the boundary of the period parallelogram, and the same argument as used in proved Theorem 1.6 now shows that the integral = 0; whence N = P.

%

T h e o r e m 1.9. A non-constant elliptic function takes on every value.

Proof. If f ( z ) is elliptic, then so is g(z) = f ( z ) - a, and f ( z ) - a has the same poles to the same multiplicities as f ( z ) ;so the result follows from Theorems 1.8 and 1.7.

8.1. Elementary properties

305

Definition 1.3. Two points zl and z2 are called congruent modulo ( w l ,w 2 ) (writz2 mod ( w l ,w 2 ) ) if there are integers m and n such that zl - z2 = ten zl mwl nw2.

+

=

Clearly any finite point is congruent modulo ( w l ,w2) to only one point in the period-parallelogram determined by wl and wa.

Definition 1.4. A set of points is called irreducible mod (wl,w2) if no two points of it are congruent to one another modulo ( w l ,w2). A fundamental system of poles S for a given elliptic function f with fundamental periods w l , w2 is an irreducible set of poles such that each pole of f is congruent modulo ( w l ,w 2 ) to some pole i n

S. Define a fundamental set of zeros for f analogously. Theorem 1.10. If a l , a2 . . .an is a fundamental set of zeros and bl, b2.. . b, a fundamental set of poles for a non-constant elliptic function f with fundamental periods w l , w ~ where , multiple zeros and poles are repeated according to multiplicity, then m = n, and

x n

a,

-

n

bv mod ( w l ,w2) .

Proof. That m = n is Theorem 1.8. As in the proof of Theorem 1.8, we write C for the boundary of the period parallelogram and may assume that no zeros or poles lie on C. We have

But by Theorem 1.4, since wl and w2 are periods of f , this last expression becomes

Now, as z traverses the straight line between zo and zo+w1, since f ( z o ) = f (zo+wl), f ( z ) traverses a closed curve C l , say, and *dz is the winding number

k JzT+wl

306

8. Elliptic Functions

o f f around C1, and hence an integer. Similarly and so the theorem follows from (1.1).

h

%dz

~ z ~ t w 2

is also an integer,

Note 1.2. The development of elliptic functions from the point of view of Problem 3 (arbitrary meromorphic doubly periodic functions) is due to Liouville and Weierstrass. In 1847 Liouville lectured on the subject of doubly periodic functions, and though his results became widely known (a summary was published in vol. XXXII of the Comptes Rendus de 1' Academie des Sciences), a "rbdaction authentique" was not published until 1879, by C. W. Borchardt 1321. Most of the results of this section are Liouville's; however, he did not use Cauchy's Theorem in his arguments. In 186211863 (Winter semester lectures in Berlin), Weierstrass gave the theory of elliptic functions a fundamentally new form; his results being partially published in 1882-83 12441, vol. 11, 245-309. For a systematic exposition, Weierstrass' theory is far preferable to earlier ones. The function Q(z) in the following Section 2, is Weierstrass' notation which has become standard. However, it is perhaps worth pointing out that some of Weierstrass' considerations are already present (independently) in work of Eisenstein published in 1847, the same year as Liouville's lectures. Weierstrass' own presentations of elliptic functions appear in lecture format in volumes V and VI of his Werke[244], where they have been deduced from student notes of a variety of sets of lectures. For more historical details, the reader may consult Fricke's already mentioned article in the Encyklopiidie [76].

8.2

Weierstrass' !J3-function

Up until now we have talked in Section 1 about the properties of a general elliptic function, if any such exist; but there is no example yet at hand of a non-constant elliptic function. From Theorem 1.7 any such function must have at least two poles in the periodparallelogram, and by Theorem 1.6 the sum of the residues at these poles must be 0. Hence there are two possible "simplest" cases: (a) A fundamental system of poles consists of a single double pole with residue 0.

(b) A fundamental system of poles consists of two simple poles with residues of the same modulus and opposite sign. Case (a) leads to Weierstrass' elliptic functions; case (b) to Jacobi's. However, we shall see that the two cases are not as independent as may seem at first. We now attempt to construct a function F in case (a), and thus have an example of an elliptic function. Since we have to have a double pole in a fundamental parallelogram, it is simplest to place it at 0. Then, if
8.2. Weierstrass ' q-function

307

+

periods of F , there must also be double poles at the lattice points mC1 nC2,m, n integers (so &/& is not real); hence by Mittag-Leffler's Theorem, the simplest possible elliptic function in case (a) will have the form 1

22

+

1 - hw (z) (z-w)2 y a lattice point m€1+n€2

C'

where the hw(z) are polynomial "convergence factors" (if necessary) and C' indicates that the lattice point 0 is omitted from the sum. for k a positive integer, summed Thus we need to investigate how sums C over lattice points converge (since the hw(z) are, in fact, by Mittag-Leffler's Theorem, partial sums of the power-series expansion of ( z - w ) ? - 2 (1-6)around 0.). In what follows, lattice point will always refer to the lattice of points {mJ1 nC2 : m, n integers). Considering the sum Cw lattice 5 , k a positive integer, we note that (by The-

5

,

+

.

~ o ~ n t

orem 1.1)the distance between any two lattice points is clearly 2 c > 0, for some constant c. Suppose, given a positive integer n , A(n) = number of lattice points w lwl < n 1. If we consider disks B(w, 6) centered at these lattice such that n points, then if 0 < E < 4 2 , these disks do not overlap, and so

<

+

<

~ ( n ) n c ~area of B(0,n = n(n

+ 1+ E) -

area of B(0,n - E )

+ 1+ E ) -~n(n - E ) =~ (1 + 2 ~ ) ( 2 n+ 1)n ,

The preceding computation is justified if n - 6 > 0, but if A(n) lattice points w' which contribute to A(n), we have

Hence,

w a lattice point

lWlk

- O
O
w a lattice point

n=l

> 0, then for all

8. Elliptic Functions

308

Since the first of these last two sums has only finitely many terms, we see that $-converges for k 2 3, and thus we may take hw(z) = and we a

5,

~Olrlt

have that the series

1 z2

-

+

1 1 -w ) w2

-1

w a lattice point

(

converges uniformly in the whole plane, except for z a lattice point, where it has double poles, and this defines our desired meromorphic function, since for lzl < Iwl,

Definition 2.1. The Weierstrass p-function associated with the periods &,& is defined bg 1 1 1 -z2 w a lattice poznt mEl+nEz For reasons that will appear shortly, (e.g. Theorem 2.2 below), it is customary to set J1 = 2w, J2 = 2w1, and to take Im(wl/w) > 0 (recall wl/w is not real by hypothesis), and we will adhere to these conventions from now on. If it is necessary to refer explicitly to a particular set of periods, one can write p(zlw, wl), but mostly this will be unnecessary, and the notation Q(z) will suffice, what the periods may be being clear in the context. We first collect a few elementary facts about p ( z ) .

Theorem 2.1. (a) p(zlw, wl) has poles of order 2 at the lattice points associated with the periods of 2w, 2w1, and the residue 0 at those poles, and no other singularities in C.

(c) Q(z) is doubly periodic with periods 2w and 2w1 and these are fundamental periods.

(d) There is a neighborhood of 0 in which

where c2n = (2n + 1)

1 E1 p

w a lattice point

8.2. Weierstrass ' !?3-function (e)

309

jJ3'(z) is an elliptic function of order 3 with fundamental periods 2w and

2w1. (f) q1(-z) = -rpl(z). Proof. (a) follows from the construction of V(z) given. (b) follows from the fact that as w runs through all lattice points, so does -w. (c) rpl(z) = -

C

w a lattice point

B1(z + 2 ~ =) -2

2 (z-w).

1 = -2 + 2w - w)3

C

w a lattice polnt

= -2

C w a lattice polnt

. So

C w a lattice po~nt

1 (2 - (W- 2~113

1 (z - w)3 = P 1 ( 4

Hence rp(2

+ 2w) = rp(2) + c

and putting z = -w, we get from (b)

and so c = 0. The same argument holds for 2w1 and so for integral m and n,

Furthermore, 2w and 2w1 are clearly fundamental periods of Q,because otherwise the fundamental parallelogram with vertices O,2w, 2w1,2w + 2w' would contain a singularity other than the one at 0 which is a contradiction.

1

w a lattice point

(z - w)2

- -1

w2

1

=-+Ckzk-I z2 k=2

If k is even, say k = 2r, we have

L

w a lattice point

=

L

w a lattice point

(-w)2r+1

C m. 1

w a lattice point

310

8. Elliptic Functions

and hence

,,,

1

= 0 . Thus, putting k = 2 n

a point

1

P(z)= ; i+ C ( 2 n + 1 ) n= 1

C'

1

+ 1, n 2 1, we have z2*

,

w a lattice point

as claimed. (e) follows from the argument proving (c). (f) follows from (b).

Theorem 2.2. I n the fundamental parallelogram (with periods 2w, 2w1)!J3I(z)has simple zeros at w , w' and w + w', and no other zeros. Proof. Suppose z is not a pole of 9 and a is any one of the points 2w, 2wt, 2w then since !J3' is an odd function (Theorem 2.1 (f)),

+ 2w';

Putting z = a / 2 , we get

P 1 ( a / 2 )= - P 1 ( a / 2 ) and so !J3'(w) = P ' ( w l ) = P ' ( w w l ) = 0. Furthermore, 9' is an elliptic function of order 3 (Theorem 2 . l ( e ) ) . Hence by Theorem 1.8, each of the zeros at w , w', w + w' must be simple and they can be the only zeros in the fundamental parallelogram.

+

Definition 2.2. If P ( z ) is a Weierstrass y-function with fundamental periods 2w and 2w1, where I m ( w l / w ) > 0 , define the numbers w l , w2, W Q by wl = w , w2 = w+wl, W Q = W' and the numbers e l , e2, es by V ( W = ~ ei, ) i = 1,2,3. With this definition, Theorem 2.2 can be restated as

Theorem 2.3. The equation P ( z ) = c, c finite, has double roots if and only if c is one of el,e2,e3. Note 2.1. As noted already in Example 4.3.3, when F is a meromorphic function, the equation F ( z ) = c can have double roots for at most four points c in C,, and since has double poles, it shows that the number four can be achieved. Theorem 2.4. ei = e j ( i ,j = 1,2,3) i f and only if i = j . Proof. Suppose for some i # j ,

Then wi would be a double root of P ( z ) = and wj would be a distinct double root of the same equation. Hence the elliptic function !J3(z) - 17 which is of order 2, would have the sum of the multiplicities of its zeros 4, contradicting Theorem 1.8.

>

8.2. Weierstrass ' P-function

311

An important fact about P ( z ) is

Theorem 2.5. P ( z ) satisfies the differential equation ( Q ' ( z ) ) ~= 4(P(z)-el)(Q(z)ez)(P(z) - e3). Proof. Y(z) - ei, i = 1 , 2 , 3 has double zeros at wi, i = 1 , 2 , 3 respectively. P1(z) has simple zeros at wi, i = 1,2,3, and so (Q'(z))~has double zeros at wi, i = 1,2,3. Furthermore, p ( z ) - ei has a double pole at all lattice points while P1(z) has a triple pole at all lattice points. Hence the function

has no poles a t lattice points or anywhere else, is clearly elliptic, and so by Theorem 1.5 is a constant, say A. By Theorem 2.1 (d), P ( z ) has the Laurent expansion around 0: -

0

0

So P1(z) has around 0 the Laurent expansion:

Thus, 00

z6((P(z)- e ~ ) ( P ( z-) e2)(P(z) - e3) = 1+

anzn n=l

and

00

for some coefficients an,pn. Taking the ratio of these two expressions and letting z + 0, we get A = 4. The differential equation of Theorem 2.5 has another equally important form:

Theorem 2.6. ( Y ' ( z ) ) ~= 4p3(z) - g2P(z) - 93 where 92 = 60 W

x'

a lattice point

1 and g3 = 140 w4

-

W

C' a lattice

1 -

w6

'

point

Proof. One could, of course, attempt a proof by multiplying in Theorem 2.5 and somehow working out the values of the ei. The following is easier: jFrom Theorem 2.1 (d), we have

8. Elliptic Functions for some coefficients an.Similarly

with some coefficients pn. Thus in a neighborhood of 0,

and consequently

in some neighborhood of 0. The right side of equation (2.1) is analytic in a neighborhood of 0; hence the left side of (2.1) is an elliptic function analytic at 0 and hence at all lattice points. But the lattice points are the only places where the left side can have poles. Thus the left side of (2.1) is an elliptic function without poles, and so equals a constant. Furthermore, since from the right side, the function is 0 at 0, this constant is 0. Thus ( V ' ( Z ) ) ~= 4y3(z) - 20c2!J3(z) - 28c4 and the substitution of the expressions for cg and c4 from Theorem 2.1 (d) gives the theorem. Note 2.2. Comparing coefficients in Theorem 2.5 and Theorem 2.6, we have

Furthermore, Theorem 2.6 shows that for a given z, the points (!J?(z),y l ( z ) ) lie on the curve defined by 92 - T Q3 (T)/2)2 = t3- Tt . The discriminant of the cubic polynomial on the right is then

On the other hand, since el, e2, e3 are the roots of

we have Q:

- 27~: = l6(el - e2l2(el - e3)2(e2 - e3)2 ,

and so, since the ei are distinct by Theorem 2.4, we have

8.2. Weierstrass ' Y-finction

313

Definition 2.3. Throughout this chapter, g2 and g3 will be defined as the quantities in Theorem 2.6. They are called the invariants of the !J3-function. We also define the discriminant A of Y(z) by

Note 2.3. Of course, A, g2,g3, like el, e2, e3 all depend implicitly on the fundamental periods 2w, 2w' of a particular p-function. In Section 5 we shall study this dependence explicitly. We come now to a theorem which essentially says that in some sense the study of elliptic functions amounts t o the study of P-functions.

Theorem 2.7. Any elliptic function f with fundamental periods 2w, 2w1 where Im(wl/w) > 0 can be expressed in the form

where R1 and R2 are rational functions and Q(z) = ~ ( z I ww,' ) . Proof. Since any function can be written as the sum of an odd and an even function, namely f (z) = ;( f (z) f (-z)) ;(f (z) - f ( - z ) ) , it is enough t o prove the theorem for odd elliptic functions and even elliptic functions. Furthermore, if g(z) is an odd is an even elliptic function, then since P1(z) is also odd (Theorem 2.1 (f)), elliptic function. So it is enough to prove that every even elliptic function h(z) is a rational function of P ( z ) with the same periods. Also, if h(z) is an even elliptic function, then a pole or a zero which it may have a t the origin necessarily has order divisible by two (the Laurent expansion around the origin contains only even powers); hence there is an integer m (positive, negative, or 0) such that h(z) (y(z))" has no zeros or poles at the origin, and so we may further assume this with no loss of generality. We need then to prove that an even elliptic function h(z) with periods (2w, 2w1), say, and with no zeros or poles a t the origin is a rational function of p ( z ) . Let a1 be a zero of h in the fundamental parallelogram. Since h is even, the point in the fundamental parallelogram congruent to -a1 mod (2w, 2w1) is also a zero. Thus, we can choose n zeros a l , . . . ,a, in the fundamental parallelogram each zero represented according to its multiplicity, so that together with points in the fundamental parallelogram congruent mod (2w, 2w1) to -a1 , . . . , -a, they form a fundamental system of zeros. Clearly, we can do the same with the poles. By Theorem 1.10, repeating each pole and zero according to its multiplicity, and denoting the poles by bv, we have, if p ( z ) = !+?(zlw,w ' )

+

+

&

is an elliptic function, which, by construction, has no zeros or poles in the fundamental parallelogram (note that since h is even, it can only have zeros or poles of even order). Hence by Theorem 1.5, k(z) is constant, and the theorem follows.

8. Elliptic Functions

314

Note 2.4. This is as good a place as any to point out that not all doubly-periodic functions are elliptic. For example, e v ( z )is not elliptic; this follows from Theorem 2.7; it also follows from the fact that ev(") has an isolated essential singularity at 0 (viz. Theorem 2.1 (d)). It is perhaps also worth mentioning that Theorem 2.7 essentially already appears in Liouville's lectures of 1847 (see Note 1.2). For simply periodic functions "addition theorems" are of great importance. We can similarly consider the possibility of an "addition theorem" for elliptic functions. We have

Theorem 2.8. Let P ( z ) = P(zlw, w'). For any two complex numbers z and w, z not congruent to fw mod (2w, 2w1), z and w not periods of !&I,

Proof. The condition that z f fw( mod 2w, 2w1),means that P ( z ) # P(w). Since also the poles of occur at the periods (and so neither z or w is a pole) it follows that the system of equations

uniquely determines A and B. Then the elliptic function

has at least two distinct roots in the fundamental parallelogram congruent mod (2w, 2w1) to .z and w respectively, as well as a triple pole at the origin. From Theorem 1.8, there is exactly one more zero of (2.3) in the fundamental parallelogram; by Theorem 1.10, the sum of these zeros must be 0, and thus this third zero is -z - w( mod 2w, 2w1)) (-2 - w may possibly be congruent to either z or w mod (2w,2w1) or there may be three simple zeros). Since P ( z ) is even, we have y ( - z - w) = P ( z w). By Theorem 2.6, we have that the cubic equation

+

where

Thus

g2

and

g3

are the constants associated with P(zIw, wl) has the three roots

8.2. Weierstrass' !?3-function Comparing the coefficients of q2 we get

Solving the equations in 2.2 simultaneously gives

and so the theorem. We should like also to have a formula where z = w, i.e. for V(2z). Theorem 2.9. If z is not congruent to -z

mod (2w, 2w1)

Proof. Take limits as w -+ z in Theorem 2.8. Note 2.5. There are many forms of the addition formula other than Theorem 2.8, not all obvious, which may be obtained by comparing other coefficients in the above proof, or other manipulations. These are discussed in Fricke's book, part two, 160-161. Definition 2.4. A function F(z) is said to have an algebraic addition theorem if there is an irreducible polynomial in three variables with constant coeficients d such that d ( F ( z w), F(z), F(w)) = 0 .

+

A consequence of Theorem 2.8 is Theorem 2.10. Elliptic functions have algebraic addition theorems.

Proof. Let E(z) = E(zlw,wl) be an arbitrary elliptic function. From Theorems 2.7 and 2.6, there is an irreducible polynomial B in two variables with constant coefficients such that B(E(C), P(C)) = 0. Thus we have the three equations (i) B(E(z

+ w), P ( z + w)) = 0,

(ii) B(E(z),P ( z ) ) = 0, (iii) B(E(w),P(w)) = 0 . But by Theorems 2.8 and 2.6 there is an irreducible polynomial & in three variables with constant coefficients such that & ( P ( z w), P ( z ) , P(w)) = 0 (i.e. satisfies an algebraic addition theorem). Hence, eliminating the P ( z + w), v ( z ) , P ( w ) from (i), (ii), (iii) through use of Q proves the theorem. It should be noted that B(u, v) and Q(u, v, w) have a t most degree 2 in u (by Theorems 2.6 and 2.8). 17

+

8. Elliptic Functions

316

What meromorphic functions have algebraic addition theorems? The answer is given by a celebrated theorem of Weierstrass:

Theorem 2.11. A function F meromorphic in @ which satisfies an algebraic addition theorem is either

( a ) a rational function, ( b ) a rational function of eXZfor some constant A, (c)

an elliptic function.

Proof. Clearly every rational function satisfies an algebraic addition theorem. Suppose, therefore, that F is not a rational function, then it has an isolated essential singularity at co. Suppose the algebraic addition theorem for F has the form G(F(z

+ w), F(z), F(w)) = 0

We will show that F must be periodic. Suppose G(u, v, w) has degree m in u. By the Casorati-Weierstrass Theorem since co is an isolated essential singularity, there is a number A such that F(w) = A has m 1 distinct roots wl, wz, . . . , w,+l, and we may choose so that F is analytic at and each of the points <+wj,j = 1 , . . . ,m+1. Let B = F ( < ) . Then the quantities

<

<

+

are roots of the mth degree polynomial

and so at least two of them must be equal. Consider now z in a neighborhood N of <. Since G(F(z wj), F ( z ) , A) = 0 for j = 1,.. . ,m 1 ,

+

+

we must have by the above argument that F(z

+ w,)

= F(z

+ WX)

for each z E N. So far, w, and wx may be well different for different z. However, there are only finitely many pairs K , X while N contains infinitely many points; hence there must be indices k and l such that

for all z, and so F is periodic with period wr, - we. By Theorem 1.2, the only periodic meromorphic functions are doubly-periodic or singly periodic. The former case is (c) above. In the latter case, suppose w is a fundamental period of F . Then we can write F(z) = H ( e e ) , and if H is

8.3. Weierstrass'

<- and a-functions

317

not a rational function, then it has an isolated essential singularity at w. Using the Casorati-Weierstrass theorem and arguing as above, except that all points lie in the open fundamental period strip, we can find a k and a e in the fundamental period strip for F such that F(z

+ ak) = F ( z + a t ) for all z .

But this means that a k - ae is a period of F ( z ) , and since a k and at both lie in the open fundamental strip, a k - ae is clearly not a multiple of the fundamental period w, which is a contradiction.

8.3

Weierstrass'

C- and a-functions

Introduction: In some sense !Q(zlw,w') is the "simplest" doubly-periodic function with double poles at 0 and the values of the periods. The analogous 'Lsimplest" simply-periodic function with double poles at "period points" on the real line is s i n h z which has double poles at the integers, or if we wish, analogously, to have an arbitrary real period 2w, s1n2(L7z) . ,, ' Furthermore, by definition

+ C' 22

1 !Q(zlw,w') = -

w a lattice

1 1 -(2 - w)2 w2

po~nt

where 2w, 2w' are arbitrary periods. Suppose we hold w fixed and let w' go to oo,then our doubly-periodic function becomes singly periodic, and formally we have

Thus, when 2w1 + cm,the function !Q(zIw,wl) "degenerates" to a linear function of its simply periodic analogue. This suggests pursuit of this analogy, indeed since J &du = - cot u c, we can pursue through integration analogues (at least as respects Mittag-Leffler partial fractions and Weierstrass product expansions) in the theory of elliptic functions of sine and cotangent. It is perhaps worth noting that the same "degeneration" takes place formally on setting w' = 0: that is, "there is no second period" can be interpreted in two ways.

+

8. Elliptic Functions

318

It is also worth noting that one way of deriving the well-known value CF=l 7r2/6 used above is through letting z + 0 in the partial fractions expansion

5=

Definition 3.1 ( T h e Weierstrass <-function). <(z) = <(zlw, w') is defined by

where the integral is taken along any rectifiable curve not passing through any lattice point. T h e o r e m 3.1. ( a ) <(z) has simple poles with residue 1 at the lattice points and no other singularities i n C. (b) <(z) is not an elliptic function.

( d ) <(z) has the Laurent series expansion around 0:

where a2n+l = -

El,

a lattice

1 i

point

(e) W i t h wi defined as i n Definitions 2.2, we have

Proof. (a) By definition and the definition of ?(Z), 1 w a lattice po~nt

1 (z-w)

+ -w1+ - w2 .Z

<

has only a single simple pole (at 0) in the fundamental parallelogram, (b) and so by Theorem 1.6, cannot be elliptic. (c) follows from Theorem 2.l(b) and the definition of <(z). (d) follows from Theorem 2.l(d) and the definition of <(z). (e) Since <'(z) = -Q(z) and the points 2wi are periods of !&l(z), we have and so <(z

+ 2wi) = <(z) + 2qi

where the r]i are constants. Putting z = -wi, we get from (c) above that

r]i

= <(wi).

8.3. Weierstrass'

C-

Definition 3.2.

17i

Theorem 3.2. (b) If i

Proof. (a)

and a-functions

= ((wi),i = 1,2,3. (a) 172 = 771

+ 773;

< j , i , j = 1,2,3

By definition

w2

= wl

+ w3; hence by Theorem 3.l(e),

(b) By Theorem 3.l(a), ((2) has a simple pole with residue 1 at each lattice point. Hence, translating a period parallelogram by a small amount, and calling the resulting positively oriented perimeter C, C contains no poles, and

Arguing as in the proof of Theorem 1.6, we also have, on using Theorem 3.1 (e),

This proves (b) in the case i = 1, j = 3, and the remaining cases follow from this on using (a).

Note 3.1. The relationships in Theorem 3.2(b) are known as Legendre's relations. Since Legendre died in 1833, when Weierstrass was 18, and long before his memoirs on elliptic functions, it is clear that this was not the form in which Legendre found the results. For Legendre's actual result, see Theorem 4.14 below. Theorem 3.3. C(r + w) = ((r)

+ ((w) + f (=)

.

8. Elliptic Functions

320

Proof. Suppose u is any complex number not a lattice point, then, by the MittagLeffler expansion of I , C(z

+ u) + ((2 - 21) - 2C(z) = -- 1 +--+

w a lpttice

1 (z+u-w)

1 + (2-21-w)

-- 2 2-20

point

=

C

w a lattice point

2u2 (2 -w)((z -w)2 -u2)

and is clearly an elliptic function of z with the same periods as the P-function associated with <. Furthermore, the above expansion shows that the only poles in the fundamental parallelogram are simple and at 0 (where the residue is -2) and the points congruent to -u and u (where the residue in each case is 1). Hence (since !$ is an even function)

is an elliptic function with no poles in the fundamental parallelogram other than a triple pole at 0. Furthermore, the principal part of the Laurent expansion around 0 of this function is Hence ( N z ) - 'T(u))(C(z + '1~)+ C(z - 2~)- 2C(z)) - ' T 1 ( ~ ) is an elliptic function which has no poles and, hence by Theorem 1.5, is a constant A. Thus we have

9.

for u not a lattice point. Interchanging the roles of z and u, we get that if z is also not a lattice point

Adding (3.1) and (3.2), and using Theorem 3.l(c), we get

if z and u are not lattice points. Clearly, if either z or u is a lattice point, both sides of this equation have poles of the same order.

Note 3.2. Theorem 3.3 is the "addition theorem" for the Weierstrass zeta-function (of course, it cannot be algebraic since C is not elliptic); it can be used, together with Theorem 2.6, to give another proof of Theorem 2.8.

8.3. Weierstrass' C- and a-functions

321

There are several different approaches to construction of Q ( z ) ;one alternative to that used in Section 2 is to construct C(z)as the "simplest" meromorphic function with singularities at the lattice points, and define p ( z ) = -C1(z). Another is after proving that converges, to construct the obvious elliptic function a po~nt

of order 3,

&

Elw

a lsttice (2-w)

,,

and then integrating and choosing the integration

point

constant appropriately to obtain - $ p ( z ) . Pursuing the analogy of the introduction to this section, we make the

Definition 3.3. The Weierstrass a-function a ( z ) = a(zlw,w') is defined by

where C is the Weierstrass C-function and the integral is taken along any rectifiable curve not passing through any lattice point. Theorem 3.4. ( a ) a ( z ) is an entire function of order 2 with simple zeros at the lattice points and Hadamard product

o ( z )= 2

JJ1

w a lattice point

(1 -

-) w Z

1

1

2

e6+~(r)

where the ' again indicates that the term w = 0 is omitted.

(b) a(-z) = -a(z). (c)

a(0) = 0,u1(O)= 1.

(d) With wi and qi, i = 1,2,3, as given b y Definitions 2.2 and 3.2, we have

Proof. (a) The product formula for a ( z ) and its convergence follow immediately from its definition (viz. Proof of Theorem 3.l(a)). That there are simple zeros at the lattice points is immediate. The argument used in constructing p ( z ) in Section 2 actually shows that Cwa l,ttice converges for a > 2 and hence, the exponent Dolnt

&

of convergence of the zeros'of a ( z ) is 2. By Theorem 3.1.4, u has order 2. (b) follows from the definition of a and Theorem 3.l(c). (c) u(0) = 0 follows from the product in (a), al(0)= 1 follows then by considering lim,,o

q.

8. Elliptic Functions (d)

= [(z), we have from Theorem 3.l(e),

Since from the definition

Hence a ( z + 2wi) = ~ ( z ) e ~ ~where i ~ +K~ is, a constant. Taking z = -wi, we get from (b) above ,K = - , 2 ~ i w i and so the result. (e) As observed, from the definition,

and so

P(2)= -C1(z) =

( ~ ~ ( 2-) D'I(Z)O ) ~ (z) (~(2))~

Theorem 3.4(e) expresses p ( z ) as the ratio of two entire functions. However, every meromorphic function has an expression as such a ratio, and hence, in particular, every elliptic function does. Weierstrass' a-function is the key to such an expression:

Theorem 3.5. Suppose E(z : w,wl) is an arbitrary elliptic function with fundamental periods 2w, 2w1. Let a ( z ) = a(zlw,wl). Suppose a j , j = 1 , 2 , . . . , r is a fundamental system of zeros and bj, j = 1 , 2 , . . . ,r is a fundamental system of poles for E(z) . Then U(Z- a j ) E(z) = a(. - bj) j=1

KH

where K is a constant. Proof. By Theorem 1.10, Cjr=la j I Cjr=lbj( mod 2w, 2w1) . Replacing bT by bT plus an appropriate period, if necessary, we obtain a pole which possibly lies outside the fundamental parallelogram, but for which we can then write

The function

8.3. Weierstrass'

C-

and a-functions

323

has the same zeros and poles as E(z) (recall that in the lists a l , . . . , a n ; bl, . . . ,b,, each zero or pole is listed according to its multiplicity). Furthermore, by Theorem 3.4(d),

and similarly for a ( z - bj

+ 2wi). Hence

by (3.3). Hence G(z) is elliptic. So 1.5, is a constant.

% is elliptic and has no poles, and so by Theorem

N o t e 3.3. It is interesting to pursue the relation of "degenerate" elliptic functions t o simply-periodic ones mentioned in the introduction to this section. Arguing as there we get 7r x2z 2w <(z~w, m) = -cot - and a(zlw, m ) = -sin 2w 12w2 ' 7r

nz 2w

m2z2

- e D

,

on using the partial fraction decomposition of the cotangent and the Weierstrass product formula for the sine. firthermore, clearly C(z 2wlw, m ) = [(zjw, m ) and so since C is an odd function, C(wlw, m ) = in conformity with Theorem 3.l(e). Similarly a ( z + 2wlw,m ) = o(z)e2(&)(~+,). Also, analogously t o Theorem 3.5, a simply-periodic meromorphic function with zeros a t the points a , j = 1,.. . , r and poles a t the points bj, j = I , . . . , s can be expressed as sin ($(z - a j ) ) ~(2) fl,sZlsin (5(z - bj)) '

&

+ &,

+

n,rZ1

where g(z) is a simply-periodic entire function without zeros. A simply-periodic meromorphic function f with real period 2w whose only singularities in the strip 0 < Re z 2w are simple poles bj, j = 1,2, . . . ,r with residues PI, . . . ,PT is an expression

<

where h(z) is a periodic entire function. This suggests the possibility of an analogous expansion involving C(z) for elliptic functions. In fact, any elliptic function can be expressed as a linear combination of <(a) and its derivatives. (Since -I1(z) = !J?(z),

8. Elliptic Functions

324

and '$I is an algebraic function of '$, this can also be viewed as a statement about the '$-function). See e.g. Whittaker and Watson, A Course of Modern Analysis [247], p. 449-450. An expression of this form is useful in dealing with elliptic functions when the principal part of the expansion at each pole is known. If we also let w + m , we get the degenerate cases

In particular, if we allow degenerate elliptic functions, Theorem 2.11 can be stated as: A meromorphic function with an algebraic addition theorem is a rational function of an elliptic function (degenerate or non-degenerate). Finally, we may note that in the degenerate case we get

&,

& &,

since C,, f = X I I = (these values may be derived from a FourierSeries argument; see also Theorem 9.3.5 (iii).) We also have A(w,m) = 923 - 279; = 0 .

Note 3.4. Theorem 3.5, in particular, says that an elliptic function is determined up to a multiplicative constant when its periods and a complete set of zeros and poles in the fundamental parallelogram are known.

Example 3.1. Suppose W is not a period of '$(z), and assume without loss of generality that W is in the fundamental parallelogram. Then

is an elliptic function with a double pole at 0 and zeros at W and -W, and these are the only zeros and poles in the fundamental parallelogram. So by Theorem 3.5

where A is a constant. Hence z2 lim z2('$(z) - '$(W)) = A lim -a ( z - W)o(z =-to zo2(z)

z-to

+ W) ,

8.3. Weierstrass'

C-

and a-functions

and so by Theorem 3.4(b) and (c),

Thus

2%)

- Y(W) = -

o(z

+ W)a(z - W) a"(.>.

(W)

Since W was an arbitrary point not a period of Y, (3.4) is sometimes called the LLaddition theorem" for the (-function. Taking logarithms in (3.4) and differentiating with respect to z, one recaptures Theorem 3.3 in the form of the proof there, with the added fact that A = 0. Weierstrass' a-function also permits an example of a non-trivial entire function for which the Phragm6n-Lindelof indicator h(B) (see Section 3.3) is constant.

Theorem 3.6. For the entire function u(z) = C (2-

1 h, i),h(0)

~/2.

Proof. Using Theorem 3.4 (d) repeatedly, one gets by induction for positive integers k

Replacing z by z - 2kwi, the same formula is seen to hold for negative integers k as well. In this case, the lattice points are just those of the usual rectangular coordinate system and wl = 112, ws = i/2 and so

But also, we have by definition

1 w a lattice 3 point

=i

(:

C'

-+

w a lattice point

C'

w

1

-2 - iw

1 +-+zw

1

1

-

2(2w)

1 1 7-+ - + L)= iC w 2w2 a lattice 5 -

(i)

= i<(w3) = iv3

polnt

since as w runs over the non-zero lattice points, so does iw.

8. Elliptic Functions Furthermore, by Theorem 3.2.(b) ni =-; 2

771W3 1'4'377

hence in this case, by (3.7),

whence

773

=

F, and 71 =

i773

= n/2. Substituting in (3.6) gives

+ + in) = ~ ( z ) ( - l ) ~ + ~ + nerz(m+in) m e2r(n2+m2) Letting reie = z + m + in, where z is a point in the fundamental parallelogram, we U(Z m

have since a(z) has order 2 (Theorem 3.4 (a)), and lzl def

- log (a(reie)l =

h(8) = lim

r2

T+OO

< 22,

*

- log Ia(z k m in)[ lim lz f m f inI2

m,n-+m

-

= lim

nm Re z

m,n+m

+ nn I m z + 4n(m2 + n2) = w/2 . m2 + n2

-

The same sort of argument shows directly that if M ( r ) = maxlzl<,(u(zll/2,i/2)), then Log M ( r ) w/2r2 as r + co and so u(z(1/2, i/2) has typev$. One can also see directly that n(r) = number of zeros of a(z11/2,i/2) in B(0,r) = number of lattice points m + i n in B(0, r ) which is nr2 as r -+ co. z-W) In Example 3.1, the formula P ( z ) - v ( W ) = u ( r+W)u u 2 ( z ) u ~ ( w ) is proved. Suppose we take W = wi in this expression. Then since by definition P(wi) = ei, we get

Also, by Theorem 3.5(d)

Hence, P ( z ) - ei = This suggests

(w) .

Definition 3.4. The associated a-functions ui (z) are defined by

Theorem 3.7.

2

(a) v ( z ) - ei =

(%), i = 1,2,3.

8.3. Weierstrass'

C- and a-functions

( c ) ai(z) is an even function, i = 1,2,3.

(e) a(2z) = 2 a ( z ) a l ( ~ ) 0 2 ( z ) f l 3 ( ~ ) .

(f) (i) ai(z

+ 2wi) = -ai(z)e2qi(Z+Wi),)i = 1 , 2 , 3

( i i ) a i ( z + 2 u j ) = a i ( z ) e 2 ~ j ( Z + W ~ ) i = 1 , 2j ,=31, , 2 , 3

i f j.

Proof. (a) is Definition 3.4 and the preceding remarks. (b) follows from Definition 3.4 (since 2wi is a period of p and hence a zero of 0). (c) Since o is an odd function (Theorem 3.4(b)), we have, by Theorem 3.4(d),

(d)

By Theorem 2.5, and (a) above

Hence,

%

= 1 (by (b) above and Theorem But lirn,,o z3p'(z) = -2, and limZ,o 3.4(c)). Hence the - sign holds in (3.8). We may note that this expresses p' in the form of Theorem 3.5. (e) From Theorem 3.3, if z is not a lattice point, taking limits as w + z, we get

Since, by Definition 3.3 of a(z),

we get from (3.9) a(2.Z) = C ( a ( z ) ) * ~ ' ( z,)

328

8. Elliptic Functions

where C is a constant and the result now follows from (d) above, since by (b) above ui(0) = 1, and by Theorem 3.4(c) lim,,, = 2, whence C = -1. (f) (i) By Theorem 3.4(d),

3

(ii) By Theorem 3.4(d), if i # j ,

by Theorem 3.2(b). Theorem 3.7(a) prompts the definition of Definition 3.5.

d-

=

s7

d-

by,

i = 1,2,3.

#

Theorem 3.8. (a) d m9 is an elliptic function with two simple poles and two simple zeros in its period parallelogram. (b) = i = l 2 3 , j=l,2,3,i<j. (c)

~ Y (+ ~Z i -)ei =

,/.j-.i,/G

m

i # j , i # k , j # k.

#

Proof. (a) = by Theorem 3.7(f)(i) and Theorem 3.4(d). Also, if i # j, Theorem 3.7(f)(ii) and Theorem 3.4(d), show that

and hence, if j # i, 4wj is a period of #. We have found only two fundamental periods this way (and by Theorem 1.3 we could not find more) since wz = wl + ws and hence one of the two possible values 4wj, j # i is a linear combination of the other and 2wi. is clearly meromorphic and with 2wi and 4wj as its fundamental periods, 4 2 ) it has poles in its period parallelogram at 2wj and 2wi + 2wj (which, being zeros of a , are simple), and simple zeros at w - i and wi + 2wj.

8.3. Weierstrass'

C- and a-functions

(b) By definition, we have for i = 1 , 2 , 3 ; j = 1,2,3,

and

Hence, using Definition 3.3 of the associated a-functions,

by Theorem 3.2(b). (c) By Theorem 2.8 with w = wi, we have (since wi are the zeros of '$3')

So, by Theorem 2.5, with j

# k, j # i, k # i,

But, as observed in Note 2.2, ei + e j

+ ek = 0.

Hence

Thus

+ =

Putting z = wj, since wj wi wk ( mod 2w, 2w1),we see that the (d) follows from Definition 3.4 and Theorem 3.7(d).

+ sign holds.

Note 3.5. Despite the long history of elliptic functions, no computation of the zeros of Q(z) was published until 1982 (though, of course, the zeros of '$3'(z) were well-known viz. Theorem 2.5). In 1982 Eichler and Zagier published a paper [65] explicitly indicating the zeros of P ( z ) . Unsurprisingly, this explicit formula is fairly complicated.

8. Elliptic Functions

8.4

Jacobi's Elliptic Functions

Introduction: We have already remarked that elliptic functions of order two with two simple poles in the fundamental parallelogram are associated with the name of Jacobi, and Theorem 3.8(a) shows that

are three examples of such functions. Instead of beginning here, however, let us return to Problem 2 of the Introduction. The reason for doing this, as will appear, is that unfortunately while Problems 1 and 2 do lead to elliptic functions of Jacobi's sort, the "naturally occurring" such functions are not i = 1,2,3, but slightly different ones. In problem 2, we were led to the integral

3,

where 0 < k2 < 1 or, equivalently,

where $ = Arcsin z, and we take z real for the moment. Suppose, as suggested in Problem 2, we consider the "inversion problem" for this integral; that is, writing the integral as F(k, $), we wish to solve the equation F ( k , +) = x, x real,

+

in terms of k and x. Since F ( k , + ) is clearly a monotone increasing and for continuously differentiable function of for 0 < k2 < 1, this problem makes sense. Jacobi called this "inverting function", the amplitude of x, which we write as

+

or if the dependence on k is inessential to the argument, simply $ = am(x). f i o m the origin of $ as Arcsin z, it is, furthermore, immediate to wish to consider the composite function sin(amx). Before proceeding further, let us return to the problem of solving

and try to simplify it still more. If we make the substitution

+

1 k2 1 sin2 9 = - and choose a = t+a' 3 '

8.4. Jacobi's Elliptic Functions

the equation takes the form

where a = -12a2

$

12a - 4,

0 = -8a3

i- 12a2 - 4a, y = -a

%.

+ sin +

and a =

Thus, if we can find a Weierstrass ?$I-function for which g2 = -a, g3 = -/3, taking sight of Theorem 2.6, we can try to make the further substitution (although justification is necessary) t = y ( u ) , which will lead to

(note that as t + oo, ?$I(u)+ 0 for u in the fundamental parallelogram). Thus, if we can also choose a = -e3, say (which is not clear), we would get for x real in some interval with lefthand end point 0, sin amx =

1

J -

.(.) = 03(x)

and this would serve to define sinamz, by analytic continuation, for all complex z. Thus, we will have established a relationship between a ?$I-function and the inversion problem of Problem 2. We now give a correct and more precise version of these heuristic remarks. We first show the existence of a ?$I-functionwith prescribed real invariants.

Theorem 4.1. If y(zlw, w') is a Weierstrass ?$I- function, with real invariants g2, g3 (Definitions 2.3), then all coeficients in the expansion of Theorem 2.1 ( d ) are real, and so, in particular, P(2) = P ( z ) -

Proof. By Theorem 2.6,

Hence, if z

9 wi(

mod 2w, 2w')

(i = 1,2,3) we get

Thus, since by Theorem 2.l(d), P ( z ) = $

+ C z l c2,z2", we have

8. Elliptic Functions Hence comparing coefficients, we get for v 2 3,

>

By using this recursion relation, we can express CZ, for r 3 in terms of c4 and cz. But, comparing Theorem 2.l(d) and Theorem 2.6, we see that c4 = E, and cz = E; hence if gz and g3 are real, all the c2, are real and consequently

Note 4.1. The recursive relation found in the proof is, of course, generally valid and shows that in any case, all c2, are polynomials in gz and g3 with rational and c8 = coefficients. For example, cs = We may also note that for v = 1, the above argument again yields c2 =

&

w.

5.

Theorem 4.2. If p ( z ) = p(zlw, w') is a Weierstrass !J?-function with real invariants g2, g3, then either

(i) A

< 0,

and w' = 0 ,

(ii) A > 0, w is real and w' a pure imaginary. Furthermore, if !J?(zlw') is any p-function such that either (i) or (ii) holds, then has real invariants.

-

+

Proof. By Theorem 4.1, if 2w is a period, then since p ( 2 ) = p ( z ) = p ( z 2w) = Y(z z ) , 2w is also a period. Hence, 2w and 2w - % are periods and so !&I has real and purely imaginary periods (though these are not necessarily 2w and 2w'). Furthermore, as observed in Note 2.2, el, e2, e3 are the roots of E3 =0 and (Definition 2.3), A = l6(el - e 2 ) 2 ( e ~ - e3)'(e2 - e3)2 . Hence, since A # 0, A > 0, if and only if el, e2, e3 are all real and A < 0 if and only if one of e l , e2, e3 is real and the other two are conjugate complex numbers with non-zero imaginary part. Thus, since ei = p(wi), i = 1,2,3, taking Theorem 4.1 into account, we have has two fundamental periods which are complex (i) A < 0, if and only if conjugates. (ii) A > 0, if and only if one fundamental period is real and the other purely imaginary. Finally, if either (i) or (ii) holds for some p-function, p(zlw, w'), then from the expressions for the invariants in terms of lattice points (Theorem 2.6) it follows that gz = and g3 =

+z

+

z.

333

8.4. Jacobi's Elliptic Functions

Theorem 4.3. Suppose p ( z ) is a Weierstrass p-function with real invariants g2,g3. Consider a real variable x such that (i) if A < 0, x E [0,w w'] (by theorem 4.2, in this case w w' is real), or if A > 0, x E [0, w]. (ii) Then the function p ( x ) = u has, in these intervals, the inverse function

+

where in case (i) u

+

> e2, and in case (ii) u 2 el.

Proof. Clearly Y(z) is real in the respective intervals by Theorem 4.1. In case (ii) as x + 0, p ( x ) -+ co along the real axis, while p(w) = p(wl) = el. Hence, if c > e l , the equation Y(x) = c has a root between 0 and w; say at a; but then p(-a 2w) = p(-a) = T ( a ) = c, and w < -a 2w < 2w. Hence the equation p ( z ) = c has two zeros in the fundamental parallelogram, and since p ( z ) and so p ( z ) - c has order 2, these are the only two zeros in the fundamental parallelogram by Theorem 1.8. It follows that !J? is strictly decreasing from +co to el as x goes from 0 to w , and so p'(x) < 0 in this interval. A similar argument holds in case (i), except here p ( w w') = !J3(w2)= ea. On the other hand by Theorem 4.1, not only 9 but p' is real on the real axis, hence (!J?'(x))~= 4q3(x) - g2p(x) - g3 > 0 and so we get

+

+

+

in the respective intervals. Since p is monotone in the intervals in question, it has an inverse whose domain is [e2,co) in case (i) and [el, co) in case (ii) and if u = p ( x ) , then

Thus

while if F is the desired inverse of p ,

Hence

where u

> e2 in case (i); u > el in case (ii) and F ( u ) -+ 0 as u + co.

Theorem 4.4. If g*' and g*** are given real numbers such that (g**)3-27(g***)2# 0, then there is a Weierstrass p-function with g** and g*** as invariants.

8. Elliptic Functions

334

Proof. Case 1: (g**)3- 27(g***)' > 0. Then all the roots of 4J3 - g**J - g*** = 0 are real and distinct, call them e*, e**,e***,where e* > e** > e***. Since

is positive for [ > e* and negative for J

< e***,if we put

then the quantities a! and p are real and we form the function !J3(zla,iP), and suppose it has invariants g2 and g3. (Note that by the proof of Theorem 4.2, g2 and g3 are real.) These definitions are motivated by the fact that in Theorem 4.3, if we take u = el in case (ii), we get a as the value of the integral

Also, from the definition of Y(z),

Thus, it follows that if g2 and g3 are the invariants of p ( a ) , then g2 and -g3 are the invariants of v(iz), and -el, -e2, -e3 are the roots of 4z3 - g2z g3 = 0. Arguing as in Theorem 4.3, we see that if g2, g3 are real, then for A > 0, and y E [0, ,8]

+

has the inverse function

where now, v

< e3. Taking v = e3, we get

Thus, if we show that the quantities a and P, as defined, uniquely determine g** and g***; then since dt a= J4t3 - g2t - g3

lr

8.4. Jacobi's Elliptic Functions and

we will have g2 = g** and g3 = g***. We have

dt

and

Making the substitution t - e***= in the first integral, and t-e*=

,*

- ,*** v2

e*** - e*

v2

in the second, the integrals become

and

where

(The reader at this point may wish to look again at Problem 2.) Hence

As k2 goes from 0 to 1, the numerator on the right is monotone increasing from 7r/2 to oo, while the denominator is monotone decreasing from oo to n / 2 , and so as k2 goes from 0 to 1, alp goes from 0 to oo monotonically. Thus, given a value of

8. Elliptic Functions

336

s.

a l p , it corresponds to a unique value of k2 = Thus, given a and /3, they determine k2,0 < k2 < 1 uniquely, and then in turn by (4.1),

Thus, e* - e*** is also uniquely determined, and so is e** - e*** = k2(e* - e***). Since e* + e** + e*** - 0 by definition of e*, e**,e***, we see that then e*** is uniquely determined and so also e* and e**. Finally, also from the definition of e*, e**,e***, 9** + ,**,*** + e*e***, -4 and

and so g** and g***are uniquely determined, as claimed. Case 2: (g**)3-27(g***)2< 0; then one of the roots e*, e**,e***of 4J3-g**t-g*** = 0 is real, say e**, and e* and e***are complex conjugates. Since 4t3 - g**<- g***= 4(5 - e*)(<- e * * ) ( E - e***)is real for real J and positive for [ > e**, negative for J < e**, we put dt 2a = J4t3 - g**t - g***

ly

and

The numbers a and ,O are real and so we may form the function

with invariants 92 and g3 (which again by Theorem 4.2 are real). Theorem 4.3 Case (i) shows that

while an argument similar to that in Case 1 above shows that

Thus, it is again enough to show that the quantities a and g***uniquely.

P

determine g** and

8.4. Jacobi's Elliptic Functions We have

In the first integral, let t - e** = v2 where v > 0 and in the second integral, let t - e** = -v2 where v > 0. Finally, write f$ = arg(e** - e*) = - arg(e** - e***) ,

and

p = le** - e*l = le** - e***l

,

(since e* and e*** are complex conjugates) to see that

and dv (v4 - 2pv2 cos f$

+ p2)i

Hence

on letting v = wJiT. Thus, the ratio alp depends only on cos4, and as f$ goes from 0 to n , the numerator on the right is monotone increasing from 7r/2 to co and the denominator is monotone decreasing from co to n/2. Thus, as f$ goes from 0 to n , a/@goes monotonically from 0 to co passing through all real values. So, given a value of a/@,it corresponds to a unique value of f$ in the interval [O, XI. Given a and @ and having determined this value of 4, we can then find p through

Thus, given values of a and @, they correspond to unique values of e** - e* = pei@ and e** - e*** = pe-i@, where 0 5 4 5 n, and now as in Case 1, we use e* +e**+e*** = 0 to determine e*,e**,e***,uniquely and these in turn to determine g** and g***.

8. Elliptic Functions Definition 4.1. For 0 < k2 < 1, the function

is known as Legendre's elliptic integral of the first kind. For x real, the equation 3 ( k ,$) = x defines qh as a function of x and k called the amplitude of x (for a given k ) (see introduction to this section) and denoted a m (. x , k .) or just arnx if the dependeice o n k is immaterial. 3 ( k , k / 2 ) is called the complete elliptic integral of the first kind. and denoted R. Theorem 4.5.

(a) am(x

+2R) = amx +n

(b) arnx is convex downward i n 0 2K.

< x 5 R, and convex upward i n R 5 x 5

Proof. ( a ) Since 3 ( k ,a m ( % ,k ) ) = x , we need t o show that F ( k , $+n) = F ( k ,$)+ 2 R . But

x=L

amx

(b) so ( 1 - k 2 s i n 2 ( a m x ) )4 = Consequently,

d8 (1-k2Sin28)$'

& (arnx).

1 d k2 d2 (arnx) = - - ( 1 - k2 sin2 a m x ) - h k2 sin(2amx)-(arnx) = - - sin(2amx)

dx2 2 dx 2 T h i s is negative i f 0 < arnx < r / 2 , i.e. i f 0 < x < IK,and positive i f r / 2 < arnx < r , de (see proof o f part a). i.e. i f IK < x < 2 R = J : ( 1 - k 2 sin2 i9)g

Definition 4.2. The functions sin(amx) and cos(amx) (see also introduction to this section) defined for real x are denoted

s n x and cnx respectively. The function & ( a m x ) is denoted d n x . Clearly s n x , c n x , d n x also depend o n k , 0 < k2 < 1, which is called the modulus of the functions.

8.4. Jacobi's Elliptic Functions Theorem 4.6. (a) sn2x + cn2x = 1 (b) dn2x + k2sn2x = 1 (c) zd( ~ n x ) = cnx dnx (d) $ (cnx) = -snx dnx (e) (dnx) = -k2snx cnx (f) as k -+ 0: am(x, k) -+ x, snx -+ sinx,cnx -+ cosx,dnx -+ l , K -+ n/2. (g) as k -+ 1 : snx -+ tanhx, cnx -+ sechx, dnx -+ sechx, K -+ oo.

2

Proof. (b) and (e) are established in the proof of Theorem 4.5(b). The formulas (a), (c), (d) follow from the properties of sine and cosine. (f) is immediate since as k -+ 0, +(k, $) -+ $. Finally, as k -+ 1, +(k, $) -+ secOd0 = Log tan(: $).

+

Hence amx -+ 2Arctan(ex) - s / 2 = i Log

(-)

snx -+ - cosh ~ o g

=-

(i-e.)

- n/2. Hence

(5) +( ) 2

1 - e2z - tan hx 1 e2"

+

The remaining formulas now follow from Parts (a) and (b).

Note 4.2. The functions sn, en, dn were written sin am, cos am, Aam by Jacobi; $), then $ and the above notation is due to Gudermann (1838). If x = logtan(: is sometimes called the gudermannian of x. Gudermann was Weierstrass' teacher.

+

Theorem 4.7. Given a value of k, 0 < k2 < 1, there is a unique Weierstrass pfunction p(zlw,wl) with w = (el - e 3 ) - 3 ~ such that in the interval

where, of course, ei = 9(wi), i = 1,2,3.

E,

Proof. Given prescribed values of k2 = 0 < k2 < 1, and of el - e3, with e l , e2, es real, we can always find a 9-function, !?3(zlw, w'), with positive discriminant, such that p(wi) = ei. This follows from Theorem 4.4 (we simply construct the invariants from el e2 e3 = 0, k, el - es and the expressions for the invariants

+ +

8. Elliptic Functions

340

in terms of the ei.) Suppose 92 and 93 are the invariants of this P-function. By Theorem 4.3, if y ( x ) = u, then

for x E [0, u]. Making the substitution we get

where sin2 4 =

e,

and k2 =

E.Hence

Thus, since P ( x ) = u we have sn((e1 - e3)ix) = sin 4 =

d e 1 - e3

J-'

The formulas for m((el - e3)4x) and dn((el - e3)3x) now follow from Theorem 4.6(a) and (b) respectively. Finally, by Theorem 4.3 w=w1=

LPQ

dt

J4t3 - 92 - 93

= (el-e3)

-'

de = (el - e 3 ) - * ~. (1 - k2 sin2 0) 5

Theorem 4.7 allows us, by analytic continuation of the right hand sides of the formulas there, to extend the definitions of sn, cn, dn to the whole plane where they represent elliptic functions (by Theorem 3.8(a)).

Definition 4.3. Given a v-function P(zlw,wl), with discriminant P(wi) = ei,O z E @,

< e3 < ez < el;

def

let X = (el - e3)? and k

m ( z , k) =

def

( ~ ( f ) - e l ) +--Ol(f) ( P ( f ) - e3)i 03(f)

dn(z, k) def =

(P(x) - e2)4 - m ( f ) 1

( P ( f ) - e3)"

-

03(f)

'

>0

and with

ef(=)', then for all

8.4. Jacobi's Elliptic Functions

341

where the second expressions follow from Definition 3.4. We will call Q(z) the associated !Q-function of the Jacobian functions. Finally, complementary to the definition of K in Definition 4.1, we define

By Weierstrass' "Principle of the Permanence of Functional Relationships", the relationships of Theorem 4.6 hold for these definitions of sn, cn, dn, since they were established by analytic continuation of the previously defined functions. Alternatively, they may be proved de novo, using where necessary, properties of the '$-function. We assume them to be known for all complex z from now on.

Theorem 4.8. The fundamental periods, and poles, and zeros in the fundamental parallelograms of the elliptic functions of order 2, snz, cnz, dnz, are respectively given by: snz cnz dnz

Periods 4K, 2 i R 4K, 2 K + 2 i K 2K, 4iK1

Poles iK1, 21K iK1 2 K + i R , 4K+iK1 iK1, 3iK1

+

Zeros 0,2K K,3K K + iK1, K

+3iR

In fact, snz, cnz, dnz all have poles in C at the same points, namely at the points 2mK + (2n + l)iK1,m, n arbitrary integers. Proof. By the argument of Theorem 4.4, Case 1, the values k =

(z)', and

X = (el - e3);, uniquely determine an associated !??-function whose periods are 2K/X and 2iR/X. Thus, by Definitions 4.3 and Theorems 3.7(f) and 3.4(d),

+ + + +

dn(z 2K) = dnz , sn(z 2iK1) = snz , cn(z 2iK1) = cnz , dn(z 2%') = dnz , (note that K = Awl and iK' = Xu3 in the notation of Theorem 3.7(f)). This verifies the statement about periods.

8. Elliptic Functions

342

The statements about poles and zeros now follow from Definitions 4.3 and Theorems 3.7(b) and 3.4(c); Awl = K, XWQ = iK1, wz = w~ WQ; and the above relations. Although the poles in the fundamental parallelogram are different for snz, cnz, dnz, this is because of the different shapes of the parallelograms and the last statement of the theorem simply follows by considering the set of periods, together with the poles in the fundamental parallelogram.

+

Since snz, cnz, dnz are elliptic, they have (by Theorem 2.10) algebraic addition theorems. Theorem 4.9.

+

snz cnw dnw snw cnz dnz 1 - k2sn2zsn2w dnz cnw - snz snw dnz dnw w, k) = 1 - k2sn2zsn2w dnz dnw - k2snz snw cnz cnw w, k) = 1 - k2sn2zsn2w

s n ( z + w,k) = cn(z

+

dn(z

+

Proof. Let w be an arbitrary but fixed complex number

+ iK1)(mod 2K, 2iK') . Let f (z) = snzsn(z + w), g(z) = cnzcn(z + w), h(z) = dnzdn(z + w) . w $ 0 , w $ f IK, w $ f iK1, w $ f ( K

By the relations 4.3-4.8 of the proof of Theorem 4.8, f , g , h have fundamental periods 2K and 2iIK1, and so by Theorem 4.8, are elliptic functions of order 2 (two simple poles in the fundamental parallelogram). In fact, the complete set of poles of each of f , g, h is given according to Theorem 4.8 by all the points congruent to iK1 or iK' - w(mod 2K, 2iK1). Thus A f (z) g(z) and B f (z) h(z) are also elliptic of order 2 with fundamental periods 2K, 2iK1. So, since f , g, h have the same poles, we can determine constants A and B such that Af (z) +g(z) and Bf (z) h(z) have no pole a t iK1 in the fundamental parallelogram, and then these functions (by Theorem 1.8) must be constant. Moreover, the value of these constants must be A f (O)+g(O) = cnw, and Bf (O)+h(O) = dn(w) respectively, since sn(0) = 0, cn(0) = 1, dn(0) = 1 by Theorems 3.4(c) and 3.7(b) and Definition 4.3). Thus we seek constants A and B (which we know exist) such that

+

+

+

A snz sn(z + w) + cnz cn(z + w) = cnw B cnz sn(z + w) + dnz dn(z + w) = dnw for all z E (C and w E C (restricted as above). Differentiating with respect to z gives A cnz dnz s n ( z + w ) + A snz cn(z+w)dn(z+w) - snz dnz cn(z+w) - cnz s n ( z + w)dn(z w) = 0 , B cnz dnz sn(z+w)+Bsnz cn(z+w)dn(z+w)-k2snz cnz dn(z+w)-k2dnz sn(z+

+

8.4. Jacobi's Elliptic Functions

343

w ) c n ( z + w ) = 0. Putting z = 0 in these last two equations, we get (since sn(0) = 0, cn(0) = 1, dn(0) = 1 )

A snw - snw dnw = 0 B snw - k2snw cnw = 0 Hence we have

s n z s n ( z + w)dnw + cnz cn(z + w ) = cnw k2snz sn(z

+ w)cnw + dnz d n ( z + w ) = dnw .

Interchanging the roles of z and w gives

s n z s n ( z + w)dnz + cnw cn(z + w ) = cnz

(4.11)

k2snw s n ( z + w)cnz + dnw d n ( z + w ) = dnz .

(4.12)

The equations 4.9-4.12 hold for all z except possibly those satisfying one of the congruences mentioned at the start of the proof. We prove the addition formulas for these z and w ; the result then follows for all z and w by continuity. Multiply (4.9) through by cnw and (4.11) through by cnz and subtract to get

sn(z

+ W ) ( s n z cnw dnw - snw cnz d n z ) = cn2w - cn2z = sn2z - sn2w .

Hence

sn(z

z - sn2w)(snz cnw dnw + snw cnz d n z ) + w ) = ( s n 2 sn2z cn2w dn2w - sn2w cn2z dn2z

Using cn2z = 1 - sn2z and dn2z = 1 - k2sn2z in each term of the denominator transforms it to ( s n 2 z - s n 2 w ) ( l - k2sn2z sn2w) and so (a) follows. Substituting (a) for s n ( z w ) in (4.10) gives

+

dnw - k 2 s n 2 zsn2w dnw - k2sn2z cn2w dnw - k 2 s n z cnz dnz snw cnw d n z ( 1 - k2sn2z s n 2 w ) dnw dn2z - k2snz cnz dnz snw cnw , d n z ( 1 - k2sn2z sn2w) on using the relationship between sn2 and cn2 and sn2 and dn2 again, and so (c) follows. Substituting (a) in (4.9) similarly gives (b).

Note 4.3. The above proof seems to be due to Hurwitz (see A. Hurwitz and R. Courant, Funktionentheorie [118]).An "elementary" proof which does not use Theorem 1.8 is also possible, by working with the real functions as originally defined. For a dynamic geometrical proof, see Greenhill, Applications of Elliptic Functions [881

8. Elliptic Functions

344

The addition formula for s n is in some sense where the theory of elliptic functions began. Stimulated by Fagnano's paper (see introduction) and the fact that the addition formula for the Arcsine is: If dt dt

JdZ&+Jd

m=Jdm ,

then z=xJ1Z-;;Z+yJC2, which is easily proved by the change of variables t = sine and using the addition formula for the sine, Euler proved the following result. If

where 0

< k2 < 1, then

which is equivalent, in a similar way, to Theorem 4.9(a). A motivated reconstruction of Euler's approach well worth perusal, is given in the beginning of C. L. Siegel, Topics in Complex Function Theory I [221]. Making appropriate algebraic manipulations and using the relations between sn, cn, dn, it is easy to see that the addition theorems of Theorem 4.9 are, in fact, algebraic addition theorems. If we let k + 0 in Theorem 4.9, we recover the addition theorems for sine, cosine, and a result valid for the constant 1. If we let k + 1 in Theorem 4.9, we recover the addition theorems for tanh and sech. (viz. Theorem 4.6(f) and (g)).

Theorem 4.10. (~n'(z)= ) ~(1 - sn2z)(1- k2sn2z)

+ k2cn2z) ( d n ' ( ~ )= ) ~(1 - dn2z)(dn2z+ k2 - 1) ( ~ n ' ( z )= ) ~(1 - cn2z)(1- k2

Proof. From Theorem 4.6(c), (d), and (e), for all complex z, we obtain immediately on using Theorem 4.6(a) and (b) these results.

Note 4.4. If k + 0, we again obtain differential equations valid for sine, cosine and the constant 1. If k + 1, we again obtain differential equations valid for tanh z and sech z. We may also make a connection between the Jacobian elliptic functions and Problem 1 of the Introduction to this Chapter. Recall that the length of the arc of the ellipse, discussed in Problem 1 of the Introduction, is a :J J1 - k2 sin2 Ode. We can now prove

8.4. Jacobi's Elliptic Functions Theorem 4.11. =alxdn2tdt,

~-8d8 where

4 = amx.

Proof. We make the substitution sin8 = snt; then d8 dB cnt dnt cos8- = cnt dnt and so - = = dnt dt dt diT&

.

Hence, the integral becomes (since s n ( 0 ) = 0 , and 1 - k2sn2t = d n 2 t )

where 4 = a m x . def

Definition 4.4. The function & ( k , z ) = So dn2t dt, where the integral is taken along any simple rectifiable Jordan arc not passing through a pole of the integrand is known as Legendre's elliptic integral of the second kind. The particular integral I

E ( k , K ) gf~ ( k is) known as Legendre's complete integral of the second kind. W h e n the dependence on k is immaterial, simply E ( z ) is written for the integral. Note 4.5. Since by definition (Definition 4.1), R = s;l2

d;-

d x , we have

~ / =2 a m K , and so

and the perimeter of the ellipse of Problem 1 with eccentricity k = ( 1 0 < k < 1, is given by 4aE(k,K ) .

Theorem 4.12.

g)',

( a ) E ( z ) is an odd function.

(b) & is meromorphic i n @. with simple poles at the poles of d n z ; all with residue 1. ( c ) E is n o t an elliptic function.

Proof. (a) Immediate from Definitions 4.4 since dn2t is an even function. (b) d n z has simple poles in its fundamental parallelogram at the points i K 1 ,3iK' (Theorem 4.8). The associated '$-function has periods 2iK/X and 2iK1/X. (Definitions 4.3); so w' = w3 = Thus

T.

lim ( z - i K 1 ) d n ( z )= lim ( z - iIK1)-

z+iK'

z -+%Kt

= X lim

az(9 03(f)

02 ( 2 )

( z - w3)-

z-+w3

0 3(2)

= = (el - e 3 ) i lim ( z - ws) Z+WQ

8. Elliptic Functions by Definitions 4.3 and 3.3. But, by Theorem 3.8(c)

and so we get that the residue of dn z at ilK1 is (el - e3)3(e3 - e2)i lim

(z -w3)(Y(z +w3) - e3)3

z-iW3

:

,

=i

(ez - e3) - (el - e3)5

by Theorem 3.8(b) and the fact that

Since the residue at iR1 is i, by Theorem 1.6 the residue at 3iR1 is -i. So dn2z is an even elliptic function with periods 21K and 2ilK1 (see proof of Theorem 4.8) and whose only singularity in its fundamental parallelogram is a double pole at iR1. Furthermore, the residue at iR1 is 0 (dn2z is an even function), and, in fact

where q5 is analytic in a neighborhood of iR1 (since i2 = -1). Thus, by Cauchy's Theorem, the integral of dn2t around any simple closed contour not passing through a pole is 0, and so E(z) is single-valued, and clearly analytic everywhere except at the poles of dn2t. At these poles, which are the points e i R ( mod 2R, 2iK1), by 4.13 &(I) has simple poles with residue 1. (c) Since at each pole of E(z) the residue is 1, no sum of the residues can be zero, and hence E(z) cannot be elliptic (Theorem 1.6).

Theorem 4.13. For all z and w

+

+

&(a w) = &(z) &(w)- k2snz snw sn(z

+ w) .

Proof. Observe that by Theorem 4.8, the function E(z) satisfies

as well as

by formula (4.8) in the proof of Theorem 4.8.

8.4. Jacobi's Elliptic Functions

347

Hence, for a fixed but arbitrary value of w which is not congruent to any of 0, fiR1,fK, f(K iK1) mod (2R, 2iR1), the function 7 ( z ) = &(z w) - &(z) is elliptic and has periods 2R and 2iR1. The poles of this function are r iK1 and -w iR1( mod 2K, 2iR1) and are simple. Clearly T(0) = &(w), and by Theorem 4.13(a), -&(-w) = E(w). Hence, the function E(z w) - E(z) - E(w) is elliptic and has (simple zeros G 0 and -w( mod 2R, 2iR1) and simple poles iR1 and -w iR1( mod 2R, 2iK'). So, though, does sn(z w)snz. Hence, by a now familiar argument,

+

+

+

+ +

E(z

+

+ w) - E(z) - E(w) = A snz sn(z + w)

where A does not depend on z. Differentiating with respect to z, we get dn2(z w) - dn2z = A(cnz dnz sn(z w) snz cn(z w) dn(z w)). Taking z = 0 gives dn2w - 1 = A snw

+ +

+

+

+

and so, A = -k2snw and the Theorem follows. As k + 0, &(z) + z, but as k + l,E(z) + tanh z, and the addition formula for &(z) becomes, after manipulation, the addition formula for tanh z (since snz + tanh z as Ic + 1).

Note 4.6. Since &(a) is not elliptic, by Theorem 2.11, the addition theorem of Theorem 4.13 is not an algebraic addition theorem. Also the function

clearly has the same addition theorem viz. Z(z

+ w) = Z(z) + Z(w) - k2snz snw sn(z + w)

and as is easily seen, Z(z) is simply periodic with period 2R. Since Z(z), however, does not have an algebraic addition theorem, it is not a rational function of epZ for some p. If we look at the special case of Theorem 4.13 when z and w are real, then writing am(z) = 4, am(w) = $, am(z w) = x (cf. Theorem 4.11) for an ellipse of eccentricity Ic, Theorem 4.13 becomes

+

E(k, X) = E(k, 4)

+ E(k, $J) - k2 sin +sin

$J

sin X, where

In terms of Problem 1 of the Introduction, this is a statement about arcs of an ellipse. In fact, from Theorem 4.9, cn(z w) snz snw dn(z w) = c n r cnw-k2sn2rsn2wcnz cnw - cnzcnw, and so, 1-k2sn2z s n Z w

+

+

+

8. Elliptic Functions

348

which gives a relationship from which the third of the angles #J,$,x can be determined when the other two and the ellipse's eccentricity are given, so that the relationship between elliptical arcs expressed by (4.14) holds. In particular, if we take x = 7r/2 in (4.15), we get 1 tan #J tan $ = --- - a l b .

-

Thus we have

Fagnano's Theorem: Suppose we are given an ellipse centered at 0 with semirespectively, of lengths a and b measured along the x and y axes axes and respectively, (0 < b 5 a). Suppose #J is an angle (measured clockwise from the y-axis as in Problem 1 of the Introduction), and $ is the angle measured similarly determined by tan $ = a l b cot 4. Then if P is the point on the ellipse corresponding to the angle #J and Q, the point corresponding to the angle 11,, we have from (4.14) The length of arc B P + the length of arc BQ

+

= the length of arcBA ak2 sin sin 11, 1 = - (the perimeter of the ellipse) ak2 sin #J sin 11, . 4 #J

+

A very special case is when $ = #J and P = Q = F, say, this happens when tan2 #J = alb, and so sin2 #J = &,and it follows since k2 = 1 - $, that length of arc BF- length of arc A F = a - b. As mentioned earlier, it is with results of this sort, that the subject of elliptic functions began.

Definition 4.5. Analogous to the definitions of R, R1,IE, we define IE' = IE1(k) = ~ : / ~ ( 1- kl2 sin2 8 ) i d 8 , w h e n kt2 = 1 - k2. The use of ' i n the definitions of K' and IE' is traditional, and should not be confused with derivatives. We have the following remarkable relation between complete elliptic integrals, discovered by Legendre.

Theorem 4.14. KIE'

+ IER' = RR' + n/2.

Proof. Let c = k2,c' = kI2 = 1 - k2 = 1 - C, 0 < c < 1, 0 < C' < 1, and consider R, IE, R', IE' as functions of the variables c, c'. Differentiation under the integral sign is justified by "Leibniz' rule" and we have,

8.4. Jacobi's Elliptic Functions Similarly, we have

Hence, dK IE 2c- = -dc 1-'C

~+-/o 1 1-c

"/21-~-(1-~sin28)2 (1-csin28)312 do

This last integral, in fact, is 0. Perhaps this is most easily seen by noting that

Hence, we get also

= -1, and interchanging c and c' interchanges Since c' = 1 - c and so K', IE and El,we also have

and -dK' = - - K' dc 2c'

IE' - K' 2c1(l - c') 2(1 - C)

from (4.16) and (4.17) respectively. Hence

IE' 2c(l - C) '

IK and

8. Elliptic Functions and similarly

d -(E dc

1

- K t ) = -IE' . 2c

Hence

d dc

-( (IE - K ) (El - K t ) ) = - IEK'

Hence $(-KE'

KE'

+ IEK'

- KK'

(IE - K ) - (IE' - K1)E dlE = IE12c 2(1 - c) dc

+ E-dlE1 . dc

+ K R ' ) = 0 and so = constant

(4.20)

for all values of k2,0 < k2 < 1. To find the constant, note that since c = k2, limc,o K = limc,o IE = ~ / 2 and , limc,o K' = limc,l K = oo by Theorem 4 . 6 ( f ) and Definitions 4.3 and 4.5, while

But, by definition, lim

c-io

IE-K - l i m i / c c-to C

~ / ( 21 - c s i n 2 8 ) $ 0

1 1 ( 1 - c sin2 8 )

and

l

lim c~ = lii

c-io

*I2

C 1

( 1 - ( 1 - c ) sin2 8 ) 5

Hence

IE-K = lim -cK1 c-io c which with (4.20),proves the theorem. lim KE'

c+O

+ IEK' - KK'

+ KE' = 7r/2 ,

Note 4.7. Theorem 4.14 is Legendre's original relation. The connection with Theorem 3.2(b) which is usually called after Legendre, may be seen as follows. From Definitions 4.4 and 4.3, we may write E ( z ) in terms of the associated P ( z ) and thus get

8.4. Jacobi's Elliptic Functions Since X = (e2 - e3)B (Definitions 4.3), we get from Theorem 3.8(c) that

5

(see the proof of Theorem 4.8). Thus, taking z = R, where wl = and WQ = we get since <(wl wg) = C(w2) = 772 = ql 773, that lE = . Similarly,

+

+

(z) '

+X

usingkl=(l-k2)B = = instead of k interchanges el and e3 and replaces X by -iX (Theorem 3.8(b)); thus we get similarly

Hence,

which gives Theorem 3.2(b). Actually, the above argument is an example of what is known as Jacobi's imaginary transformation (see Theorem 5.10 and Example 5.2). Theorem 4.14 could have been proved in this way, but a direct "calculus" proof seemed worth presenting. Since Icl < 1, (1 - csin2 0 ) - i can be expanded by the binomial theorem in a series which is uniformly convergent with respect to 8, and so can be integrated termwise. Letting c = k2, we get in the usual notation for the hypergeometric function

for lkl < 1, and the right hand side of this equation provides the analytic continuation of R as a function of k2 into the complex plane with the real axis from 1 to oo deleted (note this is in the "k2-plane"). Similarly, we have IE = lE(k2) = n/2F(-i, 1, k2).

3;

Note 4.8. There are also elliptic integrals of the third kind, going back again to Legendre. We shall not discuss these at all here, except to mention their origin. The general integral J F ( t , u), where F is a rational function of t and u and u2 a cubic or quartic polynomial in t without repeated factors can be reduced to evaluating

8. Elliptic Functions

352

one of three kinds of integrals which in "Legendre standard form" are

We have discussed the first two of these. Note that the substitution z = snu reduces ) ~ = &(u-J dn2udu) = $(u-E(u, k)). the first to J du and the second to J ( s n ~ du The same substitution in the third produces

If l = -1 or -k2, this integral can be expressed in terms of integrals of the second kind; for other values of e, it is usual to choose a parameter a such that l = -k2 sin2 a; this leads to Jacobi's fundamental integral of the third kind

For further discussion of this integral, we refer the reader to any text on elliptic functions and only mention here as as example of the innumerable identities which abound in the subject, that 96 (seemingly) different expressions for the addition formula of ll(u, a ) have been given.

Note 4.9. A comprehensive summary of results on elliptic integrals of all three kinds can be found in Erdelyi, et al., Higher Transcendental Functions, Vol. 2 [66] For a comprehensive treatment of the Jacobian Elliptic Functions, see Neville, Jacobian Elliptic Functions [172]. This book introduces an efficient, but somewhat idiosyncratic, notation. For a more elementary introduction to this material, see the posthumously published Elliptic Functions, A Primer, by Neville [173].

8.5

Theta Functions

At the end of his Fundamental Nova[123], Jacobi introduced two functions O and H; however, in his later lectures, he adopted a different notation taking four functions as basic objects of study (although, as will be apparent, one would do technically, the four are convenient) and deriving elliptic function theory therefrom. These "Jacobian Theta-functions" introduced below not only have certain theoretical advantages, they also provide a method for computation of elliptic functions by rapidly convergent series; not an unimportant feature given all the "applications"

8.5. Theta Functions

353

of elliptic functions which we cannot enter into (the most classical "application" of elliptic functions is to the mathematical theory of the pendulum, but there are a tremendous variety of others; most books on elliptic functions treat some of these; examples of books directed at applications are Oberhettinger and Magnus, Anwendung der Elliptischen Funlctionen in Physilc and Technik[l78],and Greenhill, The Applications of Elliptic functions [88]. The reader should be warned that especially with regard to theta- functions, various authors use slightly different notations and in referring to other books especial care should be taken to establish the author's definitions and avoid confusion. The connection with Jacobi's elliptic functions will become apparent. Although as indicated, the theory of elliptic functions can be developed ab initio from Jacobi's Theta-functions, we prefer to show how they arise naturally from Weierstrass' a-functions, and so return to Theorems 3.7(f) and 3.4(d) with a view to simplifying those formulas. We recall these here

Definition 5.1. ao(z) = a(z), 4, (z) =

3 L zz

(z)e 2

~

1

, a = 1,2,3,4

+

Theorem 5.1. (a) 4, (z 2w1) = f4, (z), where the - sign holds if a is 1 or 2 and the + sign if a is 3 or 4. (b) 4, (2 + 2 ~ 3 = ) &4m(z)e W 1 or 3 and the + sign if a is 1 or 4.

*(z+w3)

where now the - sign holds if a is 2

Proof. ( a )

where the - sign holds if a = 1 or 2 and the (b) Similarly,

+ sign if a = 3 or 4.

+

where the - sign holds if a is 1 or 4 and the sign if a is 2 or 3. Taking sight of Legendre's relations (Theorem 3.2 (b)), however, this becomes

where the - sign holds if a is 2 or 3 and the

+ sign if a is 1 or 4.

354

8. Elliptic Functions

Matters become even more simplified if we "correct" for the varying sign occurring in Theorem 5.1.

Definition 5.2. fl ( z ) =

(2wlz) .

f 2 ( z ) = e"izq52(2wlz) . f 3 ( z ) = #3(2wlz) . f4 ( z ) = 4 4 ( 2 ~ 1. ~ ) Definition 5.3. T def = w3/wl = ratio of periods of the !J?-function associated with the o- functions. W e recall here, because it will be important, that the labelling of the periods 2w1, 2w3 is always chosen so that Im T > 0 . Theorem 5.2. (a) f,(z (b)

+ 1) = f,(z).

Proof. Immediate from Theorem 5.1. By Theorem 5.2(a), the f a are simply-periodic functions, and so have a Fourierseries expansion which we now find. For convenience we make

Definition 5.4. q

gfexiT. Note that since Im T > 0 , lql < 1.

Theorem 5.3.

f3 ( z ) = CJ

qv2ezvxiz

8.5. Theta Functions

355

where the constants ci are determined by the fact that f,(O) = 1, a = 2,3,4, f{(O) = 1. Proof. We write for the Fourier expansions:

where the A,,, are to be determined. We have, on the one hand,

and on the other, by Theorem 5.2,

Hence Av+1,1 = -q2"Av,1 for all integers v. Writing 2v as (v

+ 1/2)2 - (v - 1/2)2, we get

for all integers v and thus that (-l)vq-(v-1/2)2~v,1= cl, a constant. The other expansions are obtained similarly. To evaluate the constants, put z = 0 in the expressions for the fa. For example, for

f2

we get f2(0) = o2(O) = ol(0) = 1, and so c2 =

(cE-,

-1

q(v-1/2)2)-1. -1

Similarly,c~= E,-,qv2) , c 4 = ( ~ ~ - ~ ( - l ) ~. Fqo r~c l ,~o ()0 ) = O , and so the argument has to be slightly modified, however,

(

and so, since fl(0) = 0, f i b ) = 2wlob(O) = 2wl f{(0) = lim z-+o

Z

CEO=_,

,

by Theorem 3.4(b). Since f{(z) = 2ricl v(-1)vq(v-1/2)e2vniz,we thus get v ( . - l / ~ ) ~ -l ) Cl = $ 4-1) s Incidentally, since Iql < 1, it is easy to see that all the series in question converge (rapidly) in C and uniformly in any compact subset of C.

(c5"=-,

356

8. Elliptic Functions

Definition 5.5. Jacobi's Theta-Functions O j ( z ) = 1,2,3,4 are the entire functions (of z ) analytic for Imr > 0, defined by

where q = errZ7,Im r > 0 . These theta-functions are functions of two complex variables, z and r where r is restricted to the upper half-plane {T : Im T > 0 ) ; however, when the dependence o n r is not material t o the argument, we will simply write B j ( z ) . Note 5.1. Tracing backwards the steps leading to these functions, we find (with w = WI, as usual)

03(z) . ( z ) = 2 w e " i ' u ; f 2 ( z ) = errir e2(2) ,. f3(z) = 0; ( 0 ) 02 ( 0 ) 03(0)) '

And, from the definition of the u j ( z )

As an example of the rapidity of convergence, consider that for

T

= i we have

8.5. Theta Functions where E is roughly of the order of 2e-4"

< 7.

Theorem 5.4. (a) 01 is an odd function; 82, &,04 are even functions. (b) The foElowing table gives values of the functions on the left at the arguments in the columns:

(c) The functions Bj(z), j = 1,2,3,4 have simple zeros at the points congruent to 0,1/2,1/2 + 7/2,7/2 respectively, with respect to the lattice points m + n7, m, n integers. Proof. (a) and the first two columns are immediate from the defining expansions. The third and fourth columns follow similarly or, alternatively, from the first two. To prove (c), 81 has a simple zero a t 0 by the second form of the defining expansion, and so the result follows from the third and fourth columns of (b). The result for O2 now follows from the first column of (b); then for 193 from the second column and now for O4 from the first and third columns. Theorem 5.5. The functions B j ( z l ~ )j , = 1,2,3,4 satisfy the partial differential equation d20 dB - = 47ri- . 8.22

a7

Proof. The defining series for the Bj are uniformly convergent with respect t o T in any compact subset of {T : I m T > 01, and the series resulting from termwise differentiation again converge uniformly in any such compact set. Remarks of the same sort apply with respect to convergence in compact subsets of the z- plane. Since the general term of (the first form of) each of the defining series has the form (aside from a sign factor): $aZeanir = ea~i(z+ar/4)where a is 2v or 2v - 1 ; and

while

the theorem follows from the defining series.

8. Elliptic Functions

358

Note 5.2. The functions ej(zlr) are special cases of the more general function

which was introduced by Hermite; here g, h, z are unrestricted complex variables, and T is restricted to I m r > 0. Og,h(zIr) satisfies the partial differential equation of Theorem 5.5, and the functional relations Og,h+2(zI7-)= Og,h(zI~) @g+2,h(~IT) = e-"ih@g,h(~l~) @g+a,h+dz~r) = e~ani(r+:)-* for arbitrary complex a and b. The Jacobi functions are given respectively by

Introducing still a further parameter (essentially replacing the 4 r i in the partial differential equation by 4rim), leads to what were classically called "Thetafunctions of mth order with arbitrary characteristic". One consequence of the differential equation of Theorem 5.5 is the important relationship given in

Theorem 5.6. ei(01r) = r82(01~)e3(Ol~)e4(Olr), where with respect to z.

'

denotes differentiation

Proof. Tracing back the Sfunctions in terms of Weierstrass' 0-functions and thereby to a p-function with periods w = wl, w' = w3 , (cf. Note 5.1) we find

Hence, since the ej are entire functions, and O1 is an odd function, while are even functions, we have

-

6'2,

e3,e4

+ az4 + higher powers . . .

8.5. Theta Functions where a! is a constant. Hence 1 'J3(2w.~)=~+ej+-

+ /?z2 + higher powers

...

where /? is a constant. But the constant term of the Laurent expansion of Q(z) is 0 by Theorem 2.l(d), and so

Since el

+ ez + e3 = 0, we thus get

On the other hand, by Theorem 5.5,

Also, differentiating with respect to z in Theorem 5.5, 0,(3) ( 0 ~ ~ ) = 4dad2 z di r- ( ~ ~ ( z ~ z=o~ =4ni-e;(zl~) ) ) ~ ar

a

.

Thus we obtain, on substituting (5.2) and (5.3) in (5.1),

and integrating with respect to

T

and exponentiating,

where C is a constant. To find the constant, consider the defining series for the theta-functions. These give, on substitution in (5.4),

where q = eniT,and I m r > 0. Multiplying both sides by q-1/4 (the reciprocal of the first term of C z l q(v-1/2)2)and letting T -+ ico (so q -+ 0), we get C = a, which proves the theorem.

8. Elliptic Functions

360

The Bj, being entire functions of z, have Weierstrass product expansions (all these were found by Jacobi before Weierstrass observed the general result). One use of Theorem 5.6 is in establishing expansions of this sort. We have

Theorem 5.7. With q = eTiT, 00

00

el (ZIT) = 2q1I4sin nz fl( I - q2v)II(I - 2q2vcos 2nz + q4")

n 00

e2

= 2q114cos nz

for all z and all T with I m T

00

fl(1 + 2q2" cos 2 n + qdv)

(1 - q2v)

> 0.

Proof. The zeros of Ol(z1~)are as observed in Theorem 5.4(c) at the points m n r , m, n integers. Also, e2"i(m+nT)= q2n. Since ELl qv converges absolutely,

+

are absolutely and uniformly convergent in any compact subset of the z-plane. Thus, the function

n . .

F ( z ) = sin xz

..

(1 - q2ve2"")

fl(1

- q2ve-2"iz)

00

= sin nz

JJ (1 - 2q2" cos 2nz + q4")

is an entire function (the factor sinnz is introduced to account for the zeros of at all integers). Furthermore, clearly

while 00

F(z

fi

+ T) = sin(n(z + T)) JJ(1- q2v+2e2niz) (1 - q2v-2e-2"iz) v=l v=l - sin n(z + T) 1 - e-2niz sin nz

-

1 - e2ni(z+~) F(z)

-e-lri(z+~) 2i sin nz

el

8.5. Theta Functions

36 1

is a doubly periodic function Comparing with Theorem 5.4(b) we see that with periods 1 and T. Furthermore, since F ( z ) and fIl(z1~)are both entire functions of z and have exactly the same zeros, it follows once more from Theorem 1.2 that

el (ZIT) = AF(z) = A sin nz

n 00

(1 - 2q2" cos 2nz + q4?

(5.6)

v=l where A depends only on T. To find A, note that by Theorem 5.4(b), 00

B2(zlr) = B1(z+1/21r) = AF(z+1/2) = Acosaz ~ ( 1 + 2 q 2 " c o s 2 a z + q 4 " ) ;(5.7) and also

- IAe-"iT/4 -

2

n 00

(1

+ 2q2v-1 cos 2az + q4v-2) ;

v=l

and finally,

We now use Theorem 5.6. Putting z = 0 in (5.7), (5.8), (5.9), gives

Dividing both sides of (5.6) by z and letting z

+ 0, gives

Comparing (5.10) and (5.11) with Theorem 5.6, we get

8. Elliptic Functions

362 However, clearly

and so from (5.12) we get

and thus

Finally, to determine the sign in (5.13), substitute it in (5.8) and note that this 00 gives as 7 + ioo (and so q + 0), lim,,iw 83(0]r) = limq+o f n v Z l ( l - q2"); but by the definition of 03, this limit is 1, and so the sign always holds. Substitution of (5.13) in (5.6)-(5.9) now gives the theorem.

+

Note 5.3. It was mentioned in the introduction that Jacobi in his first treatment used as fundamental a function he denoted O(z). Starting from Legendre's elliptic integral of the second kind E(z), and writing Z(z) = E(z) - t z , it is easy to see that Z is simply periodic with period 2K (cf. Note 4.5). We may then define

where C is a constant = O(0) to be determined (compare the way Weierstrass' u-function is constructed from his ?-function), and clearly O also is periodic with period 2K. It turns out that O is an entire function with simple zeros at the poles of Z and ( q r = wl/w, Irn r > O), that if we choose C = O(0) = ~ ~ - , ( - l ) ~ q ~=~eTi7, then

in the notation we have used. In particular, then (since = w , = w'), @(z 2iK1) = O4(z/2K + = 04(z/2K + = 6J4(z/2K + r ) = -eK"("lK+7)84(z/2K) = e-"i(Zl"+T)~(z),(by Theorem 5.4(b)). Jacobi also defined originally a function H ("Eta") by

6)

+

g)

8.5. Theta Functions Clearly, by an argument analogous to the above,

(by Theorem 5.4(b) again). It is now easy to transfer any results about theta-functions to and from Jacobi's original notation.

Theorem 5.8.

(where k and kt are as defined in Definitions 4.3).

where

& and f i

are defined as

and

where q = eTir, T = wf/w, and A is defined as in Definitions 2.3. Proof. (a) By tracing the definitions of the theta- functions back through Weierstrass' 0-functions t o a !$?-function with periods (2w, 2wt) (cf. Note 5.1), we get ( w = w l , w l = w3),

8. Elliptic Functions Taking z = 112 gives for j

# 1,

and this in turn, by Theorem 5.4(b) gives

and the results for these two expressions now follow from Theorem 5.6. Taking z = $ = 5 and j = 2 in (5.14) gives by Theorem 3.8(b),

and by Theorem 5.4(b), this last expression gives

and the result again follows from Theorem 5.6. (b) is immediate from (a) and Definitions 4.3. (c) Recalling that the $?-function associated with Jacobi's elliptic functions for a given value of k2 has w = R/X, and w' = iK/X, where X = (el - e 3 ) i (cf. Theorem 4.7 and Definitions 4.3) we have from (5.14), (a), and Definitions 4.3

The formulas for cn(2Rz) and dn(2Kz) follow similarly. (d) By definition

Hence, the first expression for A follows from (a); the second is then a conse~ Theorem 5.7, quence of Theorem 5.6. To derive the third, in the formula for 1 9 in divide both sides by z and let z -+ 0, thus obtaining

and so the third expression for A follows from the second one.

8.5. Theta Functions

365

Note 5.4. Jacobi's theta functions also satisfy an "addition theorem" (though, of course not an algebraic one) which is a special case of an extensive set of formulae discovered by Jacobi by pure,ly algebraic means. For these, see Whittaker and Watson [247], A Course in Modern Analysis, (most recently reprinted 1978), p.467469 and 487-488. For an "elementary" proof (by Cauchy) of Theorem 5.7 (for 93, whence the other expressions can be derived) see Hardy and Wright [97], An Introduction to the Theory of Numbers, Section 19.8. Sections 19.9-19.10 indicate applications to "partition problems" in number theory. From Theorem 5.7, we can also obtain rapidly convergent expressions for Weierstrass' ((z), and the coefficients czn of the Laurent expansion of P ( z ) around 0; in particular, such expansions can be given for the invariants g2 and g3 of q. Theorem 5.9. With q = eniT,T = $, I m

+

T

> 0 as usual,

+

sin(2pni), for a11 i such that (a) ( ( 2 ~ 2 )= 2 ~ 1 2 & cot nz % Crx1 IIm zl < I m T. c2,z2", in a neighborhood of 0, then (b) If p(zlw, w') = $ CzO=,

+

c2n = (2n + 1)

C'

1

w a lattice point

where B2n+2 is a Bernoulli number in an even subscript notation (see Appendix). With the invariants g2,g3 defined as usual b y (c)

we have

Proof. (a) Tracing backward the steps leading to the definitions of the thetafunctions, we find (cf. Note 5.1)

8. Elliptic Functions

hence, since [(z) =

& Log a(z), we have

By Theorem 5.7, we have 00

q2vsin 2nz = ncotnz + 4 n ~ V = I 1 2qZv cos 2nz el (2)

+q 4 ~

We may write the series term in (5.16) as 00

g2" sin 2nz

v=l

Now, if IImzl

00

= -2in

C 1- q~v(e~7riz q2"(eZaiz +- e-2xiz) e-27riz

+ q4"

v=l

< I m r , we have

and

<

Furthermore, for le2"izq21 X: < 1, for all v sufficiently large, say v I IClq12v-2 < 112, and so, for le2nizq215 X: < 1, Iq2ve27riz<

>

vo,

Hence, for le2"izq21 5 K: < 1, (i.e. by (5.18) in every closed subregion of {z : 2" 2 n i r IIm zl < I m r,} )=the " series = r C converges absolutely and uniformly. 2" - 2 n i

Similarly, from (5.19), the series Eml &&& converges absolutely and uniformly in every closed subregion of {z : IIm zl < I m T } . Thus in (5.17) (for IImzl < I m r ) , we can break the last expression into the difference of two series, expand each as a geometric progression, and interchange

8.5. Theta Functions the summation. In this way we get M

q2" sin 2nz

v=l

Putting (5.20) into (5.16) and (5.16) into (5.15) gives the result. (b), (c), and (d). Since P ( z ) = -C1(z), we have from (a) for IIm zl 2w'P(2wz) = -2111 Since, for 0

n2 + -cse2nz 2w

4n2

< I m 7,

Ccos 2pnz . 1 - q2p

--

O0

pq2p

p=1

< izl < 1,

where the B2"+2 are Bernoulli numbers in a even subscript notation (see Appendix), we have, expanding the cosine in the third term of (5.21) as well, and interchanging the order of summation,

in a deleted neighborhood of z = 0. In other words, in some deleted neighborhood of 0,

Comparing coefficients with the known Laurent expansion of V(z) around 0 (Theorem 2.l(d)), we get (b), while (d) follows from the fact that the constant term Finally, B4 = $,B6 = and by in this Laurent expansion is 0, and B2 = Theorem 2.6, g2 = 20 c2,gs = 28 e4; whence (c) follows from (b).

i.

&,

8. Elliptic Functions

368

Note 5.5. By an argument similar to that used to prove Theorem 5.9(b), one may find the ex~ansion

%

2

valid for all z with IImzl < f ~ m r Similar . expansions for and now follow from Theorem 5.4(b). These expansions are the Fourier sine expansions of the odd functions

On the other hand, they may also be viewed as Lambert series (see Note 6.1.1), and so, for example, we can write (for IIm zl < I m T)

x 00

O'(z) -

cot r r = 4a

q2" sin2vn.z

v=l 1 - 92"

el (z)

where the inner sum is over all positive divisors of k. Similarly, we have from Theorem 5.9(c), the alternative expression

There are many more fascinating expressions and identities; however, we close this section with the following important result:

Theorem 5.10.

e2 (Z7 1

--) = e=FT;e4(21T)e* 1

7

,

(ii) (iii)

Proof. The theorem emerges from an attempt to study the defining series for Oj (z) by means of the Poisson Summation Formula (see Appendix). By definition, we

8.5. Theta Functions have

is real and let w = Now consider only z such that the Poisson Summation Formula and putting T = i a , Re a

- 112. Then using

> 0,

since clearly f (v) = e-"""' satisfies the conditions in the discussion of Poisson's Theorem in the Appendix. But

and on using Cauchy's Theorem, since, as is easily verified,

lim T-tm

/

fT-"C

e-nay2dy = 0, we get

fT

since T = ia,Re a

> 0.

8. Elliptic Functions Substituting (5.24) in (5.23) gives (since r = i a and w =

- 112)

where in the penultimate line, k has been replaced by -k, and so (i) is true for all a such that is real. However, (i) now follows for all r by analytic continuation, since both sides represent entire functions. (ii) now follows from (i) and Theorem 5.4(b), since we have:

Similarly, (iii) may be proved in the same fashion as (i), and then (iv) follows from (iii) analogously.

Example 5.1. Taking z = 0 in Theorem 5.10 (iii) gives

or, by definition,

or, writing r = i a , Re a 00

> 0,

e - " ~ "= ~ a-4

a remarkable identity.

03 -nu2

e"-,

Rea

>0,

8.5. Theta Functions For example, clearly we can also write (5.26) as

If R e a is small and positive, the left side of the identity (5.27) converges extremely slowly; however, the right side extremely rapidly. Conversely, for Re a large and positive, the left side of (5.27) converges rapidly and the right side slowly.

Example 5.2. By Theorem 5.8(c), sn(2Kz, k) = zK A - i W '

&Hence, by Theorem 5.10,

& w ,where T = $ =

K '

-

i

(

1

- ) - isn(z, k') 1

0.2(&1 - T ) by Theorem 5.8(c) again, since interchanging iK' with -1 = a. 7 = K

T

K'

-

cn(z, k') '

IK and K' interchanges k and k' and

&

w.

Similarly, one can prove that en(iz, k) = and dn(iz, k) = These formulas are sometimes known as "Jacobi's imaginary transformation".

Example 5.3. The transformation r + which takes the upper half-plane onto itself is interesting when applied to Weierstrass Q-functions as well. Clearly, if 2w and 2w' are a set of fundamental periods for a Weierstrass Q-function, then so are 2w' and -2w a pair of fundamental periods for the same function, (cf. Note 1.1). Thus, if 1 Q(zlw, w') = c2,(w, w')z2"

,+ C n=l

is the Laurent expansion of !j3(z1w,w1)around 0, then by the uniqueness of Laurent expansions

Since

= d , -1 = -,: , Theorem 5.9 applied to (5.28) gives W T

8. Elliptic Functions

372

Equation (5.29) represents another remarkable relationship which holds for I m r > 0 . Let us examine some particular instances.. If T = i and n is even, we get from (5.29),

" p2n+l CeZnp-1 p=l Again, if

-

T = i/2,

B z ~ + z , for all even n . 4(n+1) we get

The left side of (5.31) can be transformed as follows:

p'even

Substituting this into (5.31) and rearranging terms gives

'+ (-4)" B ~ n + 2+ (-1)"+'

-4

( n+ 1 )

p2n+l p2n+l C _+i + C 5. p=l a

00

p= 1 p even

If in (5.32) we again take n even, and use (5.30), we get

p2n+l

xs-

p=l

-

4n - 1. 2(n

+ 1 )B2n+2 for all even n 2 2 ,

p odd

The corresponding formula for n odd, similarly turns out to be 03 p2n+l 4n+l - 1 BZn+2 for a11 odd n 2 1 . err (-1)p 4 ( n 1) p=l

C

+

+

8.6. Modular functions As examples of (5.30) and (5.33) and (5.34), we have

"

31 " 504 ' p= 1

1

p=l

C

1 --~ 1 ~ 264 '

p9 e

2

p odd

b odd and similar curious results.

Note 5.6. There is an entirely different proof of Theorem 5.10 which is less conceptual and which expresses O1(zlr) in terms of Weierstrass' a-function and an appropriate eighth root of the discriminant A (by Theorem 5.8(d) and the fact e2vlw2(cf. Note 5.1).) after which comparing (ZIT) and that o(2w.z) = 2w e l ( ~ I + ) , using Legendre's relations (Theorem 3.2(b)), and finally evaluating the eighth root of unity involved, gives the result. For a concise introduction to further various aspects of the Jacobi theta-functions, the reader should consult Richard Bellman's monograph, A Brief Introduction to Theta Functions [19], where connections with number theory are also discussed. Such connections can also be found in the chapter on Elliptic Functions in Hurwitz and Courant ([118]).

#

8.6

Modular functions

As has become clear in Section 5, what really matters in distinguishing results about elliptic functions, is not so much their periods 2w, 2w' as the ratio of the periods r = wl/w, where we assume the periods are always labelled so that I m r > 0. Indeed, in many of the formulas of Section 5, the variable r appears intrinsically, whereas w appears, in addition, in an unimportant way in the form of a power wk as a multiplier of the expression in question. In this section, we will be able to take advantage of this fact to normalize our functions so that w = $ and hence the two periods of the elliptic functions in question will be 1 and T where I m T > 0. As an example, consider the expressions for 92 (w,w') and g3 (w,w') in Theorem 5.9(c). Since A(w, w') = 923 - 27932, we have

8. Elliptic Functions

where q = eTiT,I- = w t / w , I m

T

> 0. Hence

depends only on T . This leads t o Definition 6.1. For I m T

> 0,

Theorem 6.1. (a) J ( T ) - 1 = 2 7 g 2 ( ~.) (b) J ( T ) is analytic in the upper half plane. (c) If a , b, c, d are integers such that ad - bc = 1, then J (d) J ( r + 1 ) = J ( T ). (e) J ( i ) = 1, J ( e F ) = O (e;(ol~)+e:(ol~)+e:(o~~))~ (f) J(7)= 54(0; ( 0 1 ~ ) ) ~

(

1

gi(~)

= J(T).

Proof. ( a ) is immediate since by definition A(T)= - 27 9$(r). A(T) is never 0 , g2(7) and 93(7) are analytic for I m r > 0, since the expan(b) sions o f Theorem 5.9(c) represent uniformly convergent series o f analytic functions, provided I m r > 0. (c) I f 2w and 2wt, wt/w = T , I m r > 0 are fundamental periods o f p ( z ) , then 2w* = d(2w) + c(2w); 2wt* = b(2w) + a(2w1)where a , b, c, d are integers such that ad - bc = 1, are an equivalent set o f fundamental periods for p ( z ) ( c f . Note 1.1). I f we set b(2w) + a(2w1) ar + b wl* -T* = -, then T* = w* d(2w) + c(2w1) cr + d ' and so

I, Thus I m T

T*

=I,

(-)a7 ++db CT

=

ad - bc ImT I ~ T = I C T dl2 I c r dl2

+

> 0 i f and only i f I m T* > 0 , and also,

+

'

8.6. Modular functions

375

since 2w* and 2w1* generate the same set of periods as 2w and 2w1, and hence, the values of g2, g3, A, and so J remain the same. (d) T a k e a = 1 , b = l , c = O , d = 1 in (c). , (m+ni), = - 1 4 0 ~ I 6. = -g3(i). (e) g3(i) = 140 m,n (im+n)

Hence g3(i) = 0, and so J ( i ) = 1 follows from (a) (since A is never 0).

So g 2 ( e y ) = 0, and ~ ( e q =) 0 now follows from the definition of J (since A is never 0). G, (f) By Theorem 5.8(a), squaring and combining the expressions for d & F G , dFFG (and using el e2 e3 = 0), we get

+ +

-7r2

e3 = -(e;(ol~) 12w2

+ e,4(ol~)).

By Theorem 5.8(d)

Hence, taking w = 112, since g2 = -4(ele2 (6.31, 47r4

9 2 ( ~= ) g(e:(~l~)

+ e2e3 + ele3), we get from (6.1)-

+ e:(olr) + e:(ol~) + ~ , ~ ( o ~ T ) ~ ~ ( o I T ) ~

+ e,4(ol~)e,"(ol~) - e,4(olr)e,4(olr)) . But by Theorem 5.8(b), since k2 + (kO2 = 1,

Hence, we also have

(6.5)

8. Elliptic Functions

376 Substituting (6.6) and (6.7) in (6.5) gives

The formula (f) for J(T) now follows from (6.5), (6.4) (with w = 112) and Definition 6.1.

Theorem 6.2. (a) A(T) = (2s)12 integers. (b) 1728 J(T) = e-2xiT 744 are integers.

+

Cr=l r(n)e2"inT, where the coefficients r(n) are + Cr=l a(n)e2"inT , where the coefficients a(n)

Proof. (a) By Theorem 5.9(c) with w = 112

where we have used the usual number-theoretic notations a k ( n ) = Cdln dk. (Compare Note 5.6; there should be no danger of confusing this number-theoretic notation with the Weierstrassian functions.) Similarly we have

For simplicity, we adopt the temporary notation A =

B = Cr=l05(n)e2"inT.Then

Crzl03(n)e2KinTand

Now A and B have integer coefficients; furthermore, 5A

+ 7B =

x 00

(5o3(n)

n=l

+ 705 (n))e2"'nT n= 1

(

5

+7

8.6. Modular functions But

x

5d3 + 7d5 I

-7d3

+ 7d5 = 7

x

d3(d2 - 1 ) I 0

(mod 12)

+

(one of d, d - 1, d 1 is divisible by 3 and if none of them is divisible by 4, then d is divisible by 2 and hence d2 by 4). So A(T)= (27r)12C:=l r(n)e2nin , where the coefficients r ( n ) are integers. (b) In the notation of the proof of (a) and using (a), we have

Now since u 3 ( l ) = u 5 ( l )= 1, it is clear from (6.9) that r ( 1 ) = d5 = 33, Similarly, since u3(2) = CdI2 d3 = 9 , and ~ ( 2=)

= 1.

Thus,

where the a(n)are integers. Substituting this expression in (6.10) gives

where the P(n) are integers, and the result (b) now follows on multiplying.

Definition 6.2. A function f is called a modular function if (i) f is meromorphic i n the upper half-plane {T : Im T > 0 ) ; (ii) For every T i n the upper half-plane and every set of integers a , b, c, d with ad - bc = 1.

8. Elliptic Functions

(iii) The Fourier expansion

is valid throughout the upper half-plane. Example 6.1. J and all rational functions of J are modular functions by Theorems 6.1 and 6.2. Definition 6.3. The subgroup of the group of all Mobius transformations under functional composition, with a , b, c, d integers and ad - bc = 1 is called the modular group. Note 6.1. Thus Definition 6.2 (ii) says that f is invariant under the modular group. Definition 6.2(iii) effectively says that as T -+ ico,f (7) does not grow too fast. In fact, it says that f has at worst a pole of order m at i m . The reason for the term "modular" will appear later (see Note 6.5). The function J ( T ) is called Klein's modular function or, sometimes, the absolute invariant (of the associated y-functions). The coefficients a(n) are somewhat mysterious, Peterson [I871 showed that e4fffi as n -+ co. They satisfy a large number of number-theoretic cona(n) gruences, see e.g. Atkin and O'Brien [12], and the bibliography there cited. Using a relationship between the a(n) and the partition function of number theory and a table of the latter, Zuckerman [255] computed very easily the first 24 of the a(n). For further computation of the a(n), formulas like Theorem 6.l(f) become useful. Using theta-function formulas, VanWijngaarden [239] computed a(n) up to n = 100. The number a(100) has 53 decimal digits. The a(n) have somewhat surprisingly been related to the MONSTER Fl of finite group theory (Bulletin London Math. Society (1979), 352-353 and 308-339); and this connection has been extensively studied. The MONSTER (also known as the Friendly Giant) is one of 26 "sporadic groups" which are simple, and do not seem to belong to any infinite family of simple groups. It was first constructed by .76 .112.1 3 ~17.19.23.29.31.41.47.59.71. . Robert Griess and has order 246~3~O.5' Robert Wilson (using a computer) found explicitly two 196882 x 196882 matrices that generate it. Griess has recently written a book dealing with some of the sporadic groups. The coefficients of the expansion of A(T), denoted r ( n ) above, are frequently , are consequently known as "Ramanujan's r-function" . Again, denoted by ~ ( n )and there are numerous congruences satisfied. For a summary of congruences discovered up until 1972, see section F35 of Reviews in the Theory of Numbers. As an example of the flavor of these, we may mention Ramanujan's results that for p a prime

-

r(p) r

+1

(mod 691),

8.6. Modular functions and more generally

r(n)z E d 1 '

(mod 691) .

For a survey connecting these with contemporary algebraic geometry, see JeanPierre Serre [220]. Recent deep and famous work of Deligne in this area, has established as a corollary, Ramanujan's conjecture that for p a prime, Ir(p)l 2pj2.

<

Note 6.2. If (2w, 2w1) and (2w*,2w1*) generate the same set of periods, then A(w, wl) = A(w*, wl*). Furthermore, by Theorem 5.9(c) (compare remarks at the outsetofthissection),if $ = r , I m r > 0, A(w,wl) = h A ( 1 , r ) = 1 f)= &A(r) by Definitions 6.1. Hence, taking w = l,wr = r, w* = cr+d, wl* = a r + b where ad - bc = 1 (cf. Note 1.1) we have

= 212(cr

+ d)12A(l,r) = (CT + d)12A

(i,i)

= (cr

+ d)12A(r)

Thus, A ( r ) is "not quite" a modular function. Functions which satisfy (i) and (iii) of Definition 6.2, but with (ii) replaced by

where Ie(a, b, c, d)l = 1 are called "modular forms". Thus.A(r) is a modular form of "weight 12". A ( r ) is closely related to the so-called Dedekind q-function, defined for I m r > 0 by M

In fact, it turns out that

A ( r ) = ( 2 ~ ) ' ~ ( q ( r .) ) ~ ~

The function q ( r ) satisfies "Dedekind's functional equation"

where a , b, c, d) are integers satisfying ad - bc = 1, 1e1 = 1, and €(a,b, c, d) is a fairly complicated function. We have no space to go into these and related fascinating and important matters further in this book. For an introduction, see Apostol, Modular Functions and Dirichlet Series in Number Theory [ll]. We turn our attention to the important mapping properties of J ( r ) ; first we need to know something more about the modular group. (As we proceed, the reader may wish to compare the arguments of Chapter 2, Section 2.)

8. Elliptic Functions

380

Theorem 6.3. Given any pair of complex numbers (w, w'), with wl/w not real, there is another pair, (w*,wl*) such that {mw + nw' : m , n integers) = {mw* + nwl* : m , n integers}, w*=wl+dw w'* = aw' + bw

where a d - bc = 1, and

Proof. Arrange the elements of

R = {mw + nw' : m, n integers ) in a sequence according to distance from the origin, say

where O < Iw115

IWZ~

5 ... andif lwjl= I W ~ + ~ I ,

then arg wj < arg wj+l.Let wk be the first element of this sequence which does not lie on the line determined by 0 and wl (i.e. is not a multiple of wl).Let w* = wl and wl* = wk.Then wl + wr,and wl - wk both come after wk in the sequence and so the inequalities (6.11) are fulfilled. Furthermore, the closed triangle with vertices 0, w*,wl* contains no member of R other than at the vertices, and so {mw* + nwl* : m , n , integers) = R. Hence there are integers a , b, c, d such that W*

= CW'

+ dw and wl* = awl + bw ;

(6.12)

and also there must be integers a , p , y , S such that

w = yw* + 6w1*and w' = aw* + pwl* .

(6.13)

Thus, from (6.12) and (6.13)

and

w ~ * . =(by

+ aa)w* + (b6 + ap)wl* .

Since wl*/w* is not real, (for if it were, wl/w would be real) we must consequently have

8.6. Modular functions

381

Hence we get, after some manipulations, (or by appealing to determinants), (ad - bc)(ab - By) = 1 and so (since a , b, c, d, a , ,B, y, b are integers)

Since we could repeat the argument with (w*,wl*) replaced by (w*,-wl*) if necessary, we can without loss of generality take ad - bc = 1. Definition 6.4. Two points T and T' of the upper half-plane { z : Im z > 0) will be called equivalent if there is a transformation in the modular group taking one into the other. Theorem 6.4. If I m and such that

T

> 0, then there is a point

r*,I m r *

> 0, equivalent to T,

The points T* satisfying (6.14) and (6.15) all lie in the closure of the region G bounded b y the arcs { z : Re z = $, Im z 2 { z : Re z = 1m z 2 $1 , 5 arg z 5 (see Diagram 8.4). { z : IzI = 1, No two points of G are equivalent.

7)

q),

-i,

Proof. In Theorem 6.3 we take w = i , w' = $, then there exist w* and wl* satisfying = 5 , where a, b, c, d are integers, and ad - be = 1. (6.11) such that Letting r* = 5 , (6.14) and (6.15) follow from Theorem 6.3. By (6.14), the closure of G is in the exterior of B ( 0 , l ) . Writing r* = a + i,B, a , /3 real, we see that if (6.15) is satisfied, then

>

and so 1f 2 a 0, whence I R ~ T * 5 I i , and conversely. Thus the region G is as described. Finally, suppose TI and 7 2 were both in G and equivalent. Then

and

since a d - bc = 1. Similarly

382

8. Elliptic Functions d r -b

(since 71 = - c : 2 + a ) . So, since I m 7 2

# 0, (72 E G), we get

3,

(since [Re 711 I 4,IRe 721 I 1 ~ 1 1 11,I7-21 1 1 for 71,72 E G). In (6.16), if c # 0, a and d are not both 0; furthermore, if:

(i) c # 0, a # 0, d # 0, we get 1 > ladc21 2 1 ; (ii) c # 0 , a = 0,d # 0, we get 1 > c21cdl

> 1;

(iii) c # 0 , a # 0, d = 0, we get similarly 1 > c21cal

> 1;

(iv) c = 0, we get 1 > (ad)2, and so ad = 0, whence ad - bc = 0. Since (i)-(iv) all lead to contradictions, it follows that alent.

71

and

72

are not equiv-

Theorem 6.5. Let G denote

(see Diagram 8.4). Then J takes every value exactly once in G, except that there is a triple zero at e v = i$ and a double 1-point at i.

3+

8.6. Modular functions

Diagram 8.4 Proof. Given a E @, we wish to count the zeros of J(T) - a, where to 6 . Observe first that, given a E @, if T = x iy and y c, then

+

T

is restricted

>

hence by Theorem 6.2(b), as c + co, IJ(r)l + co, and so, for sufficiently large c > 0, I J ( T ) ~> la/. So it is adequate to consider the closure of 6 truncated by a line I m T = T parallel to the real axis, where T is sufficiently large, in order to prove the theorem. We have two cases: Case 1: J ( T ) - a # 0 on the boundary of 6. We consider truncated (see Diagram 8.5).

Diagram 8.5 Writing N for the numbers of zeros in question, and C for the boundary described positively

~ J(T) ) (theorem 6.l(d)) the integrals along the straight line segments Since J ( T + = denoted L1 and L4 in Diagram 8.5 cancel.

8. Elliptic Functions

384

Also, as in Diagram 8.5, denoting by La, the circular arc {z : lzl = 1,; 5 argz and by L3, the remaining circular arc contributing to C, we see that the map T -+ -$ takes L2 into L3 with direction reversed. Since J ( T ) = J(-$) (Theorem 6.2(c) with a = 0 , b = l , c = -1,d = O), and so J'(T) = +J'(-$), we have

< 9)

Hence the integrals along L2 and L3 also cancel. Thus we are left with

The change of variable, z = e2ni(u+iT)= e-2"Te2niu , maps the line segment [-i, onto the circle CT centered at 0 with radius e-2nT described positively. Thus, using Theorem 6.2(b) we have

31

1

1 - + (a power series in z)dz = 1 ,

and so J(T) takes the value a exactly once inside 6. Case 2: J(T) - a = 0 on the boundary of 6. If a # 0, or a # 1, we describe the usual small semi-circular indentations in the boundary of 6 to avoid those points; making symmetric indentations in the symmetrically situated portion of Bdg, so that the points where J(T) - a = 0 are included in the Jordan interior of the contour the first time, and the symmetrically - a = 0 since J(T 1) = J(T) and J(-$) = J(T)) situated points (where also J(T) are in the Jordan exterior of the contour (see e.g. Diagram 8.6).

+

8.6. Modular functions L.

Diagram 8.6 Truncating again, it follows similarly to the previous case that if N denotes the number of zeros in "6 modified" = number of zeros in 6 , then

If a = 0 or a = 1, we already know that ~ ( e % )= 0 and J ( i ) = 1 (Theorem 6.1 (c)) and, in fact, from the definition of J and Theorem 6.l(a) (since A(T) # 0) it follows that the zero a t e2ni/3is a t least a triple zero and the zero of J ( T ) - 1 a t i is a t least a double zero. In the case a = 1, we again describe a small semi-circular detour around i , call it B , such that i is in the Jordan exterior of the resulting contour, and get as before (with N the number of zeros in 6 - {i})

If B' is the arc situated symmetrically to B so that B U B' is a circle oriented negatively and centered a t i, then

(since if T + - $, B + B' with the orientation reversed). By Theorem 6.l(a) the order of the zero for J(T) - 1 a t i is divisible by 2, thus writing 2v for this order

8. Elliptic Functions

386

we get, since the first integral in the expression for N is 1 as before, 1 N=1--(2~)=1-U. 2

>

Since N 0 and v 2 1, we get v = 1 and so a double 1- point in G at i. Similarly, using the fact that the order of the zero of J(T)a t e2?ri/3is divisible by 3, and making a circular arc indentation there and symmetrically a t eriI3 (each of which is of a circle), we get if 3v is the order of the zero at e2?ri/3,

and so again v = 1, and the only zero of J in G is a triple one a t e2"i/3.

Note 6.3. It is easy to see that a very similar proof will suffice to show that if F is a modular function, then F has the same number of zeros and poles in 6 ( J has a simple pole a t ioo), with appropriate modifications at e2Ki/3and i. Consequently, since if F ( T ) is modular, so is F(T) - a any non-constant modular function takes every value equally often in 6 (except that the frequency at e2Ki/3is weighted by Since a bounded modular function omits a value, it follows and that a t i by that a bounded modular function is a constant. In Section 5, we showed by a rather involved construction, that for any two real numbers a2, as, such that a; - 27a$ # 0, there was a Weierstrass !&function with g2(w,w') = a2 and g3(w,w') = a3 as its invariants. For general values of a2, a3, real or complex, J(T) and Theorem 6.5 provide an easy solution to this general "inversion" problem.

k).

Theorem 6.6. Given two complex numbers a2 and a3 such that a; - 27a$ # 0, there exist complex numbers w, w' with Im(w1/w) > 0, such that the Weierstrass p-function P(w, w') has invariants g2,g3 satisfying gz(w, w') = a2; g3(w,w') = a3. Proof. There are three cases: Case 1: a2 # 0, a3 # 0. The equations

can be satisfied if and only if g2 and g3 satisfy the equations

It is sufficient to determine w and T = w'/w. The second equation in (6.18) can be written (by Theorem 2.6)

8.6. Modular functions or, factoring w out of numerator and denominator,

Similarly, the first equation in (6.18) is equivalent to solving

but, since a; - 27ai # 0, we know this has a solution T with I m T > 0 by Theorem 6.5. Pick such a T (say the unique T E G). Then (6.19) determines that

and then w' = WT. Case 2: a2 = 0, then a3 # 0 (since a; - 27aa # 0). We can take T = eZril3 (see proof of Theorem 6.1 (e)), and get g2(r) = a2 = 0. Hence g2(w,w1)= a2 = 0, and 1 1 = 140w-~ g3(w,w1)= 1 4 0 ~ ' (mw n ~ ' ) ~ (m + ne2ri/3)6 ' m,n

+

C

m,n

Thus, setting this equal to a3, we can determine w by

and then w' = weZril3 = WT. Case 3: a3 = 0; then a2 # 0, and we can take T = i (see proof of Theorem 6.1 (e)) and in a manner similar t o the previous case, determine w by

and w' = iw = TW.

Note 6.4. It turns out that, nevertheless, there are arithmetic connections between g2,g3, w1, wg ,771, 773. The first such result was by Siege1 [222] who showed that if g2 and g3 were algebraic, then a t least one of wl, w3 must be transcendental. Schneider 773 are all transcendental. proved [216] that if g2 and g3 are algebraic, then wl ,~3,771, It is now known that, furthermore, if gz and g3 are algebraic, then wl, 771,27~iare linearly independent over the field of algebraic numbers, and moreover, that if y(zlwl, w3) "does not have complex multiplication" (which is the usual case), then 1, wl, ~3,771,r]3,2ni, are linearly independent over the field of algebraic numbers. For these and related results, see Masser, Elliptic Functions and Transcendence [I551 (note, however, that Masser uses wl,wz for the fundamental periods).

388

8. Elliptic Functions

Note that by Theorem 3.2(b) qlws - qsw1 = $ and so algebraic independence of these numbers is not true. As to the image of the "fundamental region G" defined in Theorems 6.4 and 6.5 (Diagram 8.4), a somewhat more precise result than the fact that J maps G one-to-one onto C, (except at i and e2"i/3)can be proved. Theorem 6.7. Let X denote the region bounded b y {z : R e z = -$, I m z 2 { z : IzI = 1,$ 5 argz 5 F}U{ z : R e z = 0 , I m z 2 1) (see Diagram 8.7).

$1

U

Diagram 8.7 Then J maps X (the ('lefthalf of 6 ") onto the (open) upper half plane {w : I m w > 0) and the bounding arcs onto the real axis, with the circular arc mapping onto the interval [0, 11. Furthermore, the region, call it X * , which is the reflection of in the imaginary axis, (the "right half of G") is mapped onto the lower half-plane. Proof. Suppose I-, ImI- > 0, is not on the imaginary axis, then the reflection of in the imaginary axis is -?, and we need to show

But by Theorem 6.2(b), writing

where the a(n) are integers. Hence

T

= r +is, r and s real,

I-

8.6. Modular functions -

and since ea+ib = ea-ib, and the a(n) are real, (6.20) follows. In particular, on the circular arc, 171 = 1, and so 77 = 1, and so on the arc

(by Theorem 6.2 (c)). Thus, J is real on the circular arc, and we already know that ~ ( e ~ "=~0,/ J~( i)) = 1; so the circular arc is mapped onto [O,l]. Furthermore, since there is a double 1-point a t i and a triple zero a t e2"i/3 the angle of n/2 formed by the bounding arcs at i goes into an angle of n and that of n / 3 formed by the bounding arcs a t eZKiI3similarly to an angle of n. For s real clearly J(is) is real and as s + co, J ( i s ) + co by Theorem 6.2(b). Also, by + i s , s real, Theorem 6.2(b), if we evaluate J a t

-&

which clearly is negative and monotone decreasing for all s sufficiently large. thus the line {z : Re z = -&, Irn z 2 $1 is mapped onto the negative real axis. Finally, J is clearly conformal in (except at i and e2"i/3) and as the boundary ioo to eZTil3t o i to ico, the interior of 3C lies on the of 3C is traversed from left, and these arcs map onto the real axis traversed from negative to positive. The image of the interior of 31 must consequently lie on the left, namely, therefore, in the upper half-plane, which (by Theorem 6.5) completes the proof.

-4 +

Note 6.5. The region 31 of Theorem 6.7 is the "curvilinear triangle" with vertices ezTiI3,i, co, and the angles n/3, n/2,0 respectively. J (restricted to 3C) is then the inverse of the Schwartz-Christoffel map mapping the upper half-plane onto such a region. This is only one of many connections between elliptic and related functions and conformal mappings, some of which have already been hinted at, and for details of which the reader is referred to the references in Note 1.5.8.

Note 6.6. Theorem 6.6 raises the question whether an analogous result holds for Jacobi's elliptic functions. That is, given a value of k = is there an elliptic function sn(z, k) =

+

*,

(F(I)-e3)

(z) ' < with 0

k2 < 1,

where A = (el - e 3 ) i , I has

periods 2w,2w1, wl = W , W=~ w w', wg = W' and q(wi) = e i , i = 1,2,3. (cf. Definitions 4.3.) By Theorem 4.10, sn(z, k) satisfies the differential equation

However, we now wish to reverse the procedure. That is given a value of k2, can we find a solution to the differential equation

8. Elliptic Functions

390

And, if there is a solution, is it necessarily unique (and so "identifiable" as sn(z, k))? The second question can be answered positively by appealing to a standard uniqueness theorem in differential equations derived from the Picard-Lindelof method of successive approximations (see e.g. Coddington and Levinson [48], Theory of Ordinary Differential Equations, p. 34.). As to the first question, if we can show that for a given value of a E @ - (0, I ) , there exists a complex number r with I m r > 0 such that k2(r) = a, then using that value of r , we can construct Ol(z17) and O4(21r) from their definitions, and hence by Theorem 5.8(c) a function sn(2Rz, k) = 2- 8 4 ( Z I T ) , where R by definition equals

J:~ '

,/GZx

(viz. Definitions 4.1). And so, by a change of variable, we

obtain the fuiction sn(z, k) which satisfies the differential equation

where a = k2, which is unique. Indeed, this will more than answer our question, providing, in fact, a definition of sn(z, k) for complex k with k2 # O , l , w . Thus, the question of existence of sn(z, k) for values of k2 # 0 or 1, is answered by the following theorem.

Theorem 6.8. Given a E @ , a # 0 , a k2(r)= a .

,

#

1, there is a r with Imr

>0

such that

9. + + e3 = 0, and since

Proof. Let el = F , e 2 = a, es = Then el ez el, en, e3 are to be the roots of 4z3 - g22 - g3 = 0, we have

and furthermore, clearly e l , e2, e3 are all distinct (since a

# 0, a #

1). Hence

Thus, by Theorem 6.6, there are w , and w' = rw, such that g2 and g3 are the invariants of P(zlw, w'), and Im r > 0. By Definitions 4.3 and Theorem 5.8(b),

5 Lt:j:i&$L<;!3 . By Definitions 4.3, k2( r ) = E, and so 1- k2( r ) =

Theorem 6.9. For Im r Proof. 6.1.

> 0,

J(r) =

z.

By Definitions

8.6. Modular functions

391

where el = e l ( r ) = Q ( i ) , e2 = e 2 ( r )= ~p(?), e3 = e 3 ( r )= Q ( $ ) , and Ip(z) = Q ( z l i , :). Hence

J ( T )=

+

+

+ + e;)l3 .

{(el - e2)2 (el - e3)2 (e2 - e3)2 - 2(ef ei 2(el - e d 2 ( e l - e3)2(e2- e3I2

(6.22)

+ e2 + e3 = 0 , we get on squaring e: + e; + e; = -2(ele2 + ele3 + e2e3) = (el - e2)2 + (e2 - e3)2 + (el - e3)2 - 2(e: + e; + e;) .

But, since el

Or, e:

1 + e! + e$ = -((el - e2)2+ (e2 - e3)2+ (el - e 3 ) 2 ). 3

So, from (6.22), we have

Theorem 6.10. For Im T > 0, let p ( r ) be the function constructed in Chapter 2, Section 2 (Definition 2.2. I ) . Then p ( r ) = k 2 ( 1- $).

Proof. Let e(r)= k 2 ( 1- $). The region R bounded by the three arcs (see Diagram 8.8) { z : Rez = 0 , I m z 01, { z : l z = $,o 5 argz 5 n } , { z : Re z = l,Imz>O},

>

31

Diagram 8.8 is mapped by the map z + 1 - onto itself with L1 mapping onto ( 1 : y > 0 ) = L3; L2 onto 0 ) = (1

+

$ :y >

8. Elliptic Functions

392 and L3 onto (making the substitution y = cot ):

where the boundary of the region is described in the orientation: L3, L2,L1. Note that the point exiI3 = E R goes onto itself. (, - {0,1, oo) is the image under k2 of some point in the upper Every point in C half-plane, by Theorem 6.8. Since the map z + 1 - $ takes the upper half-plane onto itself, every point in C, - {0,1, oo) is the image under C of some point in the upper half-plane. We need also to collect some information about the function k2 here. We have by Theorems 5.8, 5.7, and 5.10:

where q = exiT,I m T

> 0.

urn1(h) 1

e-f2v-l)*l~

8

Thus, for y > 0, by (6.24). C(iy) = k2(1- &) = 1, and so C maps L1 (with the real endpoint deleted) (see Diagram 8.8) onto the negative real axis. Furthermore, by (6.24) since the map z + 1 - maps L2 onto L1, C maps L2 (with the endpoints deleted) onto

Finally, using (6.25), C maps L3 (with the real endpoint deleted) onto

= (1 - k2 (1

+ iy) : y > 0) =

the open interval (1, oo) ,

393

8.6. Modular functions

since by (6.23) (or (6.24)), lc2 is periodic with period 2, and we have already seen that C maps L1 onto the negative real axis. Thus C maps the boundary of R onto the real axis (taking limits we see that lim,,o !(ir) = 0 and lim,,~ C(1+ i r ) = 1). In addition, again using the periodicity of k2, we have by (6.25)

and consequently also

Thus C is periodic with period 2. Let R* be the region obtained by translating R by 1 to the left. Then C maps R U R* into C - { z : I m z = 0) and C maps BdR - {0,1} onto the real axis -{0,1, oo), see Diagram 8.9)

Diagram 8.9 Since as T describes BdR in the counterclockwise direction, the region R lies to the right, and C(r) describes the real axis with the upper half-plane to the right R gets mapped into the upper half-plane. It follows from (6.26) that R*gets mapped into the lower half-plane. Suppose now some point in the upper half-plane were not the image under C of a point in R. Then, since every point in C , - {0,1, oo) is the image under C of some point in the upper half-plane; !is periodic with period 2; R* maps into the lower half-plane and BdR onto the real axis, such a point must be the image under C of a point either in the semi-disk {z : lz - < f ,O < argz < T} or the semi-disk {z : lz f 1 < , O < argz < n}. But any such point can be reached by analytic continuation of C by repeated reflection of R and ensuing regions over the portion of the boundaries which abut the real-axis; the images are then simply repeatedly conjugates of the original images of points in R or R*.By the uniqueness

+

3

31

394

8. Elliptic Functions

of analytic continuation this contradicts the fact that e maps the upper half-plane onto C , - {0,1, w). Thus, every point in the upper half-plane is the image under e of some point in R. Since R gets mapped onto the upper half-plane, R* is mapped by e onto the lower half-plane. In this way, we see that l and the function p constructed in 2.2 have exactly the same mapping properties. It remains to show that they are identical. However, R is the Jordan interior of the "curvilinear triangle" with three angles 0 (note that L 1 ,L2,Lg are all orthogonal to the real axis), and both e and p effect the conformal mapping of this region onto the upper half-plane. Furthermore, since limt,o [(it) = 0 = limt+o p(it) and limt,~ l ( l + i t ) = 1 = limt+o p(l+it), it follows t>O t>O t>O t>O that C and p must be identical since by the Riemann Mapping Theorem (Section 1.5) a conformal map of a simply-connected region with two boundary points onto B ( 0 , l ) (or the upper half-plane) is determined up to three real parameters.

Note 6.7. Since J ( l - f ) = J(T), it follows from Theorem 6.9 that [(T) de'k2(11 ?) = p ( ~ (of ) Chapter 2, Section 2) satisfies

Although p is not invariant under all transformations of the modular group (Definition 6.3), but only by those in the subgroup in which b and c are even, it is commonly called an elliptic modular function (despite Definition 6.2). The subgroup which leaves X invariant is a normal subgroup of the full modular group of index 6. The phrase "elliptic modular" to describe functions of the form k 2 ( s ) where is a member of the modular group, arises from the fact that k is the "modulus" (see Definitions 4.2) of the elliptic functions which arise from a Legendre elliptic integral of the first kind. In Theorem 2.1.1 (the Bloch-Landau Theorem) it was proved that iff is analytic on B ( 0 , l ) and 1 fl(0)l 1, then the image of B ( 0 , l ) under f contains an open disk however, the value was not best possible. There is, however, a of radius covering theorem" whose proof depends on the representation (6.23) of "sharp k2 (T) in the proof of Theorem 6.10.

&

&;

>

&

Theorem 6.11. Suppose f is analytic on B(0, I), f (0) = 0, and 1fl(0)l 2 1. Suppose further f (0) # 0 in B ( 0 , l ) - {0), then the image of B ( 0 , l ) under f contains and this is best possible. a disk centered at 0 of radius

&

Proof. We have by Theorems 5.8 and 5.7,

where q = e x i T ,I m T

> 0.

8.6. Modular functions Let i = *; u E B(0,l)

(Note that as T

395

where Re Log u = Log lul

< 0, and so lul < 1, and define for

+ ioo,u + 0.) Then Q(0) = 0 and

Q(u) = 1 6 . Q1(0) = lim U--to u Now k2 is periodic with period 2 by (6.23), and so, since k2 takes on every value, but 0,1, oo in the upper open half-plane, Q is analytic and takes every value but 0 and 1 in B ( 0 , l ) - (0). Let Q be a local inverse of Q in a neighborhood of 0 (which exists since Q1(0) # 0). Suppose a is a value not taken on by f (z) for z 6 B ( 0 , l ) . Then, for z E B(0, I), does not take the value 1 or 0 in B ( 0 , l ) - (0) and so the function g(z) &(*) can be (by the monodromy theorem) continued analytically throughout B(0,l). We have then g(0) = eco) = 0

%

and, for z 6 B(0, I ) , 19(z)1 < 1 , since Q, the inverse of Q, takes values in B ( 0 , l ) . By Schwarz' Lemma, we obtain

But, then,

>

&.

by (6.29) and since (fl(0)l 1 by hypothesis. Hence la1 2 The function & ~ ( z which ) omits the value (since Q omits the value 1 in B ( 0 , l ) ) shows that the theorem is best possible.

&,

Note 6.8. Theorem 6.11, but with the constant $ instead of &, was proved by Hurwitz in 1904; the best possible value & was obtained by Carath6odory in 1907, and independently rediscovered by Bochner in 1926. All these proofs use elliptic functions. Note 6.9. Modular functions and modular forms (see Note 6.2) are among the contemporaneous areas arising from elliptic functions most actively pursued at present. As indicated earlier, much of such contemporary work arising from elliptic functions has an algebraic character; indeed, the mixture of algebra and analysis is extremely

396

8. Elliptic Functions

fruitful. For an introduction to "elliptic curves" the reader might consult DuVal [63], Elliptic Functions and Elliptic Curves. An excellent introduction to modular functions and their use in analytic number theory is Apostol [ll],Modular Functions and Dirichlet Series in Number Theory.

Chapter 9

Introduction Zeta-Function

the Riemann

The Riemann zeta-function (which has no relation to the Weierstrass function of Chapter 8, and must not be confused with it) was originally of interest because of its connection with problems in prime number theory. Since then it has served as the model for a proliferation of "zeta-functions" throughout mathematics. Some mention of the Riemann zeta-function, and treatment of the prime number theorem as an asymptotic result have become a topic treated by writers of introductory texts in complex variables. This is principally because of the intrinsic interest in the result and the availability of a concise analytic proof in the form of Landau's version of Wiener's proof (reprinted as an appendix to the Chelsea reprint of Landau's Handbuch der Lehre von der Verteilung der Primzahlen [139]). More recently Donald J. Newman has given an even simpler proof of the prime number theorem [174]. Nevertheless, any such proofs often appear unmotivated to the student who knows nothing of its background. In this chapter we shall investigate the Riemann zeta-function somewhat more closely; still emphasizing the connections with prime number theory. However, the word introduction is as applicable here as in other portions of this book and, as usual, a great deal of much interest has been omitted; even from the Notes. This chapter is not a comprehensive treatment of the Riemann zeta-function (for which see Titchmarsh's excellent text, The Theory of the Riemann Zeta-function, [230]). Titchmarsh's book has been updated in two ways since his death in 1963. Aleksander Ivic published such a book in 1985 titled The RiemannZeta-Function [122]. It contains a great deal of material not in Titchmarsh's book. In the following year, a second edition of Titchmarsh's book was published by Oxford under the editorship of D.R. Heath-Brown. This is a true "update" in that at the end of each chapter there are Notes bringing the material of that chapter up-todate. Additional updated and historical information can be found in Edwards' fine expository book, Riemann's Zeta Function [64]. Neither is this chapter anything

9. Riemann Zeta-Function

398

like a reasonable introductory treatment of analytic prime number theory. The reader interested in this subject should consult Davenport, Multiplicative Number Theory [52], or Huxley, The Distribution of Prime Numbers [119]. Ingham's classic The Distribution of Prime Numbers [120], also still deserves consideration as an introduction to these topics. Prachar's Primzahlverteilung [206] is more comprehensive in the topics treated than any of the aforementioned books, though it lacks the vast contemporary development of sieve methods about which there are several books. Ordinary Dirichlet series, that is, series of the form Cr=3=1where s is a complex variable will find brief mention in this chapter; there are various monographs available dealing with one or another aspect of these or of so-called "general Dirichlet series": Cr=,ane-xns, where An t co as n t co, which include power-series (A, = n) and ordinary Dirichlet series (A, = logn) as special cases. The basic analytic facts about ordinary Dirichlet series can be found in Chapter IX of Titchmarsh, Theory of Functions [229]. The treatment in this chapter, little as it is, will be self-contained. Having spent some time saying what is not in this chapter, it is time to begin to say what is. It is assumed that the reader is familiar with the basic properties of the prime numbers, and their infinitude, as well as with the technique of partial summation. Throughout this chapter we shall use a notation traditional and idiosyncratic to analytic number theory:

s,

Notational Conventions for Chapter 9: s denotes a complex variable, Re s = a, Im s = t , p denotes a prime number (> 2). -+ 1 as x t m ; sums and Also, the expression f(x) -- g(x) will mean products over primes begin at 2, over integers begin at 1, unless otherwise mentioned. The familiar Bachmann-Landau 0, o-notation will always refer to error terms as the variable goes to co, unless otherwise specified. Ed,,means a sum over all positive divisors of n and analogous notation is used for products and for prime divisors. [XI means, as usual, the largest integer I:x.

#

9.1

Prime Numbers and [(s)

We begin with Euler's proof that there are infinitely many primes.

Theorem 1.1 (Euler). There are infinitely many primes, and to oo as x + co at least as fast as loglogx.

C,<,

diverges

Proof. By the fundamental theorem of arithmetic, if m is a positive integer,

.-

nE in a certain set of positive integers each counted once.

9.1. Prime Numbers and ((s) In fact, for a sufficiently large m (depending on a), we have

If there were only finitely many primes then, letting x + co, the right side of (1.1) remains bounded contradicting the divergence of the harmonic series. Furthermore, from (1.I ) ,

and

En,, A = logx + O(1) as x + co, proves the theorem. If, in Euler's proof, we instead consider Cnlx5 ,where Re s > 1, then a similar

which, since

argument can still be made and indeed both sides of the analogue of (1.1) above converge. Thus one might be able to study the distribution of primes by finding out more about the function defined for Re s > 1 by Er=l

+.

Definition 1.1. The Riemann zeta-function ((s) is defined for Re s

> 1 by

Theorem 1.2. For Re s > 1

P

*

P'

Proof. See proof of first part of Theorem 1.1. The arguments used in Theorem 1.1 suggest taking logarithms in Theorem 1.2. When we do this we obtain

Theorem 1.3. For Re s > 1,

9. Riemann Zeta-Function

400 where

logp,

ifn=l if n is not a prime power if n = pm,p prime.

(ii)

Proof. From Theorem 1.2, for Re s Log ((s) =

> 1,

" 1 " A(n) 1 C - log(1 - p-') = C C mp"b = C -. logn nS P

p

n=2

now follows on differentiating termwise (the resulting

The formula for series is majorized by

m=l

Crz2

which converges since u = Re s > 1).

Definition 1.2. "Von Mangoldt's Function" A(n) is henceforth defined as in Theorem 1.3. We wish to study the distribution of primes, and thus make def

Definition 1.3. ~ ( x = )

Cplx1 = the number of primes 5 x.

PSI

The relation of

T, @,

Theorem 1.4. As x

8 to one another is brought out by

-+ co,

Proof. We have, clearly, 8(x) =

C log p 5 n(x) log x PIX

Also, for 1 < y

< x,

9. I . Prime Numbers and ((s) Taking y = &,

we get from (1.3),

T(X)log x x 1 < O(x) - O(x)logx+ 1-"0'"" log x

*

We thus need an estimate from below on O(x). We have, clearly, d(x) =

x

~ ( n =)

nsx

x

log P =

p,m P" <x

x

log P

P

~

X

I-[

log x

t O(x) ,

log P

and also

where the last series in fact terminates so soon as x1im < 2, i.e. for rn But, obviously, B(x) is a non-decreasing function and

>&.

and so (1.6) yields

@ d(x) =

x

B(X~5 / ~O(x) )

m=l

x ,a (10, x ) ~ + -log - - O ( X " ~ ) 5 O(X)+ 210g2 . log 2

(1.7)

The theorem will now follow from (1.2), (1.4), (1.5) and (1.7) if we can show that O(x) > Ax for some constant A > 0, and, by (1.7), it will be enough to show that $(x) > Al x for some constant A1 . This requires a special result. Suppose we write the prime power factorization

Then it is easy to see that

I:[

(Counting each multiple of p which is 5 n once gives

multiples; however, we

need to count the multiples of p2 again and indeed there are etc. For m >

each term in the infinite series is 0.)

9. Riemann Zeta-Function Consider now the integer -

n

pop, say.

p12n

Erom (1.8), we get

and it is easy to see that

+

(In the former case, 2[x] 5 22 < 2[x] 1, and in the latter case, 2[x] 2[x] + 2). Hence, each term in (1.9) is either 0 or 1 and we get

[=I

+ 1 5 2x <

log 2n

Thus, log

(z)c

a plog P 5

=

p12n

a

c I-[

p12n

log 2n logp = a(2n) . 1 % ~

On the other hand,

and so, from (1. lo), n log 2 5 a(2n) . Taking n =

[$I,

since +(x) is non-decreasing, we have for x 2 2, +(XI t

a (2

[a]) [a 2

iog2 2

A ~ X

where A1 is a positive constant; and the theorem now follows as indicated earlier. Thus, we see that asymptotically a ( x ) and ~ ( xlogx ) have the same behavior A(n) where (by theorem 1.3) and that $(x) = -

En<,

-=c-C1(s) C(s)

00

n=l

A(n) for Re s n

>1

The study of the distribution of prime numbers thus leads to a study of series of the form on the right in this last equation.

9.2. Ordinary Dirichlet Series

9.2

Ordinary Dirichlet Series

Definition 2.1. Ordinary Dirichlet series are series of the form

x3

, a,

E @, s = o

+ i t , o, t real.

The numbers {a,) are called the coeficients of the series.

A fundamental result about such series is: Theorem 2.1. If an ordinary Dirichlet series Cr=l 3 converges for s = so, then it is uniformly convergent throughout the region

Proof. Note first that the regions described in the theorem are "angles" as in the diagram below.

Diagram 9.1

If so # 0, we can consider the series Cr=l $ where b, = % and s* = s - so. Thus, it is sufficient to consider the case when so = 0. a, converges, and so Cm2n+l a, + 0 as n -+ co. If so = 0, then Let rn = C,,,+, a,. Then r,-1 - r, = a,, and given e > 0, for n 2 no = no(€), we have lrnl < e, and

9. Riemann Zeta-Function Hence, for M

> n o ( € )+ 1,

But,

So for R e s = a

> 0, substitution in

Now, if I arg sl monotone in (-

5 $ - 6 where s $ , $) ,

I:

(2.1) gives

=a

+ it, then since tan(args)

5tan(i-6)

=

$, and tan

is

=cot6,

and so

Thus, from (2.2),

and the right hand side is independent of s and

'fr.=l3

+ 0 as E + 0.

+

Theorem 2.2. If converges for so = a0 ito, then it is convergent for s = a + it provided only that a > a,, (i.e. Ordinary Dirichlet series converge i n half-planes.)

Proof. Choose 6 sufficiently small in Theorem 2.1. Definition 2.2. B y Theorem 2.2, we may define the (extended) real number a,, by C;=, 3 converges for a > a,, diverges for a < a,. a, is called the abscissa of a, may possibly = -oo (the series convergence of the Dirichlet series C;==, converges i n the whole plane) or +m (the series converges nowhere).

s.

Theorem 2.3. Suppose f ( s ) = C;=l 3 ,for R e s = a > a,. Then f ( s ) is analytic for a > a,, and the series can be diferentiated termwise with the diflerentiated series converging for R e s = a > a, to f l ( s ) .

9.2. Ordinary Dim'chlet Series

405

>

Proof. Theorem 2.1 and the fact that is analytic for a > a, (since a uniformly convergent series of analytic functions is analytic, and may be differentiated termwise). Note 2.1. Although, (analogously to the situation of power series on the boundary of the disk of convergence) ordinary Dirichlet Series may exhibit any sort of behavior with respect to convergence, divergence, analyticity of the function f ( s ) of Theorem 2.3, etc., on the line a = a,; nevertheless it is clear from Theorem 2.1 that if Cr=l% converges for so, where Re so = a,, and f (s) = Cr=l for Re s = a > a,, then f (s) + f ( s o )as s + so on any path lying in the interior of I arg(s - so)l ;-6,0<6<

<

2.

Theorem 2.4. Given the ordinary Dirichlet Series Cr=l 3, there is an (exconverges absolutely for a > a ~ . tended) real a~ such that Cr=,

>

Proof.

Cr=l 9is a Dirichlet series, and Theorem 2.2.

Definition 2.3. The number UA of Theorem 2.4 is called the abscissa of absolute convergence of the Dirichlet series Cr'l .

>

Theorem 2.5. Given an ordinary Dirichlet series Cr=l 3 for which a, is finite, then O
>

Proof. Clearly UA a,. On the other hand, if CE1 and so is bounded; consequently

> converges, then

=

1 % 1 + 0 as n + m,

converges absolutely for every 6 > 0.

Example 2.1.

Cz=l

has a, = 0, an = 1.

Note 2.2. UA and a, may, in fact, have any values consistent with Theorem 2.5. There are formulas for a A and a, in terms of the coefficients an of the Dirichlet series, which are analogous to the Cauchy-Hadamard Theorem for power series. We will not need these, but they can be found in Chapter IX of Titchmarsh's cited text or any other introductory treatment of Dirichlet Series. Theorem 2.6 (Uniqueness Theorem for Ordinary Dirichlet Series). If h = O0 $ (both series being convergent) in some region of the s-plane, n=1 nd then a , = b, for all n.

coo

9. Riemann Zeta-Function Proof. Clearly it is enough to prove that if

in some region of the s-plane, then an = 0 for all n. If (2.3) is true, then 3 = 0 in some half-plane by Theorem 2.2. Let am be the first non-zero coefficient, if any such exist. Then,

Cr=O=,

But then for u >.uA, and 0 < S < u - U A ,

which goes t o zero as o

+ oo. Hence am = 0, a contradiction.

Theorem 2.7 (Multiplication Theorem for Ordinary Dirichlet Series). If 3 and are both absolutely convergent, then

CFZ1

Cr=l%

where

Proof.

on letting nm = k, and where the rearrangement of the series producing the last step is justified by absolute convergence.

Note 2.3. Theorem 2.7 is the analogue for ordinary Dirichlet Series of Cauchy's Theorem on the multiplication of power-series. All the standard theorems about multiplying power series have analogues not only for ordinary Dirichlet series, but for general Dirichlet series ane-xns as well. See, for example, the already cited Chelsea reprint of Landau's Handbuch [139], sections 212-220, with the updating notes thereto by P. T. Bateman.

9.2. Ordinary Dim'chlet Series

407

Note 2.4. An arithmetic function, that is, a function on the positive integers with range in (C, is called multiplicative if f (mn) = f ( m )f (n) whenever m and n are relatively prime, and f (1) = 1. Clearly multiplicative arithmetic functions are determined by their values at prime powers. If, further If f (n) is a multiplicative arithmetic function, then so is C;==, converges absolutely, then the argument of Theorem 1.1 shows that

y.

9

Expressions of the sort on the right are called "Euler products" and play an important role in analytic number theory. Theorem 1.2 is the case f (n) z 1.

Example 2.2. We list some simple examples of ordinary Dirichlet series and the preceding theorems

(i) For Re s = o

> 1, <(s) =

xT=3=1 5 is analytic.

%,where r ( n ) = Cdln1. & = np(1 - $) = En==, % , where a(1) = 1, a(p) = -1,

(ii) For R e s = o > 1, ( ( ( s ) ) ~= X;=l (iii)

00

a(pm)= 0 for m 1 2, and a ( n ) = n p l n a ( p m ) . That is a ( n ) Mobius function of number theory.

p(n) the

1 00 (iv) ('(3 - 1) = Cr=O=I = En==, 5 for R e s = a > 2. Hence, for Re s = a > 2, using (iii) above,

where $(n) is the number of integers 5 n which are relatively prime to n. (Readers unfamiliar with the above formula for $(n) can prove this result by using Note 2.4 and the fact that $(n) is multiplicative.)

(v) C(S)((S - 1) = C r = l

9for Re s = o > 2, where o(n) = xdln d.

(vi) -C1(s) = C r = l %,for

Res=o

> 1.

Since we already know (Theorem 1.3(ii)) that

we have from (iii) above, and Theorems 2.6 and 2.7

9. Riemann Zeta-Function Since Cdln p(d) =

1, 0,

ifn=l this last becomes otherwise, h(n) = -

p(d) log d . din

(The formula for 2.6 and 2.7.)

Cdln p(d) follows immediately from 1 = C(s) . & and Theorems

Note 2.5. Example 2.l(iii) allows an analytic proof of the "Mobius inversion formula" familiar in elementary number theory which now simply takes the form

if and only if 1

If we are to use Theorem 1.3 to obtain information about $(x) and thus about ~ ( x )we , need a theorem which will tell us something about Cnsx an in terms of the analytic function represented in a half-plane by Cz==l2. This and more is provided by the next theorem.

Theorem 2.8 (Perron's fomula:). Let w =u where a, = O(A(n)) as n

Then, if c T > 0,

+ iu, f(w) = C

for u > 1 ,

+ co, and A(n) is non-decreasing; and for some a > 0

> O,(T+ c > 1, x

+

mw

is not an integer, N is the nearest integer to x, and

(A(22)xLU log x

) + 0 (AT(;)Zi;),

where c may depend upon x. Proof. Let r > 0 and let R be the rectangle with vertices -r - iT, c - iT, c + iT, described positively (see Diagram 9.2).

iT,-r

+

9.2. Ordinary Dim'chlet Series

Diagram 9.2 Then if n

< x, we have

(the only singularity is a simple pole a t w = 0 with residue 1). Also, for n < x

which goes t o 0 as r

+ oa,since ;> 1. Hence (2.4) yields

Furthermore, letting w = u - iT,since n

-cm+iT

and a similar estimate holds for Jc+iT Thus, from (2.5) we get, for n < x,

1

< x,

(:)W

The idea of the proof is to multiply (2.6) by we need also to know what happens for n > x.

dw.

3 and sum, but inside the integral

9. Riemann Zeta-Function

410

For n > x, consider the rectangle R* with vertices r - i T , c - iT, c described positively (where r > c > 0) (see Diagram 9.3).

+ iT, r + i T

Diagram 9.3 Clearly,

and for n

> x, since r > 0,

(:)'-to Hence, arguing as above, we get for n

asr

+m.

> x,

Now, multiplying both sides in (2.6) by $, summing and using (2.7) we get,

To further estimate the error term in (2.8), we break it into three pieces: (i) If n < $ or n > 22, then I log($)[ > log2, and so we have

+

as a c -+1, by hypothesis. (ii) For $ 5 n < N , let n = N - m; then

9.2. Ordinary Dirichlet Series and

Hence,

A similar argument applies to the sum over the terms in (N, 2x1 and shows that we also get

Finally, (iii) if n = N ,

Substituting (2.9)-(2.12) into (2.8) gives the theorem. Note 2.6. In many applications of Theorem 2.8, one can take x as one-half an odd integer, whereupon x - N = $ and the third error term in Theorem 2.8 is absorbed into the second. There is a variant of Theorem 2.8 for x an integer, as well, which reads the same way except that the left-hand side is Cn,,-, 5 + $ 3 ,and the last error term is 0(-). This follows by the same arguments as above, except that (iii) is unnecessary, in (ii) N is replaced by x, and for n = x, note that

In order to apply Theorem 2.8 to s + 1. This is provided by

e,

we need to know how it behaves as

9. Riemann Zeta-Function

412

Theorem 2.9. (a) <(s) can be analytically continued into the region a > 0, where it is analytic except for a simple pole with residue 1 at 1. can be analytically continued into the region a > 0 where it is an(4) alytic except for a simple pole with residue 1 at 1, as well as simple poles at the zeros of < ( s ) (if any) i n c > 0. Proof. By partial summation,

and so, for a

> 1,

(A)

For CT > 0, the last integrand is 0 and so the integral converges for a > 0 and uniformly for a 2 a > 0. Thus the right side of the equation provides the analytic =1 the only singularity in continuation of C(s) into a > 0, and since this region is a simple pole with residue 1 at 1. The result for now follows.

5

+

Theorem 2.10. For x one-half an odd integer

for any c (which may depend on x) which is

> 1.

w,

Proof. In Theorem 2.8 take f (w) = s = 0, e > 1 and x one half an odd integer. By Theorem 2.9, cu = 1, and by Theorem 1.3, an = A(n). Clearly then A(n) = log n , and the theorem follows.

9.3

The Functional Equation, the Prime Number Theorem, and De La Vallke-Poussin's Estimate

The idea of an analytic proof of a theorem about the distribution of primes should now be apparent. We wish to move the contour over which the integral is taken in Theorem 2.10 to the left. The function has a simple pole at 1 with residue 1

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate

413

(Theorem 2.9); if ((w) has no zeros in some region to the left of the line Re w = 1, and if we can get an adequate estimate of on a bounding contour to the left and on the horizontal contours, then Theorems 2.9 and 2.10 and the Residue Theorem give $(x) = x E ( x , c , T )

+

where E ( x , c, T ) is an error term depending on x, c, T. If c and T can be appropriately selected as functions of x, we will get a good estimate for $(x). Theorem 1.4 then provides some indication of how this information can be translated into an estimate on n(x), the number of primes 5 x. The sine qua non of such a proof is clearly the fact that ( ( s ) has no zeros for Re s = 1. This is Theorem 3.1 below. This proof can, in fact, be developed in the absence of any analytic information about ((s) other than Theorem 2.9 to show that there is a constant A such that in a region of the form

C(s)

# 0, and in fact

1 %1

= O((1og t)'). This then yields the results that

where B and B' are positive constants. This proof is interesting, however, if we know that ((s) can be analytically continued into the whole plane (except for the simple pole a t I ) , then, as De la Vallee-Poussin showed in 1898 the much better result

for some positive constant a, can be obtained. (But see also, Notes 3.8 and 4.4.) For reasons of space, we shall take this approach immediately. It is worth noting further though that later researches have demonstrated that one can prove analytically n(x)

-

x log x

- as

z -+ co

(the "prime number theorem as an asymptotic result"; note that lim

2--too

*l " x

-dt1 log t

= 1)

9. Riemann Zeta-Function

414

knowing only Theorem 3.1 below. Furthermore, "elementary" (i.e. non-complex analytic) proofs of the prime number theorem, even with an error term, are known but these were not successful until 1947 (51 years after the first complex-analytic proof), and still cannot produce as good a result as complex-analytic methods. (See Note 4.4.)

Theorem 3.1. ((s) Proof. For a

# 0 for a = 1.

> 1, we have by Theorem 1.2,

Log ((s) = -

C Log (1 -p-')

=

O 0 1 CC -= C C mpm 00

p

P

m=l

Hence

p

00

Log IC(s)l = Re Log a s ) =

CC p

m=l

p-imt

-.

m=l mpma

cos(mt log p) mpmo

and so, I((.

+ i t ) / = exp p

cos(mt log p) mpma m=l

The idea is now to use the identity

This gives from (3.1)

3 p

+ 4 cos(mt log p) + cos(2mt logp)

(3.2)

mpmo

m=l

for u > 1. Now suppose for some t # 0, ((1 it) were 0; then would be bounded by Theorem 2.9, ((s) is analytic on a = 1, we also have ((a 2it) is bounded as u + 1+. Thus, from (3.2) we get since the pole at s = 1 is simple

9

+

+

a contradiction; and hence C(s) has no zeros with Re s = 1.

Note 3.1. Theorem 3.1 is due independently to Hadamard and De la VallCe-Poussin in 1896, for whom it was the crucial fact in proving the "prime number theorem": ~(x) as x -+ m. The identity 3 4 cos cos 24 2 0 was first used by Mertens in an adaptation of De la VallCe-Poussin's proof in 1898.

6

+

++

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate

415

Other trigonometric identities, e.g. 5+8 cos 4 + 4 cos 24+cos 3 4 = (l+cos 4)(1+ 2 cos q5)2 2 0 may be used; however, 1 cos 4 2 0 is inadequate since it only leads t o 1((o)I Ic(o it)l 1 for a > 1 and this allows the possibility of a simple zero a t o + it. French [73] has given a survey of results about such polynomials relevant in prime number theory; they affect the value of a in Theorems 3.2 and 3.10 (below). However, since for many years results which are better than any value of a in Theorem 3.10 have been known (see Note 3.7), any motivation from prime number theory for an interest in these various polynomials has more or less disappeared. It is useful to have an estimate of how the existence of a zero-free region of C(s) t o the left of o = 1 and an estimate of the growth of in such a region affect the error term in the formula for $(x) which is implied by Theorem 2.10.

+

+

>

$$/

This is provided by the conditional

Theorem 3.2. If ((s) has no rems and

% = O((10gt)~)in a region of the form

then where a is a positive constant

< D.

+& +

Proof. Theorem 1.2 shows that ((s) # 0 for o > 1. Take c = 1 in Theorem 2.10, and consider the rectangle with vertices c - iT, c iT, 6 iT, S - i T , where 6 = 1 - &, 0 < B < D , described positively. By Theorem 2.10, then, the Residue Theorem, and the hypotheses of the theorem,

+

The last integral in (3.3) is by hypothesis (writing w = 6

For the first integral in (3.3) we have by hypothesis

xc (log T ) ~

+ iv)

9. Riemann Zeta-Function

416

+

1 since we have chosen c = 1 G . A similar estimate applies to the second integral in (3.3), and thus we have from (3.3)

&,

where 6 = 1 and T may still be chosen as a function of x. The optimal choice of T in this case is that which makes the second and third error terms grow a t B approximately the same rate, i.e. T logT should be about as big as xl-" xa:I.s, and thus (10gT)~about as big as logx. Taking then T = em*, we get from (3.4) $(x) = x

+ 0(log2x e - " e ) + 0(xevB= (log x) ) + O(x(log x ) * e - " e ) = x + ~ ( x e - ~ - ) ,

on taking a = B

(3.5)

< D, where a is a positive constant < D.

Note 3.2. The motivation for the statement of Theorem 3.2 may be found in the introductory remarks to this section. The proof of the theorem shows that it is the size of the zero-free region, not the growth of which matters most. In fact, it can be shown that if ( ( s ) # 0 for 112 < Res (this is the famous unproved "Riemann hypothesis")

$$

$(x) = z

+ 0(x4 log2x) ,

a result initially proved by von Koch in 1901. See also Note 4.4. It is well to clarify beforehand also the relationship between $(x) and ~ ( x ) That . $(x) and ~ ( xlogx ) have the same asymptotic behavior is Theorem 1.4. However, more sharply, we have the conditional

Theorem 3.3. If $(x) = x for x

2 xo,

+ O ( k ( ~ ) ( l o g x ) where ~) k(x) is monotone increasing

+ 0 as x + 00, then

k x log x ) ~

and (

)(5

Proof. By (1.5) and (1.7), we have

so by the hypotheses of the theorem, we get

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate

xCxlogp, and so, by partial summation and (3.6),

But O(x) gf

def

log x x log x

- -+O(k(x)logx)+ -

=

I"

-dt loit

(

0

F d t )

+ O(k(x) logx) .

Note 3.3. It can be shown that for any k(x) satisfying the conditions of Theorem 3.3, "(")0 must 2 be unbounded as x + m. See also Note 4.4. x2 We now roughly follow De la Vallke-Poussin's approach (with some digressions) in obtaining an estimate for a zero-free region of C(s) of the sort in Theorem 3.2. First we need an important and famous result of Riemann. We give two proofs of

Theorem 3.4. ((s) can be analytically continued into the whole plane except for a simple pole with residue 1 at s = 1. For all s it satisfies the functional equation ((s) = 2sns-1sin

( y )r ( i

-s

) ~ ( l -s)

.

First Proof. The idea of this proof is to use Theorem 8.5.10. We have for u > 0, Euler's formula

and so, letting y = n2nx and replacing s by s/2, we have that for

Thus, for u

0

> 1,

> 1, on summing over n,

provided the interchange of summation and integration on the right in (3.7) can be justified. Now in the notation of Chapter 8,

9. Riemann Zeta-Function (Definition 8.5.4). Thus

But, by Theorem 8.5.10,

and so we have from (3.8)

Thus, in particular, the integral on the right in (3.7) is convergent for a > 1, and so the interchange of summation and integration may be justified by absolute and convergence (considering the pieces separately). Furthermore, breaking (3.7) into J : and substituting (3.9) in (3.7) for $, we have

fi

+ST

Or, making the change of variable x = $ in the last integral, and since a

> 1,

The last integral in (3.10) converges absolutely for any s and uniformly with respect to s in any bounded region of the plane (note that trivially CF=le-n2nx = O(e-"") as x -+ 00). Hence, the right side of (3.10) provides the analytic continuation of the left side into the whole plane.

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate

419

But the right side of (3.10) is invariant under replacement of s by 1- s, hence the left side must be also! Thus

This is the so-called symmetric form of the functional equation. To obtain the form stated in the theorem, one uses the facts from the theory of the .-function that

and

Second Proof. This proof shows that the result of the theorem is obtainable directly by contour integration. We start again from Euler's formula T(s) =

JdU

ys-'e-~dy, for a

>0,

and letting y = nx, this becomes

Thus, for a

> 1, summing both sides gives

Now consider the integral

where C is the "loop contour" which starts a t co on the positive real axis goes around the origin once positively along C(0, p), where 0 < p < 1, and then returns t o co along the positive real axis (see Diagram 9.4).

9. Riemann Zeta-Function

Diagram 9.4 Since zS-l

d-! ,(s-1)

logz

,

if the logarithm is real at the beginning of the contour, after going once around the branch point at 0, its value is augmented by 27ri on the return path to co. Furthermore, on C(0, p),

I z ~ - 1l--1

e ( s - l ) log z

- e(u-l)

I = I ,(u-l+it)(log

log 121-targ z

Izl+iargz)

< 1~10-1~2nltl -

I

(since the argument does not vary by more than 27r as we go around C(0, p) once in the positive direction). Also, for 121 < 1,

F'rom (3.14) and (3.15), for 0

and so, for

0

< p < 1,

> 1, this integral goes to 0 as p + 0.

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate Hence, for

D

> 1, we get on letting p + 0, and using

(3.12)

or, by the definition of I ( s ) in (3.13),

Also, I ( s ) is uniformly convergent in any bounded region of the s- plane and so defines an analytic function there. Thus the right hand side of (3.16) provides the analytic continuation of C(s) over the whole plane. Furthermore, the only possible singularities of the right side of (3.16) are simple poles a t the integers occasioned is entire). But at 0 and the negative by the simple zeros of e2"is - 1 (since integers has simple zeros, and so the only possible singularities are a t the positive integers. But for s = n , n an integer 2, we already know ((s) is analytic (and so I ( n ) = 0 for integers n 2); thus the only possible singularity is a t s = 1. Now

&

&

>

>

r ( 1 ) = 1 and I(1) =

1

d t = 27ri Res

(-) 1

eZ

1

/

= 2ni . t=O

Hence ((s) has a simple pole a t s = 1 and the residue there is

To prove the functional equation from (3.16), consider

where R, is the contour which starts a t co on the positive real axis, until the point (2n+ l ) n , n a positive integer, then around the square with vertices (2n+l)7r(l+i), (2n l)n(-1 i), (2n 1)7r(-1- i), (2n 1 ) ~ ( 1 i) - in the positive direction; back ; to oo along the positive real axis (see Diagram 9.5). t o (2n 1 ) ~and

+

+

+

+

+

9. Riemann Zeta-Function

Diagram 9.5 z*-l

Between C and Rn, , rhas poles at the points 2kia, where -n and the residue at 2ki7r is ( 2 k i ~ ) ~ - Hence, '.

Now, for a

< 0, limn,,

lcki

CkSn kS-'

dz = lim

n-+w

= < ( I - s). Thus, for a

e: - 1

and = O ( 1 ) as n So.

and so, for a

< 0,

+ co

> 0,

< 0, we get from (3.17)

dz - ( 2 ~ i ) ~-(eniS)<(l l - s)

But, on the square part of Rn,

5 k 5 n, k

.

(3.18)

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin7sEstimate Thus, from (3.18) and (3.16), we have for a

-

2srs-1

e

2

<0

s i n a s r ( 1 - s ) < ( l - s) = 2sas-l sin enis 1

+

(y)~

( -1s ) < ( l - s ) .

But, by the previous argument, both sides of this equation are analytic for all s (except for a simple pole at s = I ) , and thus it holds for all s.

Note 3.4. The two proofs given are both Riemann's. There are many other proofs known of Theorem 3.4; indeed Chapter 2 of Titchmarsh, Theory of the Riemann Zeta-Function [230], gives seven different methods, and many more proofs if the variants indicated are also counted. The first proof above is perhaps the most elegant (though quite unmotivated unless one has been thinking in terms of the transformation formulas for theta-functions); the second proof above shows what can be accomplished by dedicated use of the residue theorem. Formula (3.16) in the second proof is an important representation of <(s). Theorem 3.5. For m a positive integer,

6) (ii) (iii)

where Bzm is a Bernoulli number (see Appendix) and also

1 lim <(s) - -= 7 s-1 where y is Euler's constant; s-+l

Proof. From (3.16)

where C is the loop contour described in the second proof of Theorem 3.4 an integer, the integrand is single valued and so Zn-1

FT dz =

Jc(o,r)

y1-1

ez - 1

dz for integral n .

. If s is

9. Riemann Zeta-Function Furthermore, by definition (see Appendix)

where the B, are Bernoulli numbers. Hence, for integral n ,

If n = -2m where m 2 1, this integral is 0 by the residue theorem since the 1 the value of the integral in (3.20) is residue of the integrand is 0. If n = -2m by the residue theorem. Finally, if n = 0, the residue theorem gives -ni for the value of the integral. Furthermore

+

1 e-~is - e-Risr(l - s) r ( s ) (e2"is - 1) 2iI'(s) sin n s 2in

<

and if s is an integer n 0, this last expression is (-",";L~)!. Putting the values of (3.20) and (3.21) into (3.19) for n an integer 0 we get parts (i), (ii) and (iv) of the theorem. Part (iii) of the theorem now follows from part (ii) and Theorem 3.4, for putting s = 1- 2m, m 2 1, in the functional equation we get, using (ii),

<

(-lIrnB2m = c(-2m 2m

+ 1)

= 2-2rnt1 n - ~ msin

(E - a m ) r(2m)((2m)

and so (iii). For (v), we use the argument of Theorem 2.9 which shows that for a

> 0,

Hence, 00

1 lim c(s) - s+l S-1

(

1-lim lagrr--+m

(

r--too

x--)) m

r

1 = lim x - - l o g r = y , r+OOmirm

by the definition of Euler's constant y. To prove (vi), we take logarithms and differentiate in Theorem 3.4, this gives

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate Now,

and so.

and 1 ''(l - ') = - + 7 + . - - (higher powers of - s) ((1 - S) S

.

Also, from the Weierstrass product for r ( s ) ,

and so,

Substituting (3.23) and (3.24) in (3.22) and letting s

+ 0 gives

5

(since in a deleted neighborhood of s = 0, cot ?f is analytic, has a simple pole with residue 1 at 0, and the limit in (3.25) is easily computed as 0). (vi) now follows from (iv) and (3.25). Note 3.5. The reader will notice that we have not evaluated 0 ° 1

- 5(2m + 1) for integral m 2 1 .

n=l

+ 1 in Theorem 3.4 and attempt to use Theorem 3.5(i), we get lim r ( 1 - s ) < ( l - s) . C(2m + 1) = 22m+17r2m (-'Im s+2m+l

If we take s = 2m

At a = -2m, r ( z ) has a simple pole with residue

&,

and so we have

9. Riemann Zeta-Function

426

+

which is interesting, but does not give a "closed form" expression for C(2m 1). It is natural (after Theorem 3.5 (iii)) to suppose that C(2m 1) is a rational multiple + nothing ~ , about even the rationality of C(2m 1) was known until of T ~ ~ but ApBry proved that C(3) was irrational (see for the ideas involved Van der Poorten [238]). A simpler proof based on Legendre polynomials was given by F. Beukers [20le Expressions for the coefficients of the Laurent expansion of <(s) about s = 1 have been given many times, and go back at least to Hardy in 1912; a convenient reference is Chowla and Briggs [44].

+

+

The functional equation of Theorem 3.4 in the symmetric form of (3.11) leads to

Definition 3.1.

(q)

def 1 ((s) = 5 ~ - (i ) ~~ - ~ l ~ ((s) r

,

Theorem 3.6. J(s) is an entire function of order 1 satisfying ((s) = E(1 - s). Proof. That ((s) is entire is clear from Theorem 3.4 (the pole of <(s) at s = 1 is at s = 0 by the zero of s, and the cancelled by the zero of s - 1, the pole of I?(:) at s = -2m, m integral and 2 1, by the zeros of C(s) at s = -2m). poles of ):(?I ((s) = ((1 - S) is a form of the functional equation of Theorem 3.4 (viz. 3.11). In addition, for a > 0,

where K and A are positive constants by a weak form of Stirling's Theorem. Also, by the argument of Theorem 2.9, for a > 0,

and so

and thus for a

> &,say,

where K1 is a positive constant.

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate Substituting (3.26) and (3.27) in the definition of [ ( s ) gives, for a 2

427

4,

where K2 and A are positive constants. Since [ ( s ) = E(1 - s ) , (3.28) also holds for a $, and thus we have

<

and so

-

lim

Isl+m

Log Log lF(s)l < 1, loglsl

<

Thus the order of [ ( s ) is 1. On the other hand, for s real, by Stirling's Theorem

5S logs as s + m

~ o Fg( ; )

and the above arguments then show that for s real Log ( ( s )

-

S

-logs as s 2

-+

m.

Since there is a real point on each circle C(0,r ) , it follows that the order of [ ( s ) is also 1, and so [ ( s ) is of order 1.

>

Theorem 3.7. [ ( s ) has infinitely many zeros, they are all zeros of C(s), and they all lie in the strip 0 < a = R e s < 1, and none of them are real. If p is a zero of [ ( s ) so are p and ( 1 - p) zeros of [ ( s ) . Furthermore, if p runs through all such zeros, then for all s,

where a = b - logn, and b = log(2n) - 1 -

z, (y is Euler's constant).

Proof. By Theorem 3.6, [ ( s ) is entire of order 1 and < ( s )= <(1- s). It follows that S ( z ) = [(% i z ) also is entire of order 1 and S ( z ) = E(-z). So E is an even entire function of order 1; so has a power series expansion with only even powers of z appearing. Hence ~ ( z iis) also entire and of order $. By Theorem 3.1.6, Z has infinitely many zeros, each of which clearly corresponds to a zero of J . Since def 1 [ ( s ) = Z ~ - (l)x-$ ~ F($)<(s),it follows that these zeros of [ ( s ) are all zeros of C(s) since I?($) has no zeros and a pole at 0, and [ ( s ) has a pole at 1.

+

9. Riemann Zeta-Function

428

By Theorem 1.2, ('(s) has no zeros for u > 1. By Theorem 3.4, ~ ( s= ) 2'7rS-l sin

r ( 1 - S)C(I - s) .

(3.29)

Hence, C(s) has no zeros for u < 0 except those at the negative even integers since r ( l - s) has no zeros and sin has, for u < 0, zeros only at the negative even integers (and they are simple). So ((s) and ('(s) have infinitely many zeros (the same ones) in the strip 0 5 u 5 1. By Theorem 3.1, ('(s) # 0 for u = 1. Hence by (3.29), ('(s) # 0 for u = 0 except possibly at s = 0. But by Theorem 3.5(iv), ('(0) = -+. Also, since for R e s = u > 1, m

for Re s = a

.

> 1, we have

and so,

The series on the right side of this last equation converges for u > 0, and thus is a representation of the left side in this region. In particular for s real, 0 < s < 1,

&

for all positive integers k. since for s > 0, -> That along with p, 1 - p is a zero follows from the functional equation (3.29). Since for u > 0, n= 1

..

we have C(s)= ('(3) for u > 0, and so again by (3.29) for all s. Hence p is a zero along with p. Since [(s) is entire of order 1, if p runs over the zeros of ((s) in some order, C , converges for every a > 1 (by Theorem 3.1.2), and by Theorem 3.1.3,

5

where K and a are constants. Hence

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin's Estimate

429

and since [(O) = -$, K = $. Letting a + $ logx = b, from (3.30) then, Log ((s) = - Log 2+bs- Log (s-1)-

Log I' (1 + S)+C 2

P

and so

Hence, by Theorem 3.5(iv) and (vi), and since T"(1) = -7, we get on taking s = 0 in (3.31), l o g 2 r = b + l + - Y. 2 Note 3.6. Since whenever p is a zero of t(s), so is 1 - p, if we pair each zero p with the zero 1- p, we can write the product of Theorem 3.7 as

where the zeros p with I m p > 0 are arranged in non-decreasing order of their imaginary parts, say. Since, from Definitions 3.1,

the formula (3.32) gives

(where the product trivially converges since ((s) is entire of order 1 (Theorem 3.1.2)). Since ((s) = ((1 - s ) , (3.32) and (3.33) give

9. Riemann Zeta-Function

430 and so, since ((1) =

i,

p paired w i t h 1-p

We now come to De la Vallke-Poussin's estimate of a zero-free region for C(s).

Theorem 3.8. There is a constant A

> 0 such that ((8) # 0 for 1-

< a.

+

Proof. Let p run through the non-real zeros of ('(s) and put p = ,6 iy, P, y real. The use of log(lyl+ 2) and The idea of the proof is to show that P < 1log(ltl+ 2) instead of log ly( and log It1 is to ensure that all logarithms are positive. Erom Theorem 3.7,

6.

Since ('(s) has a simple pole at s = 1, we can find a to so that C(s) # 0 in the square la - 11 5 to, It1 to(s = a + i t ) . Then for a 2 1, It1 > to, we have, taking real parts in (3.34) (with p = P iy),

<

+

where Al is a constant, since ,8 = Rep

< 1 by Theorem

3.2, and

uniformly in largzl 5 7r - S. (This last can be proved by taking the logarithm of the Weierstrass product for r ( z ) , differentiating to obtain

r'o = -y - -1+ Z

lim T+oO

9.5'. The Functional Equation, the Prime Number Theorem, and De La

Valle'e-Poussin's Estimate

431

En,

and using partial summation and that lim,,,

- log T

= y, and

+

Also, if Po iyo is any particular fixed non-real zero of <(s), then similarly t o (3.35), we have (since [yo\> [to[)

On the other hand, we know that for a

> 1,

1 p

(Theorem 1.3);

m=l

and

We now argue as in the proof of Theorem 3.1 since C(s) and <'(s) are real for real s (which is completely trivial for a > 1). We have from (3.37) as in the earlier proof,

--3<'(0)

+

4Re (-<'(a

<(a) 00

=

C p

m=l

<(a

+ it))

+ it)

+

Re (-('(a <(a

+ 2it))

+ 2it)

1% P (3 + 4 cos(mt logp) + cos(2mt logp)) 2 0 ,

(3.38)

pmu

for a > 1, and all t. We now take t = yo > 0, say, (and so yo > to) and apply (3.35) t o the third term and (3.36) to the second term on the left in (3.38), and thus get

o<-- -3C(a)

[(a)

4 ( a - Po)

+ A2 log(yo + 2) ,

for a > 1, and any particular zero Po constant.

+ iyo,yo > to of <(s), where A2 is a positive

9. Riemann Zeta-Function

432 On the other hand, (Theorem 2.9)

=$#

has a simple pole at s = 1,with residue

- 5 is bounded for 1 < a < a , a > 1. Thus, on dropping the

1, and so subscripts,

where A3 is a positive constant, holds for all a 2 1 and any non-real zero P 0 < p < 1 , y > to, of C(s). The inequality (3.40) can be written as

+ iy,

and it is easy to see that the right side has its minimum at a = 1 + *,

9and , a = 1+

and so taking Aa = non-real zero ,8 iy, 0

+

A

we obtain from (3.41), for any < ,B < 1, y > to > 0, of C(s),

where A5 is a positive constant, and this proves the theorem since if p is zero, so is p by Theorem 3.7. We also need to know how fast

grows in the zero-free region.

Theorem 3.9. There is a sequence of real numbers Tm with m such that - - - O((log t)2) , C(S) Proof. The proof depends again on Theorem 3.7 We obtain analogously to (3.35)

>

for a 1 and It1 Taking s = 2

> to. + iT, T > 0, in (3.42) gives

< Tm < m + 1,

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin'sEstimate

for some positive constant C2, since

as T + oo, by Theorem 1.3. But

and so we get from (3.43)

whence (since 0 < /3

< I),

From (3.44) follow immediately: The number of zeros /3

is O(1ogT) as T

+ i y with IT - yl < I

+ oo, 1

and p=P+iy

( T - rl2

=O(logT), a s T + o o ,

and the same estimates clearly hold for negative T with T replaced by ITI. Also, by Theorem 3.7, for -1 < a < 2, and t not the ordinate of a zero, <'(s) <(s)

+

C1(2 it) - -- 1 <(2 + i t ) s-1

1 1 r l ( i + $1 + ----1+ zt 2 r ( l + $)

and so making the same estimate as before,

433

9. Riemann Zeta-Function

434

for a positive constant C3. (The rearrangement of the series is justified since - -= (2+:t+-ij)(z-p) and so the series Cpconverges absolutely, since J has order 1). The second sum in (3.46) can be estimated as follows. We have for -1 o 2, t not the ordinate of a zero,

&-

< <

and so by (3.45), the second sum in (3.46) is O(1og t) as t Furthermore, since 0 < /3 < 1

+ co.

Substituting these estimates in (3.46), thus gives

for -1 < a < 2, t not the ordinate of a zero. Now let urn be the number of non-real zeros p = P iy of [(s) with m < y = I m p < m 1. If we divide the interval (m, m 1) into urn 1 equal parts, at least one subinterval contains no ordinate of a zero, y, in its interior. Let Tm be the and SO mid-point of such an interval. Then ly - TI, 1 &,

+

C ,

+

1

P

ITm--yl
= O(1og T,)

+

max ITm-yl
+

1

+ (Tm - y)2

(3.48)

But, clearly, urn

+ 1 < 1+ { number of y with ly - ml < 1) = O(1ogm) = O(logTrn) .

Thus (3.48) becomes

9.3. The Functional Equation, the Prime Number Theorem, and De La Valle'e-Poussin'sEstimate

435

and so taking t = T, in (3.47) and substituting this estimate therein proves the theorem.

Theorem 3.10. As x

+ m,

where a is a positive constant. Proof. The result for q!~follows from Theorems 3.2, 3.8, 3.9. The result for follows from Theorem 3.3.

T

then

$$

Note 3.7. In Theorems 3.8 and 3.9 a zero-free region and an estimate for are obtained by using Theorem 3.7, and thus the analytic character of ((s) in the whole plane. There is another method for obtaining such results, originating with Landau, in which the analytic character of <(s) for a 5 0 need not be known. It depends on the Borel-Carathkodory Theorem (see Appendix). An exposition of this method with application to ((s) can be found in Titchmarsh, Theory of the Riemann Zeta-Function [230], especially in Sections 3.9-3.11. ) is known, appears to be The best estimate for ~ ( x which

obtained by Richert using complicated estimates of exponential sums due to Vinogradov and Korobov. It does not seem as though this method will allow for elimination of a log log x factor. See Walfisz, Weylsche Exponentialsummen in der Neueren Zahlentheorie 12431, especially Chapter V and pages 226-7, and on the last point, page 188. Repeated integration by parts of J . &dt produces

Lx

+ constant + ~ c !

-

(log:*+l

.

Since for fixed k the last integral ( , o ~ $ k + , as x i m, this produces an asymptotic series for J: &dt (which clearly does not converge). Theorem 3.10 gives an error estimate for a ( x ) - J," &dt which is better than truncating the series at any term. A natural question (raised already by Theorems 3.2 and 3.3) is to what extent ) &dt imply results about zero-free regions results about error estimates of ~ ( x-J: of ((s). A first step in this direction is the following observation of Ingham.

436

9. Riemann Zeta-Function

Theorem 3.11. Assuming n(x)

- &,

it follows that ((s)

# 0 on a = 1.

%

= 1 implies lim,,, = 1. Proof. By Theorem 1.4, lim,,, For a > 1, we thus have by Theorem 1.3 and partial summation

Hence, for a

> 1,

%

Since by Theorem 2,9(b), is analytic in a > 0 except for simple poles a t the zeros of ((s) and a t s = 1, and the pole a t s = 1 has residue 1, g(s) is analytic in a > 0, except for simple poles a t the zeros of ((s). As already observed, our hypothesis implies $(x) = x + o(x) as x + GO; hence given E > 0, l$(x) - XI < EX for x 2 xo = xo(e), and so (3.49) gives for a > 1,

where K is a constant, possibly depending on E, but not depending on a . So, for a > 1,

+

-('(a it)(a - 1) a +it - 1 ( a it)((a it)

+

Letting first a

+

+ 1+, and then e + 0, we get from (3.50) for t > 0, lim

+

So, f o r t

I (' ++

( a it) ( a - 1) ( a it)((a + it)

> 0,

lim

.+I+

('(a

+ it) ( a - 1) = O . ((a + i t )

+

a

+ i t - 1I = o ,

ift20.

9.4. The Riemann Hypothesis

437

+

But if c(s) had a zero at s = 1 it, t 2 0, then s = 1 it and so

+

lim

+

( a - l)C1(a it) = [(a + i t )

,-ti+

lim

,

$$/ would have a simple pole at

(s - (1

~--tl+%t ~ a r a l l e lto real axis

C1(s) + it)) C(s)

+

residue of the simple pole at 1 i t which is not 0, contradicting (3.51).

Note 3.8. Theorem 3.11 becomes more interesting after the work of Selberg and Erdijs in 1948 which established a purely elementary proof of the prime number theorem as an asymptotic result, free of any mention of complex numbers. Indeed, Theorem 3.11 was a reason which was previously advanced by those who thought no such elementary proof possible. It is natural to ask now to what further extent results like Theorems 3.2 and 3.3 can be "reversed"; this was studied by Turan [232] who proved the theorem: If there is a number a,0 < a < 1, and a positive constant a such that

then <(s) # 0 in a region of the form

K1 and Kz positive constants, K2 depending on a . Thus, if one could prove Theorem 3.10 by some other means, not involving c(s), a consequence is Theorem 3.8. The best result so far obtained by "elementary methods" (i.e. those which do not use c(s) or any functions of a complex variable) seems to be

obtained by Laurik and Sobirov in 1973. While the consequence for an area where C(s) # 0 is far weaker than what is known (or proved above), and involves a far more difficult approach; that this sort of result is obtainable by this sort of approach seems astonishing.

9.4

The Riemann Hypothesis

One of the most famous of unsolved problems of mathematics was originally posed by Riemann who considered it "sehr wahrscheinlich" ("very probable") that all the roots of Z(s) were real, but he said that he had no proof. Since 5 ( s ) iz),

sf~ (+4

9. Riemann Zeta-Function

438

Riemann's "very probable" result, known since as the Riemann hypothesis, is the same as saying that all the zeros of [(s), or, equivalently all the non-real zeros of <(s), lie on the line a = Riemann said he had failed in attempting to provide a rigorous proof, and no one else has succeeded since 1859. We present here a few basic results about the non-real zeros of <(s).

i.

Theorem 4.1. <(s) has infinitely many zeros on the line Re s = 4. Proof. As observed in the preceding introductory remarks, it is enough to show that E(z) has infinitely many real zeros. By formula 3.10 above

(The reader will recall that this formula was proved using Theorem 8.5.10 for the Jacobi Theta-function 03(01ix).) Since [(s) = i s ( s - l ) ~ - ~ / ~ r ( s / 2 ) < (and s ) , S(z) = ~ ( 4 iz), we have from (4.1)

+

-i(z)

=

-

2 2

- 1(z2 + 2

- ( 2 +

;)

)

-

00

x-3/4

c 00

+

Cr=le-n2rx

a) l

)

00

e-"'"'dx n=l

e-n2r~

cos

n=l

For convenience, let @(z) = (4.2) takes the form

1 ;(z) = - - 2 (z2 2

(x-3/4+Y + x-3/4--Y

=

(:

(4.2) log x) dx .

w, O

and take x = e2". Then

00

e"12@(e2")cos(uz)du .

(4.3)

Integrating by parts, we have

since eul2@(e2")sinuz 03

+ 0 as u -+

m, and integrating by parts again gives

-$

But,

I

00 1 d eu/2@(e2u) cos uz du = -(cos uz) -(e"/2@(e2u)) z2 du o d2 - l W ( e o suz) -(e"/2@(e2u))d~ du2 .

(4.4)

9.4. The Riemann Hypothesis ~ "u)t and e5u/2@'(e2u) cos uz Also, e ~ / ~ @ ( ecos

439

+ 0 as u + co. Thus (4.4) becomes,

But (cf. First Proof of Theorem 3.4) by Theorem 8.5.4,

Hence, also,

From (4.7) we get

Substituting (4.8) in (4.5) and then (4.5) in (4.3), gives

For convenience, let

Since @(z) is analytic if Re(z) > 0, @(u) is analytic if cos(2 I m u ) lImul < 2. Also, from (4.9) by the Fourier Integral Theorem (see Appendix)

> 0, i.e.

and, consequently, also a ( * ) ( ~=)

&

00

-

dk (cos ut) dt . du

c(t)

From (4.10), @(u)is an even function, and if we then let for

-2 < v < 2 ,

if

9. Riemann Zeta-Function

440 we have, by (4.11),

Suppose now that z ( t ) had only finitely many zeros, then Z(t) does not change sign for t > T, say, and without loss of generality, we may assume E(t) > 0 for t > T. Then 00 T+2 T (2rr)(2n)!hn = =(t)t2"dt > 2(t)t2.dt ~~(t)lt~~dt

1

ST+l

1

>

where A is a positive constant. Hence, for all n sufficiently large, say n no,c2, > 0. Thus, for n no, and -$ < v < 2 , @(2n)(iv)equals a power series in which all the powers are even and the coefficients positive. Hence, for n 2 no,

>

lim v-+n/4-

(iv) > 0

v real

By definition,

On the other hand, by the argument leading to (4.6), if Re z

Taking z = e2iw,we have 1 qe2i") = e-i"0(e-2i") + -(e-i" - 1) . 2 Let e2iv = i w, so that as v + 7 ,v real, w -+ 0. Then by (4.15),

+

n= 1

n even

n=l

> 0,

9.4. The Riemann Hypothesis

44 1

It follows easily that as w + 0 along any path in the angle 1 argwl < 5 (i.e. Re w > 0) O(i + w) + -:, and all derivatives of O(i w) + 0. Hence for v real, -2 < v < 2 , we get from (4.14) that @(iv)and all its derivatives -+0 as v -+ 2 , contradicting (4.13). Hence Z has infinitely many real zeros, which proves the theorem.

+

Note 4.1. Theorem 4.1 was first proved by Hardy in 1914 [96]; the above proof is due to Polyd in 1927 ([I991 (though Edwards in his book ascribes it to Hardy). There are now some half-dozen methods of proof (see Titchmarsh op. cit. Sections 10.1-10.6 for these). Formula (4.9) for E(z) is due to Riemann in 1859 (with u = logz). The facts about B(z) = used are also 19th century results. It is of interest to know also how "densely distributed" the non-real zeros of C(s) are (whether or not the Riemann hypothesis is true). To study this question we make Definition 4.1. N(t) and

gfthe number of zeros of C(s) with 0 < CJ < 1 , O < t < T ,

Definition 4.2. If T is not the ordinate of a zero, def 1 S(T) = -Ac(argC(s)) 7r

,

where Ac(f (z)) indicates the change in the value of f ( z ) from initial point to terminal point of the Jordan arc or curve C, and C in this case is the path consisting i T (and where arg C(2) is taken to be of straight lines joining in order 2,2 iT,

+

+

0). If T is the ordinate of a zero S ( T ) def = limt_,TS(t). sEC

Diagram 9.6 The reason for Definition 4.2 is

Theorem 4.2. T T 7 N ( T ) = - Log (-) - 27r 27r 8

1 + S ( T )+ 0(-) , as T -+ oo . T

9. Riemann Zeta-Function

442

Proof. Suppose T is not the ordinate of a non-real zero of c(s). Let R be the positively described rectangle with vertices fT f gi. Then Z(z) has 2 N ( T ) zeros in the Jordan interior of R and none on R. (If z = a iP, with - $ < P <

+

+

i,

-T < a < T , then i + iz = - ,B i a , and so since E(z) = <(; + iz), and -1 < 4 - P < 2, these are just the zeros of <,i.e. the non-real zeros of C(s) in the strip 0 < a < 1, -T < t < T . Since if p is a zero of C(s) so is p, the number of such

5

def

zeros is 2N(T).) So 1

E1(z)

dz .

Since N ( T ) is real, we have

Theorem 3.4, the functional equation for C(s), translates to E(z) = E(-z) (see is beginning of proof of Theorem 3.7). Thus Z(z) is an even function, and so ~, an odd function. and so Furthermore, since C(S) = C(s)and F(s) = r(s),we have <(s)= 1 1 '(2) = <(Z. iz) = <(z. iz) = <(k - Im z - iRez) = Z(i Im z - Re z) = Z(-Z) =

%

-

+

+

E(i). Consequently, also

m,

% = (=(.,) .

5

These properties of mean that the expression on the right in (4.16) can be expressed as four times the similar expression where the integral is taken over one-fourth of R. Forma;lly, this may be seen as follows, taking z = x iy, we have:

+

since

- is odd, and also

Similarly,

9.4. The Riemann Hypothesis and -T-3i/2

=-1

-312

=I

+ iy) dy + iy) -312 Z1(T + iy) Re 5 ( T + iy) d~ El(-T E(-T

-

=I(T iddy = Re E(T - iy)

ST

-

T-3i/2 =I (z)dz = Im = R e i l dz 5(2) T-3i/2 E(z)

.

Taking (4.17)-(4.20) together in (4.16), we get

Since -iE1(z) =
Now, by definition S

<(s) = -(s - i ) ~ - rf 2

< and putting s = $ + i z in

(5) ((s) , S

and we have:

Ac (arg(s - 1)) = arg

Thus, we have from (4.22),

a r g ( i + i ~ )+arg(-:+i~)

T -Il~gn+Ac

(a r g r (f)) + AC a%(<(s))

444

9. Riemann Zeta-Function

+

+

+

1 iT) = arg(- - T 2 ) = lr. TO compute Ac arg I?(;), since arg(H iT) arg(we need a strong form of Stirling's formula. We have

AC arg r

(i) Im =

Log I?

+ (k) iT

T +zlog(l+-&)

= Tlog(?) 4

-f

(;+o($))

-'+o(f) 2 T =-log(;) 2

T -l -r -2

8+0($),

+ (i)

+ A)

since for 1x1 > 1, arctanx = 5 0 as x -+ co,and log (1 = 0 (&) as T + co. Substituting (4.27) into (4.26) gives the theorem for T not the ordinate of a zero. For T the ordinate of a zero, the theorem follows from the definition of S(T) and since the 0 ( * ) term is easily seen to be continuous. We thus need to know the size of S(T). This is given by

Theorem 4.3. S(T) = O(1ogT) as T -+ cm. Proof. By formula (3.48),

+

C ( a it) ((u+it) -

C

1 o+it-p

+ O(l0gt) .

Now, as earlier remarked, from 3.44, we have The number of zeros P f i y , 0 < ,6 < 1 of ((s) with It-yl Now,

< 1 is O(1ogt).

(4.29)

9.4. The Riemann Hypothesis

445

For the first integral in (4.30) we have by Theorem 1.3 (and A(1) = 0)

2+iT

nslog n

= O(1) .

Thus, we have, by (4.28),

But, by (4.29)

+

on making the change of variable u = IT - ylu, and since J-wm Substituting this in (4.31) gives the theorem.

&=

T.

Combining Theorems 4.2 and 4.3 gives immediately Theorem 4.4. N ( T ) =

& log & - & + O(1og T).

Note 4.2. Since no better estimate on S(T) is known than S ( T ) = O(1ogT) the 718 in Theorem 4.2 may seem a t first a bit otiose. However, in 1922, Littlewood [146] showed that :J S(t)dt = O(logT), and independently F. and R. Nevanlinna

9. Riemann Zeta-Function

446

+

dt = K o ( Y ) where K is given explicitly. The Nevanlinna [I711 that brothers use Jensen's formula (Theorem 3.1.1) and a Green's function argument. Their result (with an undetermined K ) follows from Littlewood's. A proof that S(T) = O(1ogT) using Jensen's formula can be found in Titchmarsh (12301). Theorem 4.4 was stated by Riemann in his original 1859 memoir; the first proof was given by von Mangoldt [240]. This is essentially the proof given above (see also the Chelsea reprint edition of Landau's Handbuch [I391 (op. cit), Sections 90-92 and p. 937). Landau proves a "general" theorem in Section 92. The proof given more customarily today is due to Backlund 1131, and this is the proof in Titchmarsh's book. Pages 355-375 of Backlund's paper are devoted to the explicit estimate S ( T ) < .I37 log T .443 log log T 4.35 for all T > 2. If the Riemann Hypothesis were true, then S(T) = 0 (,$I",:, ; this and many further results on S(T) can be found in Titchmarsh, [230]. If the Riemann Hypothesis were true, then Theorem 4.1 could be greatly refined O(1ogT) zeros on the line a = 112 since then there would be &log with ordinates between 0 and T . Denoting the number of zeros on the line a = 112 by No(T), in 1921 Hardy and Littlewood proved that No(T) > AT for some positive constant A. This was improved in 1942 by Atle Selberg who showed that N,(T) > cTlogT where c is a computable (but, as it turns out, small) positive constant. In 1974, Levinson [I431 showed that more than 113 of the zeros of ((s) lie on a = 112. In a 1980 dissertation at the University of Michigan, Brian Conrey refined Levinson's method to show that at least 35.87% of the zeros of [(s) lie on a = 112, and has also established that more than 72% of the zeros of <'and more than 82% of the zeros of J" lie on a = 112. Indeed, Conrey establishes a general formula for the number of zeros of J(m)(s) with real part = 112. There are many equivalent formulations of the Riemann hypothesis known. The flavor of some of these is indicated by the fact that the Riemann hypothesis is true if and only if any of the following hold: converges for Re s > 112, where p(n) is the Miibius function (a) CTz1 (Example 2.1 (iii)). (b) Let {r,), Y = 1,.. . ,N , be the non-zero elements of the Farey Series of order N arranged in order (see Definition 2.2.2, above, and the Appendix). Let 6,(N) = r, - $. Then, for every c > 0

+

+

1

&+

9

xF=,

, then for every r > 0, F ( r ) = O(X$+'). ( e ) Let F ( x ) = (a) is due to Littlewood, (b) to F'ranel, (c) to Marcel Riesz; references to these and other conditions may be found in Titchmarsh, (2301. For (b) Chapter 9 of Huxley [119], The Distribution of Prime Numbers, may also be consulted. Some consequences of Theorem 4.4 of interest follow.

9.4. The Riemann Hypothesis

447

Theorem 4.5. For any fixed h, N ( T is a constant depending on h.

+ h) - N(T) 5 AlogT, as T + co, where A

Proof. Subtraction in Theorem 4.4. Theorem 4.6. If we arrange the zeros of c ( s ) , p = ,b + iy, with y sequence p, = B, ir, so that y,+l y,;then as n + CQ, lpnl yn

+

- >N.

>

0, in a

Proof. Clearly there are only finitely many zeros with a fixed value of y, and arranging these in any fixed order, the above sequence is well-defined (a multiple zero is counted multiply). y, as n + m . Since 0 < P, < 1, and y, is unbounded as n + co, clearly Ipnl Since by Theorem 4.4, N

N(T)

T log T 2n

-a s T + c o ,

But N(yn - 1) 5 n Hence, as n

4

5 N(yn + 1) .

co,

+

y, log y, = 2nn o(n) , log y, log logy, = log(27rn) o(1) , log(2nn) log n 1 = lim = lim n-tm logy, n-+oo logy, '

+

+

and so, substituting the last of these in the first, 2nn

Y ' n A,

il~n+o~. log n

Theorem 4.7. With the notation of Theorem 4.6, ly,+l - y,l = O(1) as n

Proof. By Theorem 4.3, there is a positive constant c such that

By Theorem 4.2,

-+co.

9. Riemann Zeta-Function

448 So, for any H

> 0,

2 27r log

(g)

- c(log(T

+ H) + log T ) + 0

Taking T = yn, and H = yn+l - yn, we get

F'rom Theorem 4.6, it now follows that %+I - Yn

<

2nc(logyn+l + logy,) + 2.ir + O ( 2 ) = O(1) , log yn - log 27r

Theorem 4.8. With the ordering of Theorem 4.6,

Proof. Using Theorem 4.5, we have,

Note 4.3. Theorem 4.5 can also be proved directly by using Jensen's Theorem, as :l diverges. Theorem in Titchmarsh [230], p.178. Theorem 4.6 shows that C= 4.7 says that the gap between the ordinates of the zeros of C(s) is bounded. Far more is known. Littlewood showed that there was a To such that for every T 2 To, <(s) has a zero 8 i y such that 1 y - TI < l o ~ , o g T . (See Titchmarsh [230], Section 9.11). In particular, it follows then that y,+l -yn + 0, and in fact, at least as fast lo:logn (take T = -). If the Riemann hypothesis should be true, then as Montgomery [I611 has shown that the stronger result, lim, ,,(T,+~ - 7,) log yn < 1.36 is true; and also that limn,,(yn+l -7,) logy, > 27r. Montgomery conjectures

+

,,,

,,,

9.4. The Riemann Hypothesis

449 -

that lim, ,,(Y,+~ -7,) logy, = 0; limn,,(yn+l -7,) logy, = m, but this is not known, even assuming the Riemann hypothesis. Titchmarsh (12301 Section 9.14) has shown that the counterpart to Theorem 4.5 is true. Namely, given h > 0, there is a positive constant B = B(h) and a To such that for T > To, N ( T h) - N(T) > BlogT.

+

Note 4.4. One might also enquire as to the further relationship between estimates for prime numbers and further information concerning the Riemann zeta-function. Von Mangoldt proved in 1895 that, with +(x) defined as in Definition 1.2, and p running over the non-real zeros of C(s) with p and 1- p paired (i.e. the numbers p are arranged in order of non-decreasing J I m pl), if x is not an integer, then +(XI = x -

c2 P

P

- log (I 2

-$)- log(2a) ,

and

def

Sox

1

where Li(x) = -dt (defined as a Cauchy principal value), although the second formula was stated by Riemann in his 1859 memoir. These are known as "explicit : &dt formulas" in prime number theory; Li(x) differs from S - by a constant,

&dt = 1.06.. . ,. Li(2) = If x is an integer, then (4.32) and (4.33) remain correct if then the left sides respectively. R o m (4.32) can are replaced by $(x) - iA(x) and P ( x ) X ) ~r ()x ) = be deduced that if O = sup, Rep, then +(a) = x O ( X @ ( ~ O ~and &dt O(xQ logx). In particular, if the Riemann hypothesis is true, a ( x ) =

e, +

St

+ J; &dt + 0 ( x ) logx). These results (which only have substance if O < 1) were proved by von Koch in 1901. In 1914, Littlewood showed for the first time that r ( x ) - Li(x) changes sign infinitely often, (it is negative for 2 < x < lo8); +(x) - a: # 0(x4 logloglogx); r ( x ) - Li(x)

#0

x B log log log x

These results are independent of any hypothesis (stronger and simpler results, due to Erhard Schmidt are, in fact, true if the Riemann hypothesis is false). Assuming the Riemann hypothesis, the gap between Littlewood's results and von Koch's has never been closed. One may also ask for the first time that a($) -Li(x) changes sign, or how often a sign change occurs. For example, it is known that ~ ( x-) Li(x) > 0 for more than 10500integers between 1 . 5 3 ~ and 1 . 6 5 ~ (Lehman [140].)

9. Riemann Zeta-Function

450

Previous to Lehman it was known that ~ ( x-) Li(x) was positive somewhere before the truly enormous "Skewes number", exp exp exp exp(7.705). Lehman's work is a combination of theoretical results and machine computation. Lehman suggests that perhaps n(x) - Li(x) < 0 for 2 5 x 5 lo2'. Applying Mobius inversion to the definition of P in equation (4.33), one has

. results, in particular, and this leads to an explicit formula for ~ ( x ) Littlewood's show that the terms arising from Li(xP) in this inversion "conspire" to L'overwhelm" the negative term - i ~ i ( x i )(this is not completely surprising since a t least one of p and 1 - p has real part Nevertheless, e ~ i ( x 4 is) up to lo7, a surprisingly good approximation to n(x), with + ~ i ( l 0 5 ) - n(107) = 88.

> i).

Appendix This Appendix contains some proofs and discussions concerning the following which are referred to as in the main body of the text. Each section is independent of the others.

1. The Area Theorem 2. The Borel-Carathkodory Lemma 3. The Schwarz Reflection Principle

4. A Special Case of the Osgood-Carathkodory Theorem

5. Farey Series

6. The Hadamard Three-Circles Theorem

7. The Poisson Integral Formula 8. Bernoulli Numbers

9. The Poisson Summation Formula

10. The Fourier Integral Formula 11. Carathkodory Convergence

Appendix

1. The Area Theorem Theorem. Suppose $ is univalent in B(0,r ) and $(0) = 0. If J is the area of the 03 image of B(0,r ) under 4, and if $ ( z ) = Cv=lavzU,say, for z E B(0,r ) , then

Proof. Let z = x

+ iy, $ ( z )= u + iv = u ( x ,y ) + i v ( x ,y ) . Then

Since $ is analytic in B(O,r), by the Cauchy-Riemann equations u, = v, and hence

u y = -21,;

Letting z = peis and changing to polar coordinates, we get

Since $'(z) = functions),

~ a ~ z "by - ~Parseval's , Theorem (which is trivial for analytic

and so

x

r m

J = 27r

[

x 03

v2/av/2p2v-1dp = 7r

vla,,12r2" .

We may note that if an area J is given, then the theorem is in effect, a condition on the mapping function $.

2. The Borel-Carathhodory Lemma Theorem. If f ( z ) is analytic in B(0,R ) , M ( r ) = maxl,l,, maxl,l,, R e f ( z ) , then for 0 < r < R ,

I f ( z )1 ,

and A ( r ) =

and, if further, A ( R ) 2 0 , (ii)

Proof. If f ( z ) is constant, trivial.

453

Appendix

Case I: f (a) non-constant, f (0) = 0. Then A(0) = 0 and A(R) > A(0) = 0. (Recall that A(r) is monotone strictly increasing, as can be proved by applying the maximum modulus principle to e f ( ~ ) . ) Let

(i)

then for z E B(0, R), $ is analytic and 4(0) = 0. Furthermore,

since, clearly, for z E B(0, R), we have 12 A(R) - Ref (z)l Applying Schwarz' Lemma to $(z) gives

for all z E B(O,r),O 5 r < R. Thus for z E B(O,r), l#(z)l

and so (since l$(z)l

Case

> Ref (2).

< 5 . By (0.1)

I 6 < I), for z E B(0, r ) ,

11: f non-constant, f (0) # 0. If we apply Case 1 to f (a) - f (0), we get

and the result now follows since -Re f (0) 5 1 f (0)l. This proves (i); to prove (ii) we have if lzl = r < R, and l? = C(z, $(R - r)),

454 Since C E C(z, ;(R - r)), gives for 1.z1 = r in (0.2)

Appendix

I< - zl = $(R - r).

Thus, since A(R)

2 0, applying (i)

As these proofs indicate, there are many variants of (i) and (ii) available, both for the bounding functions, and for inequalities involving the minimum or maximum of the real or imaginary parts of f (2). However, any bound must involve a term involving f (0) since otherwise we could make the theorem false by considering f (z) +ik with k a sufficiently large real number. Furthermore, an upper bound such as the one obtained must approach oo as r + R (for consider f (2) = -i log(1- z), for 0 5 r < R < 1; we have f(0) = O , a n d M ( r ) + oo a s r + 1, but A(R) < ,; however near R is t o 1).

3. The Schwarz Reflection Principle Theorem (Theorem A). Let r be a circle (or straight line), and D a region in the plane such that D n r = 0, but BdD n r = ro, a (non-degenerate) subarc of I?. Suppose f is analytic in D and continuous in D U ro.Suppose the values f takes on l7 all lie on a circle (or straight line) r*.Let D+ be the inverse of the region D with respect to r. Then f can be continued analytically into D+.If z+ is the point inverse to z with respect to r and w* the point inverse to w with respect to r * , then

Proof. Let g and g* be non-singular linear fractional transformations which map and r*respectively onto the real axis. We may suppose, with no loss of generality, that rois open; i.e. that g(ro) is an open interval on the real axis. The function h def zg*ofog-l is clearly analytic in g(D), continuous on g(D) U g(ro), and real on g(ro). Let

then k is analytic in g(D+); for since W E DS if and only if g-'(g(W)) E D, we have that C E g(D+) if and only if g-l(f) E D , i.e. f E g(D), and so for all

Appendix

455

z in a neighborhood of (, k(z) = h m = x z = o a n ( t Furthermore, k is continuous on g(D+) U g(I'o) and for Define the function G(z) by G(z) = h(z) k(z)

Cln = C;=O 2

cn(z - C)n. E g(ro), k(z) = h(z).

for z E g(D) U g(ro) for z E g(D+) u g(r0) .

If we can show that G(z) is analytic in g(D) Ug(r0) u g ( D + ) , then G is the analytic continuation of h into g(D+) and so since f = g*-I o G o g for z E D, we have that g*-l o G o g is the analytic continuation of f onto D + . And then for z+ E D + , f (z+) = g*-l o G o g(z+), while g*-l (g*(f (2)')) = f (2) .

f (z)* = g*-l(g*(f (z))) = g*-l o ~(g(z)) = g*-l

0

G 0 g(z+) = f (2')

.

To prove G analytic, let zo be a point of g(ro), and p so small that B(zo,p) C g(D) U g ( r o ) U g(D+), and C(zo,p) n g(I'0) consists of exactly two points. Then g ( r o ) divides the closure of B(zo,p) into two pieces, say HI and H2,of which it is the common boundary, say H1 C g(D) and Hz C g(D+). By (the strong form of) the Cauchy integral formula taken over BdHl and BdH2 and adding, we get

It now follows that in fact G(z) is not only a continuous, but an analytic function of z (cf. for example, Knopp, Theory of Functions I [131], Section 16). An important consequence of Theorem A is

Theorem (Theorem B). I f f maps a simply-connected region D onto B(0,l); if the boundary of D contains an analytic arc r o , and if f is continuous on D U Po, then f can be analytically continued across ro.

Proof. Let I denote the unit interval ( 0 , l ) . There is a region R , such that I C R, and a univalent map g of R such that g : I* 0 ro.By Theorem A, f o g can be continued analytically across I, and so f = f o g o g-l can also be so continued. Finally, we should note that by the same technique reflection can take place with respect to any analytic arc. For a fascinating and readable introduction to many ramifications of the Schwarz reflection principle and "Schwarz functions", the reader is strongly recommended t o see Davis, The Schwarz Function and its Applications [53].

Appendix

456

4. A Special Case of the Osgood-Carathkodory Theorem (special case of) In Chapter 2, we require essentially the following special case of the OsgoodCarathbodory Theorem. Theorem. Suppose D is a simply-connected region whose boundary consists of a finite number of arcs of circles or straight line segments, and such that D always lies o n the same side of the corresponding circles or straight lines. Then there is a function f univalent on D and continuous and one-to-one on D which maps D onto B ( 0 , l ) .

Proof. Suppose the arcs of circles or line segments are denoted A, : m = 1 , . . . ,n. Then for a given value of m , there is a non- singular linear fractional transformation 4 = 4, such that 4 maps A, onto the open unit interval ( 0 ,l ) ,and maps D into the upper half-plane; say 4 : D '3S . Let S* = {Z : z E S } . then E = S U ( 0 , l )U S * is simply-connected. By the Riemann Mapping Theorem, there is a unique function $J such that $J maps E onto B ( 0 , I ) , $ ( O ) = 0 , $J1(0)> 0. Clearly $1 ( z ) = + ( Z ) has the same properties. Hence $J I,- GI, and so $J must be real on the real axis. It ) ) D onto the upper unit semi-disk follows that the composite function $ J ( ~ ( zmaps { z : I m z > 0 , lzl < 1 ) also. Clearly o 4 is continuous and one-to-one on D U A, (except possibly at the endpoints of A,). Now the maD $J

maps B(O,1 ) onto the closed left half-plane { z : R e z 5 0 } , with the segment ( - 1 , l ) mapping onto the negative real axis, and the lower left quadrant { z : R e z 0 , I m z 5 0 ) , is the image of the closed upper unit semi-disk. The map 92(2) = z2

<

takes the closed lower left quadrant onto the closed upper half-plane {z : I m z 2 01, with the negative real axis mapping onto the positive real axis. Finally, the most general Mobius transformation mapping the closed upper halfplane onto B ( 0 , l ) is

Thus the map

93 92 91( 2 ) takes the closed upper semi-disk { z : I m z 2 0,Izl < 1 ) onto B ( 0 , l ) (with the boundary of the upper semi-disk going onto C ( 0 , l ) ) .

Appendix But

p2-. (z +

93 o 92 o 91 (z) = eaA'

Z-L

ezA \.-

'

*

2-1)

-/

-\--

-,

- &(z - 1)2

'

where I m a > 0, and clearly we can pick a so that a given point zl is mapped onto 0 and then X so that &(g3 o g2 o gl(z))1,=,, > 0. Call the resulting function w(z). Clearly w(z) is univalent in the open unit upper semi-disk {z : I m z > 0, lzl < 1) (analyticity is obvious and if w(z) = w(w), then =f and here if the sign holds, z = w, while if the - sign holds, zw = 1). Let f (z) = w O 4(z) Then f maps D onto B(O, I ) , is univalent on D , and for a given point zo E D, f (zo) = 0, ff(zo) > 1. Thus f is the unique function (as stated in the Riemann Mapping Theorem) with these properties. Furthermore, f is continuous and oneto-one on D U A, (except possibly at the endpoints of A,), since clearly w is continuous and one-to-one on the closed upper unit semi-disk. However, f does not depend on A, (it is, as observed, the unique Riemann Mapping Function for D with, for a given zo E D, f (20) = 0, fl(zo) > 0). Thus f is, in fact, continuous on D except possibly at the endpoints of the A,. The continuity of f at the endpoints of the A, can be seen as follows. Suppose A, and Am+1 have a common endpoint, and that a is the smaller of the two angles of intersection of A, and Am+l. If 0 < a < n, one can, arguing similarly to the above, by composing a Mobius transformation with z + a"/@, map D into the upper half-plane with A, U Am+l going onto a segment of the real axis, and adjusting parameters as before, see that, in fact, f is continuous on A, U Am+1. If a = 0, one can similarly map D into the strip {z : 0 < I m z < n), with A, going onto a segment of the real axis say, and Am+l onto a segment of the line { z : I m z = n) and their point of intersection at the point at oo; then apply the exponential map to get an image in the upper half-plane and continue as before. Finally, if a = n, the condition that D always lies on the same side of the boundary arcs or line segments means that A, and Am+1 are parts of the same circular arc or line segment, and so continuity is obvious.

5

2,

+

+

5. Farey Series In Chapter 2, the "Farey Series" of order N, F N , is defined by 3jv is the sequence of irreducible rationals a l b in [O,1] such that 0 N ; (a, b) = 1 (for a # 0) ordered in increasing order of magnitude. Use is made of the following properties of these sequences:

5a5b5

(i) If a l b and cld are successive elements of FNfor some N , then

Appendix

(ii) If E, E , are three successive terms of FNfor some N , then, after reducing to lowest terms, ? --- .m + r q n+s We now prove (i) and (ii).

Theorem (Theorem A). If (ii) holds for .TN, then (i) holds for FN Proof. By induction. (i) and (ii) are clearly true for Fl and 5. Assuming then that (ii) is true for FNand (i) is true for FN-1, we need to prove (i) holds for F N . Since we are assuming (ii), if F , $, are three successive terms of F N , then

where A is a positive integer. Also, if $ is a member of &-I, since z , f , $ are consecutive in FN,they are consecutive in 6 - 1 , and, by the induction hypothesis, But then q = N , (i) then holds. Thus we may assume $ is not a member of FNFI. and n and s are both 5 N. If n = N , then (since E and $ are consecutive members of FNand so clearly p N - 1 and m 1 p) we have

<

fi

+ <

and so comes between and $ contradicting the assumption that and $ are consecutive. Thus n N - 1. Similarly, we have s N - 1. Hence n s < 2N, and since in the case under consideration q = N , it follows from (0.4) that A = 1. So (0.3) and (0.4) become

<

<

+

whence np-mq=n(m+r)-m(n+s)

<

=nr-ms.

z,

But, since n 5 N - 1, s N - 1, q = N , and f , $ are consecutive in Fn, it follows that E , are consecutive in F N - ~ ,and so by the induction hypothesis n r - m s = 1, and so

Similarly, again from (0.5) and (0.6),

and so we have the theorem.

459

Appendix

A development of these ideas proves (ii) for all Fn and so, consequently (i) for all .Fn,as follows.

Theorem (Theorem B). (ii) is true for all F n . Proof. By induction. Clearly (ii) holds for .Fland F 2 . Assume (ii) holds for .Fn,1 5 n 5 N - 1. Suppose and are consecutive in F N - ~but , f lies between By Theorem A, (i) holds for and % in FN.We need to show that = n FN-land so, n r - ms = 1. Thus setting

E

F

s.

on solving for p and q, we get

and q

np-h - nhrfnkm-h = m m

=hs+kn.

Since p and q are relatively prime, it follows from (0.9) and (0.10) that so are h and k, and we have p- -- h r + k m q hs+kn' Let

8 = { qr qs

++nm : q, n positive relatively prime integers nn

Then f E 6. Also, every element of 8 lies between and $, and is in lowest terms. This last is true, again since n r - ms = 1 and so a common divisor of qr would divide

+ nm and qs + nn

n(qr

+ nm) - m(qs + nn) = q

r (qs

+ nn) - s(1qr + ~

as well as

m =) n

and q and n are relatively prime by hypothesis. Thus each member of 8 appears in Fn for some n > N - 1. That element of 8 which appears first must be the one with the least denominator, namely the one with q = 1 and n = 1; but this must be f , and so P- m+r -q n+s

460

Appendix

Further Notes on Farey Series: The proofs given that Farey series satisfy (i) and (ii) are due to Hurwitz [117]. There is a proof by Landau which, given a member of 3 n , actually provides a rule for constructing the succeeding member of Fn,and also a geometric proof by P6lya. It is also worth noting explicitly that it is easy t o prove directly that (i) implies (ii). These proofs all may be found along with some historical comments in Chapter I11 of Hardy and Wright, An Introduction to the Theory of Numbers [97]. A caution is in order: although if % are consecutive members of F n , then = E . it is not true in general, that m r = p and n + s = q. Consider, for n+s qr example, 0 1 1 1 2 3 1 34 : 4, 37 3' 4 ' where

F,F,

+

2 l

However, the proof of Theorem B above, in fact shows that if the fraction f makes its first appearance in Fn,then it is true that m r = p, n s = q. Thus, for example, in F4,this is true for the fractions and Farey Series turn up in various places in mathematics, in addition t o their connection with the elliptic modular function p of Chapter 2. We have mentioned in Note 9.4.2 their relation to the Riemann Hypothesis. As discussed by Hardy and Wright, they are related to approximation of irrationals by rationals. The connection with elliptic functions also plays a role in the study of the partition problem of additive number theory.

a

a.+

+

6. The Hadamard Three Circles Theorem The theorem of the title is an extremely useful result which finds application in many parts of complex analysis. We give two proofs and a brief discussion.

Theorem [(Hadamard Three Circles Theorem)] Suppose f is analytic in the anlzl p2}. Let M ( r , f ) as usual, denote maxl,i,, If (z)l. Then nulus A = {z : pl log M(r, f ) is a convex function of log r for pl 5 r 5 p2.

< <

First Proof: By a well-known theorem of Jensen, if g is a real-valued continuous function of a real variable, then g is convex in the interval [a, b] if and only if

+

for all h > 0 such that a 5 x - h < x h 5 b. Since If (z)l is continuous in A, log M ( r ) is a continuous function of logr in the interval pl r 5 p2, and so it is

<

461

Appendix

only necessary to prove (replacing the variables x and h in (0.11) by log r and log k and putting g(1og r ) = log M (r))

or for k

> 1 and any r such that T

P15-
> 1,

and for a given r and Ic

r -
> 1, the transformation 7-

A = {z : k

z -+

$ takes the annulus

< 1x1 < rk}

onto itself, leaving C(0, r ) pointwise fixed. The function

is clearly analytic in A (e.g. Consider the Laurent expansion of f in A). Also, M ( f , f ) = M(rIc.9) and M(rk, f) = M ( f , g )

.

(0.12)

The function F ( z ) = f (z)g(z) is analytic in A and so by the maximum modulus principle IF ( z ) 1 attains its maximum on the boundary of A. But

and also by (0.12) on each of the circles bounding A,

Thus, for all points of C(0, r),

which proves the theorem.

462

Appendix Our statement and proof of the theorem do little to explain its name. Suppose we let rl = pl, 7-3 = p3 and let 7-2 be a point with rl < r 2 < 7-3. F'urther,

let Ml = M ( T l , f ) , M2 = M(7-2,f), M3 = M ( r 3 , f ) .

< r 5 7-3, amounts

Then the fact that log M ( r , f ) is a convex function of log r for r l log M2

,

log M1

(e)

log

(2)

log

(5)+ log M3log ( 2 ) '

log

This explains the name "three-circles". The following proof, which does not depend on a general theorem about convex functions, leads directly to this form of the theorem.

Second Proof: Using the notation of the preceding paragraphs, let g(z) = zXf (z), where X is a constant to be determined later. Then g is not necessarily a single-valued analytic function in the annulus {z : rl 5 lzl 5 r3}, but Ig(z)l is single-valued and continuous in the annulus. By an easy extension of the maximum modulus theorem, it follows that lgl takes its maximum on the bounding circles, and so 1g(r2eie)l 5 max

If

max 1g(z)1, max 1g(z)1 = m a x ( r t ~7~-, 2 ~ ,~ )

IzI=Tl

lzI=~3

(r2eie)15 max

((z)'

I.

(2)'

M.)

.

The optimal choice of X is the one in which the two expressions on the right are equal, i.e.

X= With this value of A, and since M2 is taken on by If (z)l for some z E C(0,r2), we have log M2

< log Ml + log (2)log ( 2 ) 1% ( 2 ) r2 1% ( 2 ) log ( 2 ) = Log M1 log ):( + log M31% ( 2 )

'

Appendix

463

and so the theorem. The first proof given is due to Collingwood [49] and does not seem to be as well known as it should be. The second proof given is a variant of the standard one which originates with Hadamard. The theorem seems to have been discovered independently by Hadamard, Blumenthal, and Faber. There is a three-circles theorem for harmonic functions; also an analogous result is true for the quantity

namely, under the hypotheses of the Hadamard Three-Circles Theorem, log Ip(r, f ) f))'lp = M ( r , f ), the Hadamardis a convex function of log r. Since lim,,,(Ip(r, Three Circles Theorem can be considered as the case p = oo of this result. These may all be proved by Collingwood's method, as in his paper; see also, P6lya and Szego, Aufgaben und Lehrsatze aus der Analysis [200], Section 111, problems 303310. We may also note that lim (Ip(rlf))'lp = exp

P+O

(a12=

log if(rea? 1d6)

,

c"

and Jensen's formula (Theorem 3.1.1) shows that .& log If (reis)ld6 is a convex function of logr. Questions of this sort were also studied systematically from the point of view of subharmonic functions by Monte1 [163], who proves all the above mentioned results, as well as related ones like T ( r ) (the Nevanlinna characteristic, see Definition 4.1.4) being a convex function of log r (Theorem 4.1.4): see especially Section I11 of Montel's paper.

7. The Poisson Integral Formula At various places we require the Poisson Integral Formula for the real part of a function analytic in B(0, R). Theorem. If f (2) is analytic in B(0, R), and for 0 R e f (reis), then for 0 5 r < R,

<

Proof. We give a proof of "Fourier series type". We have for 00

f (z) =

CO

anzn = x ( a n

+ i&)rne"'

r

5 R, u(r,6)

It1 = r, 0

,

5 r 5 R,

=

Appendix where a, = an + iPn. Thus, for 0

< r < R,

x 00

(a, cos n9 - p, sin n9)rn . n=O This series is uniformly convergent in 9 and so multiplying respectively by cos k9 and sin k9 and integrating termwise, we get for 0 r R, and k 2 1, ~ ( r9), = Re(f (rei") =

1

1 2" ;

and

< <

U(T,9) cos kgdg = a k r k

:

u(r, 9) sin k9d9 = -Pkrk 12* since all the terms with n # k integrate out to 0, and

,

~ 2 ' ( C o sk9)'d9 = Also, clearly, i

P ~ K

Taking r = R in these formulae and replacing the dummy variable 8 by 00

4, we have

a

u(r, 8) = x ( a k cos k8 - sin k9)rk k=O rk = ,(ak Rk cos k8 - R' sin kg) R k=O

100

+ ;k=l

rk

1

2rr

u(R, 4) sin kq5 dm sin k8

We can change interchange sum and integral by uniform convergence, but for Olr
Appendix

465

and the theorem now follows. One can also give a proof involving Cauchy's Theorem; (see e.g. Titchmarsh, Theory of Functions, p. 124) [229] but not a proof simply by trying to separate real and imaginary parts in Cauchy's Theorem.

8 . Bernoulli Numbers The Bernoulli numbers B2, may be defined as the coefficients of the power series expansion

which is the definition used in the main part of the text. However, the reader should be warned that other notations are in use. For example, some authors incorporate the power of -1 in (0.13) into the definition, or denote B2, by B,. Thus, when dealing with Bernoulli numbers, it is well to know what definition is in use. The above definition tacitly assumes the fact that

is an even function, which is easily verified. The power series (0.13) in fact, converges for z E B(O,27r), there being poles of & a t z = f27ri. Bernoulli numbers occur in quite varied contexts in mathematics. For example, they are important in the Euler-Maclaurin expansion for Cz=lf (m); they have important number-theoretic properties, for example, the divisibility of their numerators once played a prominent role in the study of "Fermat's Last Theorem", they have interesting congruential and recursive properties (e.g. the fractional part of B2, is given by the formula CP-112m $); they have even occurred in homotopy p prime

theory. While there is a recursive formula for the B2,, there is no explicit compuB6 = tational one. The first few are given in this notation by B2 = B4 = I 30 , 1 B8 = =, . . . . Euler computed the first thirty B2,, and the first one hundred and twenty-five B2, can be found in Knuth and Buckholtz [132], Mathematics of Computation. Introductory material on Bernoulli numbers can be found in Hardy and Wright, An Introduction to the Theory of Numbers [97], sections 7.9-7.10. The standard treatise is Nielsen, Daitd Eldmentaire des Nombres de Bernoulli [176]. Material on the Euler-Maclaurin Sum Formula or Fermat's Last Theorem can be found in Hardy, Divergent Series [94], Chapter XIII, and Ribenboim, 13 Lectures on Fermat's Last Theorem [209], respectively.

i,

&,

Appendix

9. The Poisson Summation Formula This formula says that under suitable conditions on f

Irn

03

v=-03

LW 03

e2"'"" f (x)dx .

f (v) = v=-00

There is also a variant in which the sum on the left and integral on the right are finite. Various sufficient conditions on f exist in the literature. We prove the theorem in the following form which is adequate for the purpose of the text.

Suppose

Theorem.

(i) f (x) is twice continuously differentiable in (-co, co)and f (x) and fl(x) both + 0 as 1x1 + co; (ii)

S-wOOf (x)dx and STm If

Proof. For v

Since J:-

# 0, integrating by

I1(x)ldx both converge. Then

parts, and using (i) and (ii),

I fll(x)ldx converges by (ii) and

not only do we have from (0.14) that

converges, but since interchange of summation and integration is justified,

Appendix Now Q(x) %

c"

eZvaix

-=

c"

cos 2v7rx v= 1 2v27r2 '

v=-00 4v27r p#O is clearly periodic with period 1. Moreover, for 0 5 x sin 2v7rt

00

< 1,

1

v=l

v=l

on using the well-known Fourier series expansion of t - in (O,l), and Crzl $ = $. (Alternatively, one can find directly the (obviously quadratic) polynomial whose Fourier series is C ~ ~ the ~ value " , of the constant will, in this approach, be seen simply to "work out".) Thus (0.15) becomes

Juz,

&

But, by hypothesis (i), C F - , ( f l ( k

+ 1) - f l ( k ) ) = 0, and so we have

and integrating again by parts, this last is equal to

But

5 Jdl

f(x

+ k)dz = k=-ca

k=-co

which converges by hypothesis. In this way we get, f (x)e2""'"dx = v=-w

k=-00

f (k) + f (k + 1) .

468

Appendix

+

+

+

+

But CEO=_, f (k) f (k 1) = CEO=_, 2f (k) f (k 1) - f (k) , and by hypothesis (i), CEO=_, f (lc + 1)- f (k) = 0; hence CEO=_, f (k) also converges and so we have the theorem. It is clear that the Poisson Integral Formula is really a result involving Fourier series. The proof above follows Mordell [165], whose conditions are, in fact, slightly weaker. For other proofs, and formulations under other conditions, see Linfoot [145], Dixon and Ferrar [57]; Titchmarsh [229], p. 443, Problem 26.

10. The Fourier Integral Theorem This result says that under suitable conditions on f , formulas such as the following hold:

1

w

f (x) =

w

f (x) =

I" I"

C O S ( X U ) ~ U f (t) cos(ut)dt sin(xu)du

f (x) sin(ut)dt

There has been a great deal of work on suitable conditions. An important theory, of these and related integrals, valuable in applications of mathematics as well, has been built up. The reader interested in Fourier integrals should consult any standard work such as Titchmarsh, Fourier Integrals 12291. For use in Chapter 9, we need only a very simple version of the cosine formula above, and so limit ourselves to it.

Theorem. Suppose f is a continuous function of bounded variation in (0, oo), and f (t) -+ 0 as t + oo, then

Proof. Clearly it is enough to prove the theorem for f positive and monotone nonincreasing. By the second mean-value theorem, for h > 0 and some T E (T, T h),

+

cos ut dt

+ f ( T + h)

cosutdt

I

5 -. ,

"1"

and so the inner integral in (0.16) converges and the outer integral converges uniformly with respect to u for 0 < X u p. Thus,

< <

Appendix

iP l " cos(xu)du

f (t) cos ut dt =

1" iP f (t)dt

cos xu cos ut du

+

+

sin p(x - t) +

sin p(x t) - sin X(x - t) - sin X(x t) x+t x -t x+t

) dt.

Now again by the second mean-value theorem, since f (t) + 0 as t -+ m , for some 7- > T, sinn(x z t t) sina(x f t) dt = f(T) x ft which tends to 0 as T -+ m . Thus, given E > 0, we can choose T so large that

I I lT

sin p(x - t)

+ t) - sin X(x - t) - sin X(x + t) ) d t l + sin p(x x+t x-t x+t

<e

(0.18) for T > To = To(E),for all values of X and p , 0 Fixing such a T > x, we also have,

< X < p.

Also, by the Riemann-Lebesgue Lemma

and

Thus breaking the last integral in (0.17) at T, taking sight of (0.18) and letting E + 0, we get from (0.17)

X -+ 0 , p -+ m , and then

I"

I" Jpx

cos(xu)du

1 = -f (x) lim 2 pj"

1 f (t) cosut dt = - lim f (x) 2 P+" " sin u sinu 1 ~r -du=-f(x) -du=-f(x), p(x-T) 2 2

dt

I,

and so the theorem. Admittedly, we have, by appealing to the second mean value theorem, implicitly used the fact that a monotone function is differentiable almost everywhere. The reader to whom this is unfamiliar, need only observe that in the application at hand in Chapter 9, the function in question is, in fact, continuously differentiable.

Appendix

11. Carath6odory Convergence

5

Suppose D l , D 2 , . . . , D, is a sequence of domains, D, C, 0 E D,. Let f,(z) be the conformal mapping of B(0,.1) onto D, with f,(O) = 0 and fL(0) > 0. Suppose some disk {w : Iwl < p) lies in D, for all n. Define the kernel D of the {D,) as the largest domain such that 0 E D and each compact subset of D lies in all but a finite number of the D,. The sequence {D,) is said to Carathe'odory-converge t o D if D is the kernel of every subsequence of {D,). We will write this as D, + D. CarathBodory convergence agrees with the usual (topological) convergence of the sequence {D,) if {D,) is an increasing sequence but not in general, not even if {D,) is a decreasing sequence. (See Duren [70]). Theorem. Let {D,) be a sequence of simply-connected domains with 0 E D C .f C , n = 1 , 2 , . . . . Suppose f, (univalent) maps B ( 0 , l ) conformally onto D , (wzth f,(O) = 0 and fA(0) > 0). Let D be the kernel of {D,), and some disk {w : Iwl < p) D n for all n . Suppose f, + f uniformly on compact subsets of B ( 0 , l ) . Then f is univalent on B ( 0 , l ) ; f (0) = 0, f'(0) > 0. Let A = f (B(0,l)). Then A = D and D, + D . Conversely, suppose D, + D(# @) then, f, + f uniformly on compact subsets of B(0, 1). Proof. Let E be an arbitrary compact subset C A. Let I' be a rectifiable Jordan curve in A - E so that E c Jordan interior of r , and 6 the positive distance from E to r. Fix a point wo E E , then for all z E f - ' ( r ) , If(z) - wol 2 6 Since convergence is uniform I f,(z) - f (z)l < 6 for all z E f-'(r) for all n 2 some N. But f, (z) - wo = (f (z) - WO) (f,(z) - f (z)). SOby RouchB's theorem f, (a) - wo has the same number of zeros in the Jordan interior of f -'(I?) as does f (z) - wo, namely one. So wo E D, for all n N and N does not depend on wo. So E c D, for all n N. So A, which is the image of B ( 0 , l ) under f , is contained in D. It follows that for n 2 N, the inverse functions $, = f;l are defined on E and 1. NOWchoose a sequence of uniformly bounded there, in fact clearly I&(w)l I compact sets { E n ) ,with Em C Em+land all Em C A. We can apply a diagonal a t the (4,) that converges uniformly on argument t o obtain a subsequence {$,) 0. compact subsets of A to a function $ analytic on A with $(O) = 0 and $'(0) we have In fact, since f,(O) = 0 and f i l ' ( 0 ) consequently = *,

+

>

>

>

f,(

$'(I)) = lim &(O) = lim n+cc

n+cc

1 1 fzl'(0) = n+wfA(O> lim -- f'(0)

>0 .

To show that $ = f-', fix zo E B ( 0 , l ) with w, = f (z,). Consider the circle C(zo, E) where E is chosen so small that the circle, call it C lies in D. Let r = f (C) and 6 the distance from wo to I?. Then I fnk(z) - f (z)l < 6 for z E C for all k some ko, while If (z) - wol 2 6. Hence, it follows from Rouche's Theorem that for some zk in B(zo, E ) , fnk(zk)= wo SO zk = $nk (WO)and Izk - zol < E. SO

>

471

Appendix

for all k 2 ko. Letting E + 0, we get ~ ( w o=) zo, where zo was an arbitrary fixed point in B ( 0 , l ) . So 4 = f Repetition of the above argument shows that 4, + f - l uniformly on each compact subset of A. In fact, it shows that 4, converges uniformly on compact subsets of D to a bounded (by 1) univalent function which must be an analytic continuation o f f from A to D (since A C D). But since f -l maps A conformally onto B(0, I), this is only possible if A = D. The whole argument can be repeated for any subsequence {D,,}of {D,} with the conclusion that f maps B ( 0 , l ) onto the kernel of {D,,}, which must be the kernel of {D,}. Hence D, + D. Conversely, if D, + D(# C ) , then the sequence {fA(O)) is bounded, so the functions f n are uniformly bounded on compact subsets of B ( 0 , l ) . Consequently, the f, are a normal family. By Vitali's theorem we need only prove point-wise convergence t o prove uniform convergence on compact subsets of D. Furthermore (since the f, form a normal family) if there are two subsequences with different limits a t a point to E B ( 0 , l ) we would have subsequences converging uniformly on compact sets to two different functions and these would have corresponding sequences of simply-connected domains with different kernels. This contradicts Dn + D. For further information on Carathkodory convergence, the reader should consult Duren 1621.

-'.

-'

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Index abscissa of absolute convergence, 405 abscissa of convergence, 404 accessible boundary points, 18 addition theorem for elliptic functions, 314 admissible, 137 Ahlfors-Shimizu characteristic, 141 algebraic addition theorems (elliptic functions, 315 amplitude, 330, 338 area theorem, 27, 258 Arzel&Ascoli theorem, 14 associated a-functions, 326 asymptotic value, 155 automorphic function, 50 Bernoulli numbers, 367, 423 Bloch's constant, 37, 60 Bloch-Landau Theorem, 36 Bohr's theorem, 102 Borel exceptional value, 79 Borel summability, 211, 214 Borel's theorem on entire functions, 78 Borel-Carathkodory lemma, 72,435,451, 452 Carlson's Theorem, 100, 227 central index, 88 chief star, 216 chordal derivative, 52 chordal metric, 15 conformal, 3 congruent points, 305

contiguous paths, 158 corner of a chief star, 216 curves of Julia, 161 Darboux's theorem, 12 De la Vallke-Poussin's estimate for ~ ( x ) , 430 deficiency, 124 deficient value, 136 deficient values, 107 Denjoy-Carleman- Ahlfors theorem, 169 deviation, 154 directions of Julia, 161 Dirichlet series, 403 discriminant of the Q-function, 313 elliptic function, 312 elliptic integral, 338, 345 elliptic modular functions, 45, 394 equicontinuous, 13 equivalent points, 381 Euler products, 407 exact order of a fixed point, 148 exponent of convergence, 71 Fabry gap theorem, 227 Factorization of Meromorphic Functions, 106 Fagnano's Theorem, 348 Farey series, 47, 451, 457 Fatou's theorem, 200 Fatou-PolyBHurwitz theorem, 240 Fatou-Riesz theorem, 196

486 fixed point of order u , 148 Fourier Integral Theorem, 468 fundamental parallelogram, 303 fundamental periods, 303 fundamental sets of zeros or poles, 305

Index Legendre's elliptic integrals of the third kind, 351 Legendre's relations, 319 linear fractional transformations, 5 lines of Julia, 161

Mobius inversion formula, 408 Mobius transformation, 5 mapping radius, 247 maximum term, 84 modular function, 377 Hadamard Gap Theorem, 201, 204, 208 modular group, 378 Hadamard multiplication theorem, 215 modulus (of an elliptic function), 338 Hadamard Product Theorem, 105 Multiplication Theorem for Ordinary Dirichlet Series, 406 Hadamard Three Circles Theorem, 89, 117, 206, 460 natural boundary, 189 Hankel determinants, 247 Nevanlinna characteristic, 144 Hurwitz's theorem, 16 Nevanlinna order, 119 Nevanlinna type, 119 inaccessible boundary points, 18 Nevanlinna's First Fundamental Theoindex of multiplicity, 138 rem, 150 Ingham's theorem on x ( x ) and ((s), 435 Nevanlinna's Second Fundamental Theinvariants of a p-function, 313 orem, 124, 138 inverse of a point, 6 non-contiguous paths, 158 Jacobi's elliptic functions, 330 normal at a point, 152 normal families, 13, 51, 220 Jensen's Formula, 67, 108, 116 normal function, 273 Julia exceptional functions, 166 Julia's Theorem, 159 order (of an elliptic function), 304 Julia, lines of, 161 order (of an entire function), 70 order of a fixed point, 148 Klein's modular function, 378 Ordinary Dirichlet series, 403 Koebe function, 263 Osgood-Carathkodory Theorem, 45, 456 Koebe's Theorem, 253 Ostrowski's Theorem, 206, 214 Kronecker's Theorem, 243 overconvergent series, 206 Lambert series, 192 P6lya-Carlson Theorem, 250 Landau's constant, 37, 63 paths of finite or infinite determination Landau's Theorem, 40, 50, 103 or indetermination, 155 lattice points, 303 Legendre's elliptic integrals of the first period parallelogram, 303 kind, 338 Perron's formula, 408 Legendre's elliptic integrals of the second Phragmkn-Lindelof indicator function, 89, kind, 345 98

genus, 77 Gross' function (example for asymptotic values), 182 growth of an entire function, 82

Index

Phragm6n-Lindelof Principle, 90 Wigert's Theorem, 226 Picard exceptional value, 79, 166, 174 Wirtinger's inequality, 175 Picard's "Big" Theorem, 50, 53, 55 Picard's "Little" Theorem, 49, 50 Poisson Integral Formula, 463 Poisson Summation Formula, 368, 451, 466 Poisson- Jensen Formula, 108, 128 primitive periods, 303 product star, 216 Ramanujan's T-function, 378 ramification index, 138 regular point, 189 Riemann Hypothesis, 437 Riemann Mapping Theorem, 17 Riemann zeta-function, 399 Robertson's conjecture, 293 schlicht, 10 Schottky's theorem, 42, 43 Schwarz Reflection Principle, 28 Schwarz reflection principle, 45, 46, 454 Schwarz-Christoffel formula, 27, 30 simple, 10 singular point, 189 sinusoid, 93 star, 216 theta functions, 352, 356 tract of determination, 158 transfinite diameter, 118 type (of an entire function), 83 typically real, 293 Uniqueness theorem for ordinary Dirichlet Series, 405 univalent, 10 Valiron-deficiency, 153 Von Mangoldt's Function, 400 Weierstrass !J?-function, 371 Weierstrass <-function, 321 Weierstrass' c-function, 358

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