Nine Introductions in Complex Analysis

NINE INTRODUCTIONS IN COMPLEX ANALYSIS REVISED EDITION NORTH-HOLLAND MATHEMATICS STUDIES 208 (Continuation of the Not...

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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

First edition 2008 Copyright © 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-51831-6 ISSN (Series): 0304-0208 For information on all Elsevier publications visit our website at books.elsevier.com

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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

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

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


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


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 ) .


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


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.

+

+


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
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.


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 [ 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 .


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,

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


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).


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


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.


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


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


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


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


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.

<
or (1-Q)If1(C)l.

&,

and if f'( 0} is given by

2

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)

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

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


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

0,

max

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

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.

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


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 -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
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.)

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,

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

0 such that

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 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.

+

(ii) COSe)
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 % 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 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


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.


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 )

O.

P>" la,l 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.

+


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,

po) of measure 5 2. So,

+ h)
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.

_ 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).


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

<
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 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


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

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

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

(

')
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.


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( pl = M (rl). Consider the set of all such points S = {[ : lg( 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


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


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

<
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

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 ))

+

+ +

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.


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

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 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 (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)


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,


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


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


{z: Iz(1- z)I = 2)

Diagram 6.2

<

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,

-11, and no further. 101Ok+l

Note 2.4. In Example 2.4, the coefficients an are 0 for lo4,10k+,olok 1 + S, then

Diagram 6.3

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 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


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

>

>


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

+

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


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

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,


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,

j=1

j=1

<
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 )


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


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

_ 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

$


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


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.


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
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.

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

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 ,

t ) ?) t.) It follows that and so &(f (z, t ) ) exists and is continuous. Let x

which equals &g(

&

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)

.


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


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) & ~ ~ (

-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


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


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

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.

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 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.


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


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)


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

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


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


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)


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.


(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.

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.)


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


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+


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.


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


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


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


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 0,

J(r) =

z.

By Definitions


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 >


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


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


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


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, 1,


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
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.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.


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, 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

+


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


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


412

Theorem 2.9. (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)


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

+

++


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)


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


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.


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).


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

+

+

+

+

+


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) 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

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


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


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),

1,

1 p

(Theorem 1.3);

m=l

and

We now argue as in the proof of Theorem 3.1 since C(s) and 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 1, and any particular zero Po constant.

+ iyo,yo > to of 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, 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 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


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 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

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.

<
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

1 and any r such that T

P15- 1,

and for a given r and Ic

r - 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

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