Complex Analysis in One Variable Second Edition

Raghavan Narasimhan Yves Nievergelt

Springer Science+Business Media, LLC

Raghavan Narasimhan Department of Mathematics University of Chicago Chicago, IL 60637 U.S.A.

Yves Nievergelt Department of Mathematics Eastern Washington University Cheney, WA 99004 U.S.A.

Library of Congress Cataloging-in-Publication Data

Narasimhan, Raghavan. Complex analysis in one variable.-2nd ed. / Raghavan Narasimhan and Yves Nievergelt.

p. cm. Includes bibliographical references and index. ISBN 978-1-4612-6647-1 ISBN 978-1-4612-0175-5 (eBook) DOI 10.1007/978-1-4612-0175-5

1. Functions of complex variables. 2. Mathematical analysis. I. Nievergelt, Yves. II. Title.

QA331.N27 2000 515'.9-dc21 00-051906

CIP

AMS Subject Classifications: 30-01, 30A05, 30A99, 30D30, 31A05, 32A10

Printed on acid-free paper. ©2001 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 2001 Softcover reprint of the hardcover 2nd edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, L L C , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 978-1-4612-6647-1 SPIN 10749282

Typeset in IATEX2E by T^Xniques, Inc., Cambridge, M A .

9 8 7 6 5 4 3 2 1

Contents

Preface to the Second Edition

Preface to the First Edition

Notation and Terminology

I Complex Analysis in One Variable Raghavan Narasimhan

1 Elementary Theory of Holomorphic Functions 1 Some basic properties of C-differentiable

and holomorphic functions . . . . . . . . . 2 Integration along curves. . . . . . . . . . . 3 Fundamental properties of holomorphic functions 4 The theorems of Weierstrass and Montel 5 Meromorphic functions . . . . . 6 The Looman-Menchoff theorem . . . .

2 Covering Spaces and the Monodromy Theorem 1 Covering spaces and the lifting of curves . . . 2 The sheaf of germs of holomorphic functions 3 Covering spaces and integration along curves 4 The monodromy theorem and the homotopy form

of Cauchy's theorem .. . . . . . . . . . 5 Applications of the monodromy theorem . . .

3 The Winding Number and the Residue Theorem 1 The winding number . . . . . . . . 2 The residue theorem ....... . 3 Applications of the residue theorem

4 Picard's Theorem

ix

xi

xiii

1

3

4 10 22 32 36 43

53 53 55 57

60 63

69 69 73 79

87

vi

5 Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem 1 Partitions of unity . . 2 Th . au A.. e equatIOn az = 'I' . . . . . . . . . . .

3 Runge's theorem . . . . . . . . . . . . . 4 The homology form of Cauchy's theorem

6 Applications of Runge's Theorem 1 The Mittag-Leffler theorem . . . . . . . . . 2 The cohomology form of Cauchy's theorem 3 The theorem of Weierstrass . 4 Ideals in 'H.(0.) .............. .

7 Riemann Mapping Theorem and Simple Connectedness in the Plane 1 Analytic automorphisms of the disc and of the annulus 2 The Riemann mapping theorem. . 3 Simply connected plane domains . .

8 Functions of Several Complex Variables

9 Compact Riemann Surfaces 1 Definitions and basic theorems 2 3 4 5 6

Meromorphic functions . . . . The cohomology group HI (11, 0) A theorem from functional analysis The finiteness theorem . . . . . . . Meromorphic functions on a compact Riemann surface

10 The Corona Theorem 1 The Poisson integral and the theorem of F. and M. Riesz 2 The corona theorem . . . . . . . . . . . . . . . .

11 Subharmonic Functions and the Dirichlet Problem 1 Semi-continuous functions . . . . . . . . . . 2 3 4 5

6 7 8

Harmonic functions and Harnack's principle . . . Convex functions . . . . . . . . . . . . . . . . . Subharmonic functions: Definition and basic properties Subharmonic functions: Further properties and application to convexity theorems .......... . Harmonic and subharmonic functions on Riemann surfaces The Dirichlet problem. . . The Rad6-Cartan theorem

Appendix: Baire's Theorem

Contents

97 97 99

103 111

115 115 119 121 127

139 139 143 145

151

161 161 166 167 171 176 179

187 188 197

209 209 212 215 219

227 237 237 244

253

Contents Vll

II Exercises Yves Nievergelt 255

Introduction 257

o Review of Complex Numbers 259 1 Algebraic properties of the complex numbers 259 2 Complex equations of generalized circles 261 3 Complex fractional linear transformations 262 4 Topological concepts . . . . . . . . . . . 265

1 Elementary Theory of Holomorphic Functions 267 1 Some basic properties of ((>differentiable

and holomorphic functions . . . . . . . . . . . . . . . . . . . 267 1.1 Complex derivatives and Cauchy-Riemann equations . 267 1.2 Differentials and conformal maps . . . 269 1.3 Conformal maps . . . . . . . . . . . . . . . . . . . . 270 1.4 Radius of convergence of power series. . . . . . . . . 275 1.5 Exponential, trigonometric, and dilogarithm functions 277

2 Integration along curves. . . . . . . . . . . . 278 2.1 Complex line integrals . . . . . . . . . . . 278 2.2 Complex derivatives of line integrals. . . . 279 2.3 Remainder of complex Taylor polynomials 281 2.4 H. A. Schwarz's reflection principle . . . 281

3 Fundamental properties of holomorphic functions 282 3.1 The complex exponential function . . . . 282 3.2 Holomorphic functions . . . . . . . . . . 284 3.3 Bounds on the size of roots of polynomials 285 3.4 Principal branch of the complex square root . 287 3.5 Complex square roots in celestial mechanics 288

4 Theorems of Weierstrass and Montel . 290 5 Meromorphic functions . . . . . . . . . 290

5.1 A complex Newton's method. . 291 5.2 Sequences of complex numbers 293

2 Covering Spaces and the Monodromy Theorem 297 1 Covering spaces and the lifting of curves . . . 297

1.1 Examples of real or complex manifolds 297 1.2 Covering maps . . . . . . . . . . . . 299

2 The sheaf of germs of holomorphic functions . 299 3 Covering spaces and integration along curves . 300 4 The monodromy theorem and the homotopy form

of Cauchy's theorem '" . . . . . . . . 303 5 Applications of the monodromy theorem . . . . . 303

viii

3 The Winding Number and the Residue Theorem 1 The winding number . . . . . . . . 2 The residue theorem ....... . 3 Applications of the residue theorem

4 Picard's Theorem

Contents

305 305 307 310

313

5 The Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem 315 315 316 316 321 322 323

1 Partitions of unity . . . . . . . . . . 2 The equation au/az- = ¢ ..... .

2.1 Complex differential fonns . 2.2 RouchC's theorem ..... . 2.3 Inhomogeneous Cauchy-Riemann equations

3 Runge's theorem ............ . . . . . . .

6 Applications of Runge's Theorem 1 The Mittag-Leffler theorem . . . . . . . . . 2 The cohomology fonn of Cauchy's theorem 3 The theorem of Weierstrass . 4 Ideals in 1t(Q) .............. .

7 The Riemann Mapping Theorem and Simple Connectedness in the Plane 1 Analytic automorphisms of the disc

and of the annulus. . . . . . . . . 2 The Riemann mapping theorem. . 3 Simply connected plane domains .

8 Functions of Several Complex Variables

9 Compact Riemann Surfaces 1 Definitions and basic theorems 3 The cohomology group HI (it, 0) . . 6 Meromorphic functions on a compact

Riemann surface .......... .

10 The Corona Theorem 1 The Poisson integral and the theorem of

F. and M. Riesz . . . . . . . . . . . . .

11 Subharmonic Functions and the Dirichlet Problem

Notes for the exercises

References for the exercises

Index

331 331 332 332 335

337

337 340 342

343

351 351 355

358

361

361

365

369

373

379

Preface to the Second Edition

The original edition of this book has been out of print for some years. The appear­ance of the present second edition owes much to the initiative of Yves Nievergelt at Eastern Washington University, and the support of Ann Kostant, Mathematics Editor at Birkhauser.

Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. In 1997, Yves Nievergelt informed me that, for a decade, he had regularly taught a course at Eastern Washington based on the book, and that he had systematically compiled exercises for his course. He kindly put his work at my disposal.

Thus, the present edition appears in two parts. The first is essentially just a reprint of the original edition. I have corrected the misprints of which I have become aware (including those pointed out to me by others), and have made a small number of other minor changes.

The second part of the book, authored by Yves Nievergelt, consists of exercises and relevant references. Most of the exercises are based on his course at Eastern Washing­ton, but it also includes several problems from a set that I sent him. This set was a selection from problems that Kevin Corlette, Madhav Nori and I prepared when we taught the first year graduate course on one complex variable here at Chicago. We hope that the addition of this Part 2 will enhance the usefulness of the book.

The first edition of this book was dedicated to K. Chandrasekharan. The reasons, professional and personal, for doing this have only grown stronger. I should like, therefore, to dedicate Part 1 of this Second Edition once again to him.

Raghavan Narasimhan 2000

Preface to the First Edition

This book is based on a first-year graduate course I gave three times at the University of Chicago. As it was addresed to graduate students who intended to specialize in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Complex analysis in one variable is ideally suited to this attempt. Of course, the branches of mathematics one chooses, and the connections one makes, must depend on personal taste and knowledge. My own leaning towards several complex variables will be apparent, especially in the notes at the end of different chapters.

The first three chapters deal largely with classical material which is available in the many books on the subject. I have tried to present this material as efficiently as I could, and, even here, to show the relationship with other branches of mathematics.

Chapter 4 contains a proof of Picard's theorem; the method of proof I have cho­sen has far-reaching generalizations in several complex variables and in differential geometry.

The next two chapters deal with the Runge approximation theorem and its many applications. The presentation here has been strongly influenced by work on several complex variables.

Although Chapter 8 is entitled "Functions of Several Complex Variables," the book as a whole is about a single variable. This chapter is meant to constrast the behavior in higher dimensions with that in the plane.

Chapter 9, on Riemann surfaces, is meant to serve as an introduction to tools which are of great importance, not only in the modem study of Riemann surfaces, but also in algebraic geometry, in several complex variables, and elsewhere.

Chapter 10 presents Tom Wolff's proof of the corona theorem. It is meant to demonstrate the use of real variable methods in complex analysis, and could be used as an introduction to the study of H P spaces, a subject that has been much in evidence in recent works in Fourier analysis.

The last chapter is a return to classical material. Subharmonic functions and their generalizations to several variables are of great importance.

There are notes at the end of each chapter which are partly historical and pardy an attempt to point out some directions in which the material of the chapter has developed.

Chapters are divided into sections (§), and, in each section, definitions, theorems,

xu Preface to the First Edition

propositions and lemmas are each numbered consecutively. References such as Theo­rem 1 are to the same section; §2, Lemma 1, for example, refers to the second section of the same chapter. A reference such as Chapter 3, § 1, Definition 1 is self-explanatory. References to books and papers in any chapter (which are few in text, but more numer­ous in the notes) are to the list given at the end of the chapter in which they occur.

As for prerequisites, it is assumed that the reader is well acquainted with calculus in several variables and with point set topology (properties of locally compact spaces, of connected components and the like). Some basic definitions and results from linear algebra and the theory of rings and ideals are needed, as well as elementary properties of Lebesgue measure and the standard convergence theorems for Lebesgue integrals. Finally, from functional analysis, the reader needs the Hahn-Banach theorem and the closed graph theorem (for Banach spaces) and a few other elementary and easily accessible facts (such as the finite dimensionality of a locally compact Banach space).

I prepared a handwritten version of the course I gave at Chicago. A few people who saw this version CW. Beckner, K. Chandrasekharan, I. Kaplansky among them) found it useful and suggested that it might be of some general use. I am very grateful to Klas Diederich who saw my handwritten notes and suggested to Klaus Peters that Birkhauser might be interested in publishing this book.

To K. Chandrasekharan and Irving Kaplansky, lowe special thanks. Without the suggestions and encouragment of the former, this book would never have been com­pleted. As for the latter, anyone glancing through the notes on Chapters 6 and 9 will find acknowlegment for specific results and proofs he showed me. But he also read my notes very carefully, compared the proofs given there with others in the literature and was always most helpful with suggestions, references, and answers to questions.

Raghavan Narasimhan 1985

Notation and Terminology

We shall denote by N the set of nonnegative integers, by Z the set of all integers, by lR the set of real numbers and by e the set of complex numbers.

We shall use the standard notation of set theory throughout. Thus, if A, B are sets, A C B means that A is contained in B, B :J A that B contains A. {x E A I ... } stands for set of elements of A satisfying the conditions specified in ... ; V x E A stands for "for every element x of A"; 3 x E A stands for "there exists an element x of A." As usual, p :::} q stands for "p implies q" (p and q being statements).

If Z E e, we usually write Z = x + iy where x and yare real; Izl = +Jx2 + y2 is the absolute value of z.

If n ~ 1, lRn, respectively en, will denote the set of n-tuples of real, respectively, complex, numbers. If z = (Zl, ... ,Zn) E en, we write Izi for the positive square root of IZl12 + ... + IZn 12. The function (z, w) 1-+ Iz - wi defines a metric on en, and the topology we shall consider on en will be the one induced by this metric. We use these conventions also in the case of lRn (which we may consider as a subspace of en).

If X is a locally compact topological space and if A, B are subsets of X with A c B, we write A <s B to mean that A is relatively compact in B, i.e., that the closure of A in B is a compact set.

o If X is a topological space and A C X, we denote by A the interior of A (i.e.,

the largest open subset of X that is contained in A), by A the closure of A and by _ 0

vA = A- A the boundary of A. (This last notion is used most often when A is an open set in X.)

A limit point (or a point of accumulation) of a set A in a topological space X is the set of points x E X such that for any neighborhood U of x in X, there is a point y in UnA with y =1= x. A point x E A is an isolated point of A if it is not a limit point of A. A closed set without limit points is called a discrete set.

If X and Y are sets and f: X -+ Y is a map, we use the following standard terms:

f is injective if whenever x, y E X and x =1= y, we have f(x) =1= fey).

f is surjective if for every y E Y, 3 x E X with f(x) = y.

f is bijective if it is both injective and surjective.

An open connected subset of a topological space is called a domain.

XIV Notation and Terminology

An open covering of a topological space X is a family it = {Ui }iEI of open subsets of X such that X = U Ui.

iEI If a, b are real numbers, a :::: b, the closed interval [a, b] is the set {t E ]R I a ::::

t :::: b}; the open interval (a, b) is the set {t E ]R I a < t < b}. The half open intervals [a, b) and (a, b] are, respectively, the sets {t E]R I a :::: t < b}, {t E]R I a < t :::: b}.

Let a E <e and R > O. The disc with center a and radius R is the open set D(a, R) = {a E <e I Iz - al < R}; D(a, R) is also called the open disc with center a and radius R. The closed disc with center a and radius R is the closed set D(a, R) = {z E <e I Iz - al :::: R}. Throughout this book, the symbols D(a, R) and D(a, R) are reserved exclusively for the open and closed discs, respectively, with center a and radius R; D(a, R) is the closure of D(a, R).

Let Q be an open set in]Rn (n ::: I), and let I be a complex valued function defined on Q. The partial derivative (aflaxj)(a) of I at a, where a E Q and 1 :::: j :::: n, is the limit, when it exists,

We denote by C l (Q) the set of functions I on Q such that (aflaxj )(a) exists for every a E Q and 1 :::: j :::: n, and such that the partial derivatives a f-+ (a fI a x j ) (a) are continuous functions on Q for each j (1 :::: j :::: n). For kEN, k ::: 2, we define Ck(Q) inductively by Ck(Q) = {f E CI(Q) I aflaxj E Ck-I(Q) for j = 1, ... , n}; Ck(Q) is the set of k-times continuously differentiable functions (= the set of functions for which partial derivatives of all orders:::: k exist and are continuous).

00

We denote by COO(Q) the set n Ck(Q); COO(Q) is the set of infinitely differen­k=l

tiable functions. If 1 :::: k :::: 00 and I E Ck(Q), we shall also say that I is Ck onQ.

We often identify <e with ]R2 by means of the map z = x + iy f-+ (x, y), X, Y E R If Q is open in <e and I is a complex valued function on Q, we often write I(x, y) for I(z) (with the above identification in mind). We use a similar identification of <en with ]R2n and use the analogous notation I(x, y) for I(z) with Z = (Zl, ... , Zn), Zk = Xk + iYk, xt, Yk E R If Q is open in <e", we may therefore speak of Ck(Q), the partial derivatives aflaxk. aflayk. and so on.

Let X be a topological space and I a complex valued function defined on X. We define the support 01 I to be the closure in X of the set{x E X I I (x) i= O} and denote this set by supp(f). When it is necessary to emphasize the space X, we shall speak of the support of I in X and write suPPx(f).

If Q is an open set in ]Rn or in <en and 1 :::: k :::: 00, we denote by C~(Q) the set of IE Ck(Q) such that sUPPn(f) is compact.