Raghavan Narasimhan - Springer978-1-4757-1106...Raghavan Narasimhan Complex Analysis in One Variable...

Raghavan Narasimhan

Complex Analysis in One Variable

Springer Science+Business Media, LLC

Professor Dr. Raghavan Narasimhan The University of Chicago Department of Mathematics 5734 University Avenue Chicago, IL 60637 (USA)

Library of Congress Cataloging in Publication Data

Narasimhan, Raghavan Complex analysis in one variable.

Includes bibliographies and index. 1. Functions of complex variables. 2. Mathematical

analysis. 1. Title. II. Title: Complex analysis in 1 variable. QA331.N27 1985 515.9 84-11151

CIP-Kurztitelaufnahme der Deutschen Bibliothek

Narasimhan, Raghavan: Complex analysis in one variable ! by Raghavan Narasimhan. -Springer Science+ Business Media, LLC ISBN 978-0-8176-3237-3 ISBN 978-1-4757-1106-6 (eBook) DOI 10.1007/978-1-4757-1106-6

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 prior permission of the copyright owner.

© 1985 Springer Science+Business Media New York

Originally published by Birkhăuser Boston, Inc in 1985.

Softcover reprint ofthe hardcover Ist edition 1985

Dedicated to K. CHANDRASEKHARAN

vii

Contents

Preface.. .. .. .. .. .. .. . .. .. .. . .. ... . ... . .. .. .. .... .. .. .. .. .. . .. .. xi Notation and Terminology ........................................ XIV

Chapter 1. Elementary Theory of Holomorphic Functions. . . . . . . . . . . . . . 1 1. Some basic properties of <C-differentiable and holomorphic functions 2 2. Integration along curves ....................................... 9 3. Fundamental properties of holomorphic functions. . . . . . . . . . . . . . . .. 21 4. The theorems of Weierstrass and Montel. . . . . . . . . . . . . . . . . . . . . . . .. 32 5. Meromorphic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36 6. The Looman-Menchoff theorem ................................ 43

Notes on Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 References: Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51

Chapter 2. Covering Spaces and the Monodromy Theorem ............ 53 1. Covering spaces and the lifting of curves. . . . . . . . . . . . . . . . . . . . . . . .. 53 2. The sheaf of germs of holomorphic functions. . . . . . . . . . . . . . . . . . . .. 55 3. Covering spaces and integration along curves. . . . . . . . . . . . . . . . . . . .. 58 4. The monodromy theorem and the homotopy form of

Cauchy's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60 5. Applications of the monodromy theorem. . . . . . . . . . . . . . . . . . . . . . . .. 64


viii Contents

Chapter 3. The Winding Number and the Residue Theorem. . . . . . . . . . .. 70 1. The winding number .......................................... 70 2. The residue theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75 3. Applications of the residue theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81

Notes on Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87 References: Chapter 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87

Chapter 4. Picard's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89


Chapter 5. The Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem . ............................................... 100 1. Partitions of unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100

2. The equation ~~ = ¢ .......................................... 102

3. Runge's theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106 4. The homology form of Cauchy's theorem. . . . . . . . . . . . . . . . . . . . . . .. 115

Notes on Chapter 5 .............................................. 117 References: Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 118

Chapter 6. Applications of Runge's Theorem ........................ 119 1. The Mittag-Leffler theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119 2. The cohomology form of Cauchy's theorem ...................... 123 3. The theorem of Weierstrass .................................... 126 4. Ideals in Yf (Q) ............................................... 131

Notes on Chapter 6 .............................................. 141 References: Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 142

Chapter 7. The Riemann Mapping Theorem and Simple Connectedness in the Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 144 1. Analytic automorphisms of the disc and of the annulus. . . . . . . . . . .. 144 2. The Riemann mapping theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 148 3. Simply connected plane domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150

Notes on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. 153 References: Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 154

Contents IX

Chapter 8. Functions of Several Complex Variables 156

Notes on Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 164 References: Chapter 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 165

Chapter 9. Compact Riemann Surfaces ............................. 166 1. Definitions and basic theorems ................................. 166 2. Meromorphic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 171 3. The cohomology group Hl(U,CD) ................................ 173 4. A theorem from functional analysis ............................. 176 5. The finiteness theorem ......................................... 183 6. Meromorphic functions on a compact Riemann surface. . . . . . . . . . .. 185

Notes on Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 190 References: Chapter 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191

Chapter 10. The Corona Theorem . .................................. 193 1. The Poisson Integral and the theorem of F. and M. Riesz .......... 194 2. The corona theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204

Notes on Chapter 10 ............................................. 214 References: Chapter 10 . .......................................... 215

Chapter 11. Subharmonic Functions and the Dirichlet Problem . ........ 216 1. Semi-continuous functions ...................................... 216 2. Harmonic functions and Harnack's principle ..................... 219 3. Convex functions .............................................. 223 4. Subharmonic functions: Definition and basic properties. . . . . . . . . . .. 227 5. Subharmonic functions: Further properties and application to

convexity theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 236 6. Harmonic and subharmonic functions on Riemann surfaces . . . . . . .. 246 7. The Dirichlet problem ......................................... 246 8. The Rad6-Cartan theorem ..................................... 255

Notes on Chapter 11 ............................................. 260 References: Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 261

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 263

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 264

xi

Preface

This book is based on a first-year graduate course I gave three times at the University of Chicago. As it was addressed 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 the 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 contrast the behavior in higher dimension with that in the plane.

xii Preface

Chapter 9, on Riemann surfaces, is meant to serve as an introduction to tools which are of great importance, not only in the modern 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 HP spaces, a subject that has been much in evidence in recent work 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 partly 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, 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 the 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). He will need some basic definitions and results from linear algebra and the theory of rings and ideals, as he will elementary properties of Lebesgue measure and the standard convergence theorems for Lebesgue integrals. Finally, from functional analysis, he will need 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 (W. 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 them for their encouragement and for the many suggestions they made.

I am also 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 encouragement of the former, this book would never have been completed. As for the latter, anyone glancing through the

Preface xiii

notes on Chapters 6 and 9 will find acknowledgment 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.

xiv

Notation and Terminology

We shall denote by N the set of nonnegative integers, by 7L the set of all integers, by JR 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::::> A that B contains A. {x E AI . .. } stands for set of elements of A satisfying the conditions specified in ... ; 'if x E A stands for "for every element x of A"; j 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; Izi = + V x 2 + y2 is the absolute value of z.

If n ~ 1, JRn, respectively <en, will denote the set of n-tuples of real, respectively complex, numbers. If Z=(ZI' ... , zn) E <en, we write Izi for the positive square root of IZ112+ ... +lznI2. The function (z, w)f---+lz-wl 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 JRn (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 ~ B to mean that A is relatively compact in B, i. e., that the closure of A in B is a compact set.

If X is a topological space and A c X, we denote by AO in the interior of A (i.e., the largest open subset of X that is contained in A), by A the closure of A

Notation and terminology xv

and by 0 A = A - A the boundary 01 A. (This last notion is used usually only when A is an open set in X.)

A limit point (or a point 01 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 U ("\ A with y =F 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 Yare sets and I: X - Y is a map, we use the following standard terms:

I is injective if whenever x, y e X and x =F y, we have I(x) =F I(y).

lis surjective if for every ye Y, 3 xeX with/(x)=y.

I is bijective if it is both injective and surjective.

An open connected subset of a topological space is called a domain. An open covering of a topological space X is a family U = { U;} i E 1 of open

subsets of X such that X = U Ui • iel

If a, b are real numbers, a 5 b, the closed interval [a, b] is the set {te1R1 a5 t5b}; the open interval (a, b) is the set {te 1Rla< t< b}. The half open intervals [a,b) and (a,b] are, respectively, the sets {te1Rla5t<b}, {te1Rla< t5b}.

Let a e CC and R> O. The disc with center a and radius R is the open set D(a, R)={zeCCllz-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)={zeCCllz-aI5R}. 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 1R n (n ~ 1), and let I be a complex valued function defined on Q. The partial derivative (ol/ox)(a) of I at a, where aeQ and 1 5j 5 n, is the limit, when it exists,

lim h- 1 (j(a 1 , ••• ,aj- 1 , aj+h, aj+l' ... ,an)-/(a1 , ••• ,an))· h-O h~O

We denote by C1 (Q) the set of functions Ion Q such that (ol/oxj)(a) exists for every a e Q and 1 5j 5 n, and such that the partial derivatives a 1-+ (01/0 x) (a) are continuous functions on Q for eachj (15j5n). For kelN, k~2, we define Ck(Q) inductively by Ck(Q) = {Ie C1 (Q)1 ol/oxj e Ck- 1 (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 5 k exist and are continuous).

00

We denote by COO(Q) the set n Ck(Q); COO(Q) is the set of infinitely k=1

XVI Notation and terminology

differentiable funtions. If 1 ~ k ~ 00 and f E Ck(Q), we shall also say that f is Ck

on Q.

We often identify <e with IR2 by means of the map z=x+iYf--+(x,y), X,YEIR. If Q is open in <e andfis a complex valued function on Q, we often write f(x, y) for f(z) (with the above identification in mind). We use a similar identification of <en with IR 2n and use the analogous notation f(x, y) for f(z) with Z=(Zl' .. . ,zn), Zk=xk+iYk> xk, hEIR. If Q is open in <en, we may therefore speak of Ck(Q), the partial derivatives 8f/8xb 8f/8Yb and so on.

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

If Q is an open set in IR n or in <en and 1 ~ k ~ 00, we denote by C~ (Q) the set of fE Ck(Q) such that sUPPQ(f) is compact.

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