Graduate Texts in Mathematics 103 Editorial Board F. W. Gehring P. R. Halmos (Managing Editor) C. C. Moore

Graduate Texts in Mathematics

I TAKEUTUZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 3 SCHAEFFER. Topological Vector Spaces. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 5 MACLANE. Categories for the Working Mathematician. 6 HUGHEs/PIPER. Projective Planes. 7 SERRE. A Course in Arithmetic. 8 TAKEUTUZARING. Axiometic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory.

10 COHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FuLLER. Rings and Categories of Modules. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings and Their Singularities. 15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 ROSENBLATT. Random Processes. 2nd ed. 18 HALMOS. Measure Theory. 19 HALMOS. A Hilbert Space Problem Book. 2nd ed., revised. 20 HUSEMOLLER. Fibre Bundles. 2nd ed.

HUMPHREYS. Linear Algebraic Groups. 21 22 23

BARNEs/MACK. An Algebraic Introduction to Mathematical Logic. GREUB. Linear Algebra. 4th ed. HOLMES. Geometric Functional Analysis and its Applications. HEWITT/STROMBERG. Real and Abstract Analysis. MANES. Algebraic Theories. KELLEY. General Topology.

24 2S' 26 27 28 ZARISKUSAMUEL. Commutative Algebra. Vol. I. 29 ZARISKUSAMUEL. Commutative Algebra. VoL II. 30 JACOBSON. Lectures in Abstract Algebra I: Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II: Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III: Theory of Fields and Galois Theory. 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk. 2nd ed. 35 WERMER. Banach Algebras and Several Complex Variables. 2nd ed. 36 KELLEy/NAMIOKA et al. Linear Topological Spaces. 37 MONK. Mathematical Logic. 38 GRAUERT/FRITZSCHE. Several Complex Variables. 39 ARVESON. An Invitation to C*-Algebras. 40 KEMENy/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 42 SERRE. Linear Representations of Finite Groups. 43 GILLMAN/JERISON. Rings of Continuous Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 LOEVE. Probability Theory I. 4th ed. 46 LOEVE. Probability Theory II. 4th ed. 47 MOISE. Geometric Topology in Dimensions 2 and 3.

continued after Index

Serge Lang

Complex Analysis Second Edition

With 132 Illustrations

Springer Science+Business Media, LLC

Serge Lang Department of Mathematics Yale University New Haven, CT 06520 U.S.A.

Editorial Board

P. R. Halmos Managing Editor Department of

Mathematics University of Santa Clara Santa Clara, CA 95053 U.S.A.

F. W. Gehring Department of

Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A.

AMS Subject Classification: 30-01

Library of Congress Cataloging in Publication Data Lang, Serge

Complex analysis. (Graduate texts in mathematics; 103) Includes index. 1. Functions of complex variables. 2. Mathematical

analysis. I. Title. II. Series. QA331.L255 1985 515.9 84-21274

c. C. Moore Department of

Mathematics University of California

at Berkeley Berkeley, CA 94720 U.S.A.

The first edition of this book was published by Addison-Wesley Publishing Co., Menlo Park, CA, in 1977.

© 1977, 1985 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc. in 1985. Softcover reprint of the hardcover 2nd edition 1985 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC

Typeset by Composition House Ltd., Salisbury, England.

9 8 7 6 543 2 I

ISBN 978-1-4757-1873-7 ISBN 978-1-4757-1871-3 (eBook) DOI 10.1007/978-1-4757-1871-3

Foreword

The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. Somewhat more material has been included than can be covered at leisure in one term, to give opportunities for the instructor to exercise his taste, and lead the course in whatever direction strikes his fancy at the time. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students.

In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc.) and I would recom­mend to anyone to look through them. More recent texts have empha­sized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex anal­ysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues. The systematic elementary development of formal and convergent power series was standard fare in the German texts, but only Cartan, in the more recent books, includes this material, which I think is quite essential, e.g., for differential equations. I have written a short text, exhibiting these features, making it applicable to a wide variety of tastes.

The book essentially decomposes into two parts. The first part, Chapters I through VIII, includes the basic properties

of analytic functions, essentially what cannot be left out of, say, a one­semester course.

I have no fixed idea about the manner in which Cauchy's theorem is to be treated. In less advanced classes, or if time is lacking, the usual

vi FOREWORD

hand waving about simple closed curves and interiors is not entirely in­appropriate. Perhaps better would be to state precisely the homological version and omit the formal proof. For those who want a more thorough understanding, I include the relevant material.

Artin originally had the idea of basing the homology needed for com­plex variables on the winding number. I have included his proof for Cauchy's theorem, extracting, however, a purely topological lemma of independent interest, not made explicit in Artin's original Notre Dame notes (cf. collected works) or in Ahlfor's book closely following Artin. I have also included the more recent proof by Dixon, which uses the winding number, but replaces the topological lemma by greater use of elementary properties of analytic functions which can be derived directly from the local theorem. The two aspects, homotopy and homology, both enter in an essential fashion for different applications of analytic functions, and neither is slighted at the expense of the other.

Most expositions usually include some of the global geometric prop­erties of analytic maps at an early stage. I chose to make the prelimi­naries on complex functions as short as possible to get quickly into the analytic part of complex function theory: power series expansions and Cauchy's theorem. The advantages of doing this, reaching the heart of the subject rapidly, are obvious. The cost is that certain elementary global geometric considerations are thus omitted from Chapter I, for instance, to reappear later in connection with analytic isomorphisms (Conformal Mappings, Chapter VII) and potential theory (Harmonic Functions, Chapter VIII). I think it is best for the coherence of the book to have covered in one sweep the basic analytic material before dealing with these more geometric global topics. Since the proof of the general Riemann mapping theorem is somewhat more difficult than the study of the . specific cases considered in Chapter VII, it has been postponed to the second part.

The second part of the book, Chapters IX through XIV, deals with further assorted analytic aspects of functions in many directions, which may lead to many other branches of analysis. I have emphasized the possibility of defining analytic functions by an integral involving a parameter and differentiating under the integral sign. Some classical functions are given to work out as exercises, but the gamma function is worked out in detail in the text, as a prototype. The chapters in this part are essentially logically independent and can be covered in any order, or omitted at will.

In particular, the chapter on analytic continuation, including the Schwarz reflection principle, and/or the proof of the Riemann mapping theorem could be done right after Chapter VII, and still achieve great coherence.

As most of this part is somewhat harder than the first part, it can easily be omitted from a course addressed to undergraduates. In the

FOREWORD VII

same spirit, some of the harder exercises in the first part have been starred, to make their omission easy.

In this second edition, I have rewritten many sections, and I have added some material. I have also made a number of corrections whose need was pointed out to me by several people. I thank them all.

I am much indebted to Barnet M. Weinstock for his help in correcting the proofs, and for useful suggestions.

SERGE LANG

Prerequisi tes

We assume that the reader has had two years of calculus, and has some acquaintance with epsilon-delta techniques. For convenience, we have recalled all the necessary lemmas we need for continuous functions on compact sets in the plane.

We use what is now standard terminology. A function

f:S -. T

is called injective if x t= y in S implies f(x) t= f(y). It is called surjective if for every z in T there exists XES such that f(x) = z. If f is surjective, then we also say that f maps S onto T. If f is both injective and surjective then we say that f is bijective.

Given two functions f, 9 defined on a set of real numbers containing arbitrarily large numbers, and such that g(x) ~ 0, we write

f« 9 or f(x) «g(x) for x -. 00

to mean that there exists a number C > 0 such that for all x sufficiently large, we have

If(x) I ~ Cg(x).

Similarly, if the functions are defined for x near 0, we use the same symbol « for x -. 0 to mean that there

If(x) I ~ Cg(x)

x PREREQUISITES

for all x sufficiently small (there exists {) > 0 such that if I x I < {) then If(x) I ~ Cg(x». Often this relation is also expressed by writing

f(x) = O(g(x»),

which is read: f(x) is big oh of g(x), for x --+ 00 or x -+ 0 as the case may be.

We use ]a, b[ to denote the open interval of numbers

a < x < b.

Similarly, [a, b[ denotes the half-open interval, etc.

Contents

Prerequisites

PART ONE

Basic Theory

CHAPTER I

Complex Numbers and Functions

§1. Definition . . . . . . . . §2. Polar Form. . ..... . §3. Complex Valued Functions . §4. Limits and Compact Sets.

Compact Sets. . . . . . §5. Complex Differentiability. . §6. The Cauchy-Riemann Equations §7. Angles Under Holomorphic Maps

CHAPTER II

Power Series

§ 1. Formal Power Series. . . §2. Convergent Power Series . §3. Relations Between Formal and Convergent Series .

Sums and Products . Quotients. . .... Composition of Series

§4. Analytic Functions . . §5. The Inverse and Open Mapping Theorems §6. The Local Maximum Modulus Principle §7. Differentiation of Power Series . . . . .

ix

3

3 8

12 17 21 28 32 34

38

38 49 62 62 66 67 69 72 79 82

xii CONTENTS

CHAPTER III

Cauchy's Theorem, First Part

§l. Holomorphic Functions on Connected Sets. Appendix: Connectedness . . . . . . .

§2. Integrals Over Paths . . . . . . . . . . . §3. Local Primitive for a Holomorphic Function. §4. Another Description of the Integral Along a Path. §5. The Homotopy Form of Cauchy's Theorem . . . §6. Existence of Global Primitives. Definition of the Logarithm

CHAPTER IV

Cauchy's Theorem, Second Part

§l. The Winding Number ..... §2. Statement of Cauchy's Theorem. §3. Artin's Proof . . . . . . . . .

CHAPTER V

Applications of Cauchy's Integral Formula.

§l. Cauchy's Integral Formula on a Disc §2. Laurent Series. . . . . . §3. Isolated Singularities. . .

Removable Singularities Poles ....... . Essential Singularities .

§4. Dixon's Proof of Cauchy's Theorem.

CHAPTER VI

Calculus of Residues

§l. The Residue Formula . . . . . §2. Evaluation of Definite Integrals .

Fourier Transforms .. Trigonometric Integrals Mellin Transforms. . .

CHAPTER VII

Conformal Mappings .

§ 1. Schwarz Lemma. . . . . . . . . . §2. Analytic Automorphisms of the Disc §3. The Upper Half Plane . . . . . . §4. Other Examples. . . . . . . . . §5. Fractional Linear Transformations.

87

87 93 94

103 109 lIS 1I8

123

124 128 137

144

144 151 155 155 156 158 162

165

165 180 182 185 187

196

198 200 203 206 215

CHAPTER VIII

Harmonic Functions

§l. Definition . . . .

CONTENTS

Application: Perpendicularity . Application: Flow Lines . . .

§2. Examples ......... . §3. Basic Properties of Harmonic Functions §4. Construction of Harmonic Functions . §5. The Poisson Representation. . . . . .

PART TWO

Various Analytic Topics

CHAPTER IX

Applications of the Maximum Modulus Principle.

§l. The Effect of Zeros, Jensen-Schwarz Lemma §2. The Effect of Small Derivatives . . . .

Hermite Interpolation Formula . . . . . §3. Entire Functions with Rational Values. . . §4. The Phragmen-Lindelof and Hadamard Theorems. §5. Bounds by the Real Part, Borel-Carath odory Theorem

CHAPTER X

Entire and Meromorphic Functions

§l. Infinite Products. . . . . §2. Weierstrass Products. . . . . . . . §3. Functions of Finite Order . . . . . §4. Meromorphic Functions, Mittag-Leffler Theorem

CHAPTER XI

Elliptic Functions

§l. The Liouville Theorems . §2. The Weierstrass Function. §3. The Addition Theorem. . §4. The Sigma and Zeta Functions

CHAPTER XII

Differentiating Under an Integral.

§l. The Differentiation Lemma .. §2. The Gamma Function . . . .

Proof of Stirling's Formula.

xiii

224

224 228 230 234 241 244 249

253

255

255 260 261 262 268 273

276

276 280 286 290

292

292 295 299 302

307

308 311 316

xiv CONTENTS

CHAPTER XIII

Analytic Continuation

§l. Schwarz Reflection . §2. Continuation Along a Path .

CHAPTER XIV

The Riemann Mapping Theorem

§l. Statement and Application to Picard's Theorem. §2. Compact Sets in Function Spaces . . . §3. Proof of the Riemann Mapping Theorem §4. Behavior at the Boundary . . . . . .

Appendix

Cauchy's Formula for cro Functions

Index ............... .

324

324 330

340

340 344 347 351

359

365