Ahlfors复分析英文版.pdf

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Cover

S Title

COMPLEX ANALYSIS: An Introduction to the Theory of Analytic Functions of One Complex Variable, Third Edition

ISBN 0-07-000657-1

QA331.A45 1979 515' .93

Dedicated To Erna

Contents

Preface

1 COMPLEX NUMBERS

1. THE ALGEBRA OF COMPLEX NUMBERS

1.1. Arithmetic Operations

1.2. Square Roots

1.3. Justification

1.4. Conjugation, Absolute Value

1.5. Inequalities

2. THE GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS

2.1. Geometric Addition and Multiplication

2.2. The Binomial Equation

2.3. Analytic Geometry

2.4. The Spherical Representation

2 COMPLEX FUNCTIONS

1. INTRODUCTION TO THE CONCEPT OF ANALYTIC FUNCTION

1.1. Limits and Continuity

1.2. Analytic Functions

1.3. Polynomials

1.4. Rational Functions

2. ELEMENTARY THEORY OF POWER SERIES

2.1. Sequences

2.2. Series

2.3. Uniform Convergence

2.4. Power Series

2.5. Abel's Limit Theorem

3. THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS

3.1. The Exponential

3.2. The Trigonometric Functions

3.3. The Periodicity

3.4. The Logarithm

3 ANALYTIC FUNCTIONS AS MAPPINGS

1. ELEMENTARY POINT SET TOPOLOGY

1.1. Sets and Elements

1.2. Metric Spaces

1.3. Connectedness

1.4. Compactness

1.5. Continuous Functions

1.6. Topological Spaces

2. CONFORMALITY

2.1. Arcs and Closed Curves

2.2. Analytic Functions in Regions

2.3. Conformal Mapping

2.4. Length and Area

3. LINEAR TRANSFORMATIONS

3.1. The Linear Group

3.2. The Cross Ratio

3.3. Symmetry

3.4. Oriented Circles

3.5. Families of Circles

4. ELEMENTARY CONFORMAL MAPPINGS

4.1. The Use of Level Curves

4.2. A Survey of Elementary Mappings

4.3. Elementary Riemann Surfaces

4 COMPLEX INTEGRATION

1. FUNDAMENTAL THEOREMS

1.1. Line Integrals

1.2. Rectifiable Arcs

1.3. Line Integrals as Functions of Arcs.

1.4. Cauchy's Theorem for a Rectangle

1.5. Cauchy's Theorem in a Disk

2. CAUCHY'S INTEGRAL FORMULA

2.1. The Index of a Point with Respect to a Closed Curve

2.2. The Integral Formula

2.3. Higher Derivatives

3. LOCAL PROPERTIES OF ANALYTIC FUNCTIONS

3.1. Removable Singnlarities. Taylor's Theorem

3.2. Zeros and Poles

3.3. The Local Mapping

3.4. The Maximum Principle

4. THE GENERAL FORM OF CAUCHY'S THEOREM

4.1. Chains tind Cycles

4.2. Simple Connectivity

4.3. Homology

4.4. The General Statement of Cauchy's Theorem

4.5. Proof of Cauchy's Theorem

4.6. Locally Exact Differentials

4.7. Multiply Connected Regions

5. THE CALCULUS OF RESIDUES

5.1. The Residue Theorem

5.2. The Argument Principle

5.3. Evaluation of Definite Integrals

6. HARMONIC FUNCTIONS

6.1. Definition and Basic Properties.

6.2. The Mean-value Property

6.3. Poisson's Formnla

6.4. Schwarz's Theorem

6.5. The Reflection Principle

5 SERIES AND PRODUCT DEVELOPMENTS

1. POWER SERIES EXPANSIONS

I .1. Weierstrass's Theorem

1.2. The Taylor Series

1.3. The Laurent Series

2. PARTIAL FRACTIONS AND FACTORIZATION

2.1. Partial Fractions

2.2. Infinite Products

2.3. Canonical Products

2.4. The Gamma Function

2.5. Stirling's Formula

3. ENTIRE FUNCTIONS

3.1. Jensen's Formula

3.2. Hadamard's Theorem

4. THE RIEMANN ZETA FUNCTION

4.1. The Product Development

4.2. Extension of t(s) to the Whole Plane

4.3. The Functional Equation

4.4. The Zeros of the Zeta Function

5. NORMAL FAMILIES

5.1. Equicontinuity

5.2. Normality and Compactness

5.3. Arzela's Theorem

5.4. Families of Analytic Functions

5.5. The Classical Definition

6 CONFORMAL MAPPING. DIRICHLET'S PROBLEM

1. THE RIEMANN MAPPING THEOREM

1.1. Statement and Proof

1.2. Boundary Behavior

1.3. Use of the Reflection Principle

1.4. Analytic Arcs

2. CONFORMAL MAPPING OF POLYGONS

2.1. The Behavior at an Angle

2.2. The Schwarz- Christoffel Formula

2.3. Mapping on a Rectangle

2.4. The Triangle Functions of Schwarz

3. A CLOSER LOOK AT HARMONIC FUNCTIONS

3.1. Functions with the Mean-value Property

3.2. Harnack's Principle

4. THE DIRICHLET PROBLEM

4.1. Subharmonic Functions

4.2. Solution of Dirichlet's Problem

5. CANONICAL MAPPINGS OF MULTIPLY CONNECTED REGIONS

5.1. Harmonic Measures

5.2. Green's Function

5.3. Parallel Slit Regions

7 ELLIPTIC FUNCTIONS

1. SIMPLY PERIODIC FUNCTIONS

1.1. Representation by Exponentials

1.2. The Fourier Development

1.3. Functions of Finite Order

2. DOUBLY PERIODIC FUNCTIONS

2.1. The Period Module

2.2. Unimodular Transformations

2.3. The Canonical Basis

2.4. General Properties of Elliptic Functions

3. THE WEIERSTRASS THEORY

3.1. The Weierstrass rf'-function

3.2. The Functions t(z) and u(z).

3.3. The Differential Equation

3.4. The Modular Function

3.5. The Conformal Mapping by A(r).

8 GLOBAL ANALYTIC FUNCTIONS

1. ANALYTIC CONTINUATION

1.1. The Weierstrass Theory

1.2. Germs and Sheaves

1.3. Sections and Riemann Surfaces

1.4. Analytic Continuation along Arcs

1.5. Homotopic Curves

1.6. The Monodromy Theorem

1.7. Branch Points

2. ALGEBRAIC FUNCTIONS

2.1. The Resultant of Two Polynomials

2.2. Definition and Properties of Algebraic Functions

2.3. Behavior at the Critical Points

3. PICARD'S THEOREM

3.1. Lacunary Values

4. LINEAR DIFFERENTIAL EQUATIONS

4.1. Ordinary Points

4.2. Regular Singular Points

4.3. Solutions at Infinity

4.4. The Hypergeometric Differential Equation

4.5. Riemann's Point of View

Index

COMPLEX ANALYSIS

COMPLEX ANALYSIS An Introduction to the Theory of Analytic Functions of One Complex Variable Third Edition Lars V. Ahlfors Professor of Mathematics, Emeritus Harvard University McGraw-Hill, Inc. New York St. Louis San Francisco Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto

COMPLEX ANALYSIS Copyright© 1979, 1966 by McGraw-Hill, Inc. All rights reserved. Copyright 1953 by McGraw-Hill, Inc. All rights reserved. Printed in the United States of America. 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. 2223 BRBBRB 9876543 This book was set in Modern SA by Monotype Composition Company, Inc. The editors were Carol Napier and Stephen Wagley; the production supervisor was Joe Campanella. Library of Congre~~~~ Cataloging in Publication Data Ahlf ors, Lars Valerian, date Complex analysis. (International series in pure and applied mathematics) Includes index. 1. Analytic functions. I. Title. QA331.A45 1979 ISBN M7-000657-1 515' .93 78-17078

To Erna

Contents Preface CHAPTER 1 COMPLEX NUMBERS 1 The Algebra of Complex Numbers 1.1 Arithmetic Operations 1.2 Square Roots 1.3 Justification 1.4 Conjugation, Absolute Value 1.5 Inequalities 2 The Geometric Representation of Complex Numbers 2.1 Geometric Addition and Multiplication 2.2 The Binomial Equation 2.3 Analytic Geometry 2.4 The Spherical Representation CHAPTER 2 COMPLEX FUNCTIONS 1 Introduction to the Concept of Analytic Function 1.1 Limits and Continuity 1.2 Analytic Functions 1.3 Polynomials 1.4 Rational Functions 2 Elementary Theory of Power Series 2.1 Sequences 2.2 Series xiii 1 1 1 3 4 6 9 12 12 15 17 18 21 21 22 24 28 30 33 33 35 vii

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