The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra reading material for students on their own. 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.

此书为英文版。

Foreword

Prerequisites

PART ONE

Basle 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. Differentiation of Power Series

6. The Invelse and Open Mapping Theorems

7. The Local Maximum Modulus Principle

CHAPTER III

Cauchy's Theorem, First Part

1. Holomorphic Functions on Connected Sets Appendix: Connectedness

2. Integrals Oer Paths

3. Local Primitive for a Holomorphic Function

4. Ancther Description of 1he Integral Along a Path

5. The Homotopy Form of Cauchy's Theorem

6. Existence of Global Primitives. Definition of the Logarithm

7. The Local Cauchy Formula

CHAPTER IV

Winding Numbers and Cauchy's Theorem

1. The Winding Number

2. The Global Catchy Theorem Dixon's PIocf of Theorem 2.5 (Cauchy's Formula)

3. Artin's Proof

CHAPTER V

Applications 1 Cauchy's Integral Formula

1. Uniform Limits of Analytic Functions

2. Lament Series

3. Isolated Singularities

Removable Singularities

Poles

E sential Singularities

CHAPTER Vl

Calculus ot Residues

1. The Residue Formula

Residues of Differentials

2. Evaluation of Definite Integrals

Fourier Transforms

Trigonometric Integrals

Mellin Transforms

CHAPTER VII

Conlormsl Mappings

1. Schwarz Lemma

2. Analytic Automorphisms of the Dic

3. The Upper Half Plane

4. Olher Examples

5. Fractional Linear Transformations

CHAPTER VIII

Harmonic Functions

I. Definition

Application: Perpendicularity

Application: Flow Lines

2. Examples

3. Basic Prol;erties of Harmonic Functions

4. The Poisson Formula

The Poisson Integral as a Convolution

5. Construction of Harmonic Furctions

6. Appendix. Differentiating Under the Int(gral Sign

PART TWO

Geometric Function Theory

CHAPTER IX

Schwarz Reflection

t. Schwarz Reflection (by Complex Conjugation)

,2. Reflection Across Analytic Arcs

3. Application cf Schwatz Reflection

CHAPTER X

The Riemann Mapping Theorem

1. Statement of the Theorem

2. Compact Sets in Function Spces

3. Proof cf the Riemann Mapping Theorem

4. Behavior at the Boundary

CHAPTEA Rnalytic ContinuatiXl on Along Curves

1. Continuation Along a Curve

2. The Dilogarithm 

3. Application lo Picard's Theorem

PART THREE

Various Analytic Topics

CHAPTER XII

Applications of the Maximum Modulus Principle and Jensen's Formula

1. Jensen's Formula

2. The Picard-Borel Theorem

3. Bounds by the Real Part, Borel-Carathrodory Theorem

4. The Use cf Three Circles and the Effect of Small Derivatives Hermite Interpolation Formula

5. Entire Functions with Rational Valves

6. The Phragmen-Lindelrf and Hadamard Theorems

CHAPTER XIII

Entire and Meromorphic Functions

1. Infinite Products

2. Weierstrass Products

3. Functions of Finite Order

4. Meromorphic Functions, Mittag-Leffler Theorem

CHAPTER XIV

Elliptic Functions

1. The Liouville Theorems

2. The Weierstrass Function

3. The Addition Theorem

4. The Sigma and Zeta Functions

CHAPTER XV

The Gamma and Zeta Functions

1. The Differentiation Lemma

2. The Gamma Function

Weierstrass Product

The Gauss Multiplication Formula (Distribution Relation)

The (Other) Gauss Formula

The Mellin Transform

The Starling Formula

Proof of Starling's Formula

3. The Lerch Formula

4. Zeta Functions

CHAPTER XVl

The Prime Number Theorem

1. Basic Analytic Properties of the Zeta Function

2. The Main Lemma and its Application

3. Proof of the Main Lemma

Appenflix

1. Summation by Parts and Non-Absolute Convergence

2. Difference Equations

3. Analytic Differential Equations

4. Fixed Points of a Fractional Linear Transformation

5. Cauchy's Formula for C Functions

6. Cauchy's Theorem for Locally Integrable Vector Fields

Bibliography

Index