Elias M.Stein、Rami Shakarchi所著的《复分析》由在国际上享有盛誉普林斯大林顿大学教授Stein等撰写而成，是一部为数学及相关专业大学二年级和三年级学生编写的教材，理论与实践并重。为了便于非数学专业的学生学习，全书内容简明、易懂,读者只需掌握微积分和线性代数知识。与本书相配套的教材《傅立叶分析导论》和《实分析》也已影印出版。本书已被哈佛大学和加利福尼亚理工学院选为教材。

Foreword

Introduction

Chapter 1. Preliminaries to Complex Analysis

 1 Complex numbers and the complex plane

1.1 Basic properties

1.2 Convergence

1.3 Sets in the complex plane

 2 Functions on the complex plane

2.1 Continuous functions

2.2 Holomorphic functions

2.3 Power series

 3 Integration along curves

 4 Exercises

Chapter 2. Cauchy's Theorem and Its Applications

 1 Goursat's theorem

 2 Local existence of primitives and Cauchy's theorem in a disc

 3 Evaluation of some integrals

 4 Cauchy's integral formulas

 5 Further applications

5.1 Morera's theorem

5.2 Sequences of holomorphic functions

5.3 Holomorphic functions defined in terms of integrals

5.4 Schwarz reflection principle

5.5 Runge's approximation theorem

 6 Exercises

 7 Problems

Chapter 3. Meromorphic Functions and the Logarithm

 1 Zeros and poles

 2 The residue formula

2.1 Examples

 3 Singularities and meromorphic functions

 4 The argument principle and applications

 5 Homotopies and simply connected domains

 6 The complex logarithm

 7 Fourier series and harmonic functions

 8 Exercises

 9 Problems

Chapter 4. The Fourier Transform

 1 The class ξ

 2 Action of the Fourier transform on ξ

 3 Paley-Wiener theorem

 4 Exercises

 5 Problems

Chapter 5. Entire Functions

 1 Jensen's formula

 2 Functions of finite order

 3 Infinite products

3.1 Generalities

3.2 Example: the product formula for the sine function

 4 Weierstrass infinite products

 5 Hadamard's factorization theorem

 6 Exercises

 7 Problems

Chapter 6. The Gamma and Zeta Functions

 1 The gamma function

1.1 Analytic continuation

1.2 Further properties of τ

 2 The zeta function

2.1 Functional equation and analytic continuation

 3 Exercises

 4 Problems

Chapter 7. The Zeta Function and Prime Number Theorem

 1 Zeros of the zeta function

1.1 Estimates for 1/ζ(s)

 2 Reduction to the functions ψ and ψ1

2.1 Proof of the asymptotics for ψ1

 Note on interchanging double sums

 3 Exercises

 4 Problems

 Chapter 8. Conformal Mappings

 1 Conformal equivalence and examples

1.1 The disc and Upper half-plane

1.2 Further examples

1.3 The Dirichlet problem in a strip

 2 The Schwarz lemma; automorphisms of the disc and upper half-plane

2.1 Automorphisms of the disc

2.2 Automorphisms of the upper half-plane

 3 The Riemann mapping theorem

3.1 Necessary conditions and statement of the theorem

3.2 Montel's theorem

3.3 Proof of the Riemann mapping theorem

 4 Conformal mappings onto polygons

4.1 Some examples

4.2 The Schwarz-Christoffel integral

4.3 Boundary behavior

4.4 The mapping formula

4.5 Return to elliptic integrals

 5 Exercises

 6 Problems

 Chapter 9. An Introduction to Elliptic Functions

 1 Elliptic functions

1.1 Liouville's theorems

1.2 The Weierstrass p function

 2 The modular character of elliptic functions and Eisenstein series

2.1 Eisenstein series

2.2 Eisenstein series and divisor functions

 3 Exercises

 4 Problems

 Chapter 10. Applications of Theta Functions

 1 Product formula for the Jacobi theta function

1.1 Further transformation laws

 2 Generating functions

 3 The theorems about sums of squares

3.1 The two-squares theorem

3.2 The four-squares theorem

 4 Exercises

 5 Problems

Appendix A: Asymptotics

 1 Bessel functions

 2 Laplace's method; Stirling's formula

 3 The Airy function

 4 The partition function

 5 Problems

Appendix B: Simple Connectivity and Jordan Curve Theorem

 1 Equivalent formulations of simple connectivity

 2 The Jordan curve theorem

2.1 Proof of a general form of Cauchy's theorem

Notes and References

Bibliography

Symbol Glossary

Index