本书是Cambridge《应用数学系列丛书》之一，内容相当精辟，巧妙地展示了复变量在数学科学中的核心地位以及其在工程和物理科学应用中的关键性作用。复变量的引入不仅增加数学理论本身的完美性，更重要的是提供了一种解决一些数学疑难问题的途径，甚至可以说是解决有些问题的唯一途径。

本书的内容分为两大部分。第一部分是整个课程的引入，包括：解析函数，积分，级数和残数积分等初等理论以及一些过渡性方法：复平面的普通微分方程、数值方法等。第二部分包括保形映射，渐近扩张以及Riemann-Hilbert问题。每章节都提供了大量的应用、图例以及练习，这些可以帮助读者加深对复变量的基本概念和基本定理的理解。

新版本做了全新的改进，是研究生以及分析方向本科生的理想教程。

Sections denoted with an asterisk (*) can be either omitted or read

independently.

Preface

Part I Fundamentals and Techniques of Complex Function Theory

1 Complex Numbers and Elementary Functions

1.1 Complex Numbers and Their Properties

1.2 Elementary Functions and Stereographic Projections

 1.2.1 Elementary Functions

 1.2.2 Stereographic Projections

1.3 Limits, Continuity, and Complex Differentiation

1.4 Elementary Applications to Ordinary Differential Equations

2 Analytic Functions and Integration

2.1 Analytic Functions

 2.1.1 The Cauchy-Riemann Equations

 2.1.2 Ideal Fluid Flow

2.2 Multivalued Functions

*2.3 More Complicated Multivalued Functions and Riemann Surfaces

2.4 Complex Integration

2.5 Cauchy's Theorem

2.6 Cauchy's Integral Formula, Its a Generalization and Consequences

 2.6.1 Cauchy's Integral Formula and Its Derivatives

 *2.6.2 Liouville, Morera, and Maximum-Modulus Theorems

 *2.6.3 Generalized Cauchy Formula and a Derivatives

*2.7 Theoretical Developments

3 Sequences, Series, and Singularities of Complex Functions

3.1 Definitions and Basic Properties of Complex Sequences,Series

3.2 Taylor Series

3.3 Laurent Series

*3.4 Theoretical Results for Sequences and Series

3.5 Singularities of Complex Functions

 3.5.1 Analytic Continuation and Natural Barriers

*3.6 Infinite Products and Mittag-Leffler Expansions

*3.7 Differential Equations in the Complex Plane: Painleve Equations

*3.8 Computational Methods

 *3.8.1 Laurent Series

 *3.8.2 Differential Equations

4 Residue Calculus and Applications of Contour Integration

4.1 Cauchy Residue Theorem

4.2 Evaluation of Certain Definite Integrals

4.3 Principal Value Integrals and Integrals with Branch Points

 4.3.1 Principal Value Integrals

 4.3.2 Integrals with Branch Points

4.4 The Argument Principle, Rouche's Theorem

*4.5 Fourier and Laplace Transforms

*4.6 Applications of Transforms to Differential Equations

Part II Applications of Complex Function Theory

5 Conformal Mappings and Applications

5.1 Introduction

5.2 Conformal Transformations

5.3 Critical Points and Inverse Mappings

5.4 Physical Applications

*5.5 Theoretical Considerations - Mapping Theorems

5.6 The Schwarz-Christoffel Transformation

5.7 Bilinear Transformations

*5.8 Mappings Involving Circular Arcs

5.9 Other Considerations

 5.9.1 Rational Functions of the Second Degree

 5.9.2 The Modulus of a Quadrilateral

 *5.9.3 Computational Issues

6 Asymptotic Evaluation of Integrals

6.1 Introduction

 6.1.1 Fundamental Concepts

 6.1.2 Elementary Examples

6.2 Laplace Type Integrals

 6.2.1 Integration by Parts

 6.2.2 Watson's Lemma

 6.2.3 Laplace's Method

6.3 Fourier Type Integrals

 6.3.1 Integration by Parts

 6.3.2 Analog of Watson's Lcmma

 6.3.3 The Stationary Phase Method

6.4 The Method of Steepest Descent

 6.4.1 Laplace's Method for Complex Contours

6.5 Applications

6.6 The Stokes Phenomenon

 *6.6.1 Smoothing of Stokes Discontinuities

6.7 Related Techniques

 *6.7.1 WKB Method

 *6.7.2 The Mellin Transform Method

7 Riemann-Hiibert Problems

7.1 Introduction

7.2 Cauchy Type Integrals

7.3 Scalar Riemann-Hilbert Problems

 7.3.1 Closed Contours

 7.3.2 Open Contours

 7.3.3 Singular Integral Equations

7.4 Applications of Scalar Riemann-Hilbert Problems

 7.4.1 Riemann-Hilbert Problems on the Real Axis

 7.4.2 The Fourier Transform

 7.4.3 The Radon Transform

*7.5 Matrix Riemann-Hilbert Problems

 7.5.1 The Riemann-Hilbert Problem for Rational Matrices

 7.5.2 Inhomogeneous Riemann-Hilbert Problems and Singular Equations

 7.5.3 The Riemann-Hilbert Problem for Triangular Matrices

 7.5.4 Some Results on Zero Indices

7.6 The DBAR Problem

 7.6.1 Generalized Analytic Functions

*7.7 Applications of Matrix Riemann-Hilbert Problems and Problems

Appendix A Answers to Odd-Numbered Exercises

Bibliography

Index