Chapter 1 Complex Numbers and Functions
1 Complex Numbers
1.1 Complex Number Field
1.2 Complex Plane
1.3 Modulus, Conjugation, Argument,and Polar Representation
1.4 Powers and Roots of Complex Numbers
2 Regions in the Complex Plane
2.1 Some Basic Concepts
2.2 Domain and Jordan Curve
3 Functions of a Complex Variable
3.1 The Concept of Functions of a Complex Variable
3.2 Limits and Continuous
4 The Extended Complex Plane and the Point at Infinity
4.1 The Spherical Representation,and the Extended Complex Plane
4.2 Some Concepts in the Extended Complex Plane
Chapter 2 Analytic Functions
1 The Concept of the Analytic Function
1.1 The Derivative of Functions of a Complex Variable
1.2 Analytic Functions
2 Cauchy-Riemann Equations
3 Elementary Functions
3.1 Exponential Functions
3.2 Trigonometric Functions
3.3 Hyperbolic Functions
4 Multi-valued Functions
4.1 Logarithmic Functions
4.2 Complex Power Functions
4.3 Inverse Trigonometric and Hyperbolic Functions
Chapter 3 Complex Integration
1 The Concept of Contour Integral
1.1 Integral of a Complex Function over a Real Interval
1.2 Contour Integrals
2 Cauchy-Goursat Theorem
2.1 Cauchy-Goursat Theorem
2.2 Caucby Integral Formula
2.3 Derivatives of Analytic Functions
2.4 Liouville' s Theorem and the Fundamental Theorem of Algebra ~
3 Harmonic Functions
Chapter 4 Series
1 Basic Properties of Series
1.1 Convergence of Sequences
1.2 Convergence of Series
1.3 Uniform Convergence
2 Power Series
3 Taylor Series
4 Laurent Series
5 Zeros of Analytic Functions and Uniquely Determined Analytic Functions
5.1 Zeros of Analytic Functions
5.2 Uniquely Determined Analytic Functions
5.3 Maximum Modulus Principle
6 Three Types of Isolated Singular Points at a Finite Point
7 Three Types of Isolated Singular Points at an Infinite Point
Chapter 5 Calculus of Residues
1 Residues
1.1 Residues
1.2 Cauehy' s Residue Theorem
1.3 The Calculus of Residue
2 Applications of Residue
2.1 The Type of Definite Integral ∫2π0 F(sinθ,cosθ)dθ
2.2 The Type of Improper Integral□（数理化公式）
2.3 The TypeoflmproperIntegral□（数理化公式）
3 Argument Principle
Chapter 6 Conformal Mappings
1 Analytic Transformation
1.1 Preservation of Domains of Analytic Transformation
1.2 Conformality of Analytic Transformation
2 Rational Functions
2.1 Polynomials
2.2 Rational Functions
3 Fractional Linear Transformations
4 Elementary Conformal Mappings
5 The Riemann Mapping Theorem
Appendix 1 Common Mathematical English Vocabulary
Appendix 2 Mathematical English Vocabulary Related to Complex Variable Function Theory
References

本书是一本关于复分析的英文书。复分析是研究复函数，特别是亚纯函数和复解析函数的数学理论。复分析的应用领域较为广泛，在其他数学分支和物理学中也起着重要的作用，包括数论、应用数学、流体力学、热力学和电动力学。复分析研究的主要内容包括复数与复变函数、解析函数、复变函数的积分、级数、留数及其应用和共形映射等。复分析研究的函数定义在复平面上，其值为复数，而且可微。研究中常用的理论、公式及方法包括柯西积分定理、柯西积分公式、留数定理、洛朗级数展开等。本书的出版具有重要的学术价值和经济社会效益。