曹怀信编著的《复变函数引论(第2版)》简介：For several years, I have been conducting courses in Complex Analysis, Real Analysis and Functional Analysis in a so-called "bilingual" way. That is, the lessons are given with Chinese textbooks, but mainly teached in English. The main purpose of teaching in this way is to improve the undergraduate students' ability to read and write English. Using a Chinese textbook in such "bilingual" courses is not, however, useful for training students' ability of English-thinking. Consequently, although there are a number of books on complex analysis in Chinese, in order to meet the requirements of bilingual teaching, it is necessary to write a textbook on complex analysis in English for Chinese undergraduate students. This is just the main aim of compiling the present book.

Preface

Chapter Ⅰ Complex Number Field

1.1 Sums and Products

1.2 Basic Algebraic Properties

1.3 Further Properties

1.4 Moduli

1.5 Conjugates

1.6 Exponential Form

1.7 Products and Quotients in Exponential Form

1.8 Roots of Complex Numbers

1.9 Examples

1.10 Regions in the Complex Plane

Chapter Ⅱ Analytic Functions

2.1 Functions of a Complex Variable

2.2 Mappings

2.3 The Exponential Function and its Mapping Properties

2.4 Limits

2.5 Theorems on Limits

2.6 Limits Involving the Point at Infinity

2.7 Continuity

2.8 Derivatives

2.9 Differentiation Formulas

2.10 Cauchy-Riemann Equations

2.11 Necessary and Sufficient Conditions for Differentiability

2.12 Polar Coordinates

2.13 Analytic Functions

2.14 Examples

215 Harmonic Functions

Chapter Ⅲ Elementary Functions

3.1 The Exponential Function

3.2 The Logarithmic Function

3.3 Branches and Derivatives of Logarithms

3.4 Some Identities on Logarithms

3.5 Complex Power Functions

36 Trigonometric Functions

3.7 Hyperbolic Functions

3.8 Inverse Trigonometric and Hyperbolic Functions

Chapter Ⅳ Integrals

4.1 Derivatives of Complex-Valued Functions of One Real Variable

4.2 Definite Integrals of Functions W

4.3 Paths

4.4 Path Integrals

4.5 Examples

4.6 Upper Bounds for Integrals

4.7 Primitive Functions

4.8 Examples

4.9 Cauchy Integral Theorem

4.10 Proof of the Cauchy Integral Theorem

4.11 Extended Cauchy Integral Theorem

4.12 Cauchy Integral Formula

4.13 Derivatives of Analytic Functions

4.14 Liouville's Theorem

4.15 Maximum Modulus Principle

Chapter Ⅴ Series

5.1 Convergence of Series

5.2 Taylor Series

5.3 Examples

5.4 Laurent Series

5.5 Examples

5.6 Absolute and Uniform Convergence of Power Series

5.7 Continuity of Sums of Power Series

5.8 Integration and Differentiation of Power Series

5.9 Uniqueness of Series Representations

5.10 Multiplication and Division of Power Series

Chapter Ⅵ Residues and Poles

6.1 Residues

6.2 Cauchy's Residue Theorem

6.3 Using a Single Residue

6.4 The Three Types of Isolated Singular Points

6.5 Residues at poles

6.6 Examples

6.7 Zeros of Analytic Functions

6.8 Uniquely Determined Analytic Functions

6.9 Zeros and Poles

6.10 Behavior of f Near Isolated Singular Points

6.11 Reflection Principle

Chapter Ⅶ Applications of Residues

7 I Evaluation of Improper Integrals

7.2 Examples

7.3 Improper Integrals From Fourier Analysis

7.4 Jordan's Lemma

7.5 Indented Paths

7.6 An Indentation Around a Branch Point

7.7 Definite Integrals Involving Sine and Cosine

7.8 Argument Principle

7.9 Rouche's Theorem

Chapter Ⅷ Conformal Mappings

8.1 Conformal mappings

82 Unilateral Functions

8.3 Local Inverses

84 Affine Transformations

85 The Transformation W = 1/z

8.6 Mappings by 1/z

8.7 Fractional Linear Transformations

8.8 Cross Ratios

8.9 Mappings of the Upper Half Plane