复变函数及应用

Preface

1 Complex Numbers

Sums and Products

Basic Algebraic Properties

Further Algebraic Properties

Vectors and Moduli

Triangle Inequality

Complex Conjugates

Exponential Form

Products and Powers in Exponential Form

Arguments of Products and Quotients

Roots of Complex Numbers

Examples

Regions in the Complex Plane

2 Analytic Functions

Functions and Mappings

The Mapping w = zz

Limits

Theorems on Limits

Limits Involving the Point at Infinity

Continuity

Derivatives

Rules for Differentiation

Cauchy-Riemann Equations

Examples

Sufficient Conditions for Differentiability

Polar Coordinates

Analytic Functions

Further Examples

Harmonic Functions

Uniquely Determined Analytic Functions

Reflection Principle

3 Elementary Functions

The Exponential Function

The Logarithmic Function

Examples

Branches and Derivatives of Logarithms

Some Identities Involving Logarithms

The Power Function

Examples

The Trigonometric Functions sin z and cos z

Zeros and Singularities of Trigonometric Functions

Hyperbolic Functions

Inverse Trigonometric and Hyperbolic Functions

4 Integrals

Derivatives of Functions w (t)

Definite Integrals of Functions w (t)

Contours

Contour Integrals

Some Examples

Examples Involving Branch Cuts

Upper Bounds for Moduli of Contour Integrals

Antiderivatives

Proof of the Theorem

Cauchy-Goursat Theorem

Proof of the Theorem

Simply Connected Domains

Multiply Connected Domains

Cauchy Integral Formula

An Extension of the Cauchy Integral Formula

Verification of the Extension

Some Consequences of the Extension

Liouville's Theorem and the Fundamental Theorem of Algebra

Maximum Modulus Principle

5 Series

Convergence of Sequences

Convergence of Series

Taylor Series

Proof of Taylor's Theorem

Examples

Negative Powers of (z - z0)

Laurent Series

Proof of Laurent's Theorem

Examples

Absolute and Uniform Convergence of Power Series

Continuity of Sums of Power Series

Integration and Differentiation of Power Series

Uniqueness of Series Representations

Multiplication and Division of Power Series

6 Residues and Poles

Isolated Singular Points

Residues

Cauchy's Residue Theorem

Residue at Infinity

The Three Types of Isolated Singular Points

Examples

Residues at Poles

Examples

Zeros of Analytic Functions

Zeros and Poles

Behavior of Functions Near Isolated Singular Points

7 Applications of Residues

Evaluation of Improper Integrals

Example

Improper Integrals from Fourier Analysis

Jordan's Lemma

An Indented Path

An Indentation Around a Branch Point

Integration Along a Branch Cut

Definite Integrals Involving Sines and Cosines

Argument Principle

Rouche's Theorem

Inverse Laplace Transforms

Mapping by Elementary Functions

Linear Transformations

The Transformation w = 1/z

Mappings by 1/z

Linear Fractional Transformations

An Implicit Form

Mappings of the Upper Half Plane

Examples

Mappings by the Exponential Function

Mapping Vertical Line Segments by w=sin z

Mapping Horizontal Line Segments by w=sin z

Some Related Mappings

Mappings by z2

Mappings by Branches of z1/2

Square Roots of Polynomials

Riemann Surfaces

Surfaces for Related Functions

9 Conformal Mapping

Preservation of Angles and Scale Factors

Further Examples

Local Inverses

Harmonic Conjugates

Transformations of Harmonic Functions

Transformations of Boundary Conditions

10 Applications of Conformal Mapping

Steady Temperatures

Steady Temperatures in a Half Plane

A Related Problem

Temperatures in a Quadrant

Electrostatic Potential

Examples

Two-Dimensional Fluid Flow

The Stream Function

Flows Around a Comer and Around a Cylinder

11 The Schwarz-Christoffel Transformation

Mapping the Real Axis onto a Polygon

Schwarz-Christoffel Transformation

Triangles and Rectangles

Degenerate Polygons

Fluid Flow in a Channel through a Slit

Flow in a Channel with an Offset

Electrostatic Potential about an Edge of a Conducting Plate

12 Integral Formulas of the Poisson Type

Poisson Integral Formula

Dirichlet Problem for a Disk

Examples

Related Boundary Value Problems

Schwarz Integral Formula

Dirichlet Problem for a Half Plane

Neumann Problems

Appendixes

Bibliography

Table of Transformations of Regions

Index

《华章数学原版精品系列：复变函数及应用（英文版·第9版）》阐述了复变函数的理论及应用，还介绍了留数及保形映射理论在物理、流体及热传导等边值问题中的应用。这本经典的教材对复变函数的教学影响深远，被美国斯坦福大学、加州理工学院、加州大学伯克利分校、佐治亚理工学院、普度大学、达特茅斯学院、南加州大学等众多名校采用。新版对原有内容进行了重新组织，增加了更现代的示例和应用，更加方便教学。