布朗、丘吉尔编著的《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。

布朗、丘吉尔编著的《复变函数及应用（英文版第9版）》是复分析入门教材，内容丰富，写作精炼，论证严密。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。第9版对第8版做了全面修订，重新组织了内容，增加了很多新的示例和习题，更加方便教学。

这本畅销全世界的经典教材初版于20世纪40年代，被国外众多名校广泛采用，如美国斯坦福大学、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。前几版曾被译成日语、西班牙语、阿拉伯语、希腊语、韩语等众多版本，对复变函数的教学影响深远。

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