复变函数及应用（英文版·第7版）

本书始印于20世纪40年代，是本科数学方面为数不多的几本典范教材之一。James Ward Brown在Ruel V. Churchill原作的基础上加以扩充，编写出了复变函数课程基础教材的最新版本。

本书的首要目标是阐述复变函数应用方面的一些重要理论；另外，提供有关于残数及共形映射应用方面的介绍。

与第6版相比，第7版提供了更多的新例子、图片和补充材料；专门为讲解例子增设了许多新的小节；重新绘制了部分插图。

本书的首要目标是阐述复变函数应用方面的一些重要理论；另外，提供有关于残数及共形映射应用方面的介绍。

与第6版相比，第7版提供了更多的新例子、图片和补充材料；专门为讲解例子增设了许多新的小节；重新绘制了部分插图。

James Ward Brown is Professor of Mathematics at The University of Michigan Dearborn. He earned his A.B. in Physics at Harvard University and his A.M. and Ph.D. in Mathematics at The University of Michigan in Ann Arbor, where he was an Institute of Science and Technology Predoctoral Fellow. He was coauthor with Dr. Churchill of the sixth edition of Fourier Series and Boundary Value Problems. A past director of a research grant from the National Science Foundation, he is the recipient of a Distinguished Teaching Award from his institution, as well as a Distinguished Faculty Award from the Michigan Association of Governing Boards of Colleges and Universities. He is listed in Who's Who in America.

Ruel V. Churchill is Late Professor of Mathematics at The University of Michigan, where he began teaching in 1922. He received his B.S. in Physics from the University of Chicago and his M.S. in Physics and Ph.D. in Mathematics from The University of Michigan. He was coauthor with Dr. Brown of the recent sixth edition of Fourier Series and Boundary Value Problems, a classic text that he first wrote over sixty years ago. He was also the author of Operational Mathematics, now in its third edition. Throughout his long and productive career, Dr. Churchill held various offices in the Mathematical Association of America and in other mathematical societies and councils.

James Ward Brown密歇根大学迪尔本分校数学系教授，美国数学学会会员。1964年于密歇根大学获得数学博士学位。他曾经主持研究美国国家自然科学基金项目，获得过密歇根大学杰出教师奖，并被列入美国名人录。

Ruel V．Churchill 已故密歇根大学知名教授。早在60多年前，就开始编写一系列经典教材。除本书外，还与James Ward Brown合著有《Fourier Series and Boundary Value Problems》、《Selections from Complex Variables，5th ed》等。另外独自著有《Operational Mathematics》一书。他曾在美国数学学会等研究机构担任过多项职务。

Ruel V. Churchill is Late Professor of Mathematics at The University of Michigan, where he began teaching in 1922. He received his B.S. in Physics from the University of Chicago and his M.S. in Physics and Ph.D. in Mathematics from The University of Michigan. He was coauthor with Dr. Brown of the recent sixth edition of Fourier Series and Boundary Value Problems, a classic text that he first wrote over sixty years ago. He was also the author of Operational Mathematics, now in its third edition. Throughout his long and productive career, Dr. Churchill held various offices in the Mathematical Association of America and in other mathematical societies and councils.

James Ward Brown密歇根大学迪尔本分校数学系教授，美国数学学会会员。1964年于密歇根大学获得数学博士学位。他曾经主持研究美国国家自然科学基金项目，获得过密歇根大学杰出教师奖，并被列入美国名人录。

Ruel V．Churchill 已故密歇根大学知名教授。早在60多年前，就开始编写一系列经典教材。除本书外，还与James Ward Brown合著有《Fourier Series and Boundary Value Problems》、《Selections from Complex Variables，5th ed》等。另外独自著有《Operational Mathematics》一书。他曾在美国数学学会等研究机构担任过多项职务。

This book is a revision of the sixth edition, published in 1996. That edition has served, just as the earlier ones did, as a textbook for a one-term introductory course in the theory and application of functions of a complex variable. This edition preserves the basic content and style of the earlier editions, the first two of which were written by the late Ruel V. Churchill alone.

In this edition, the main changes appear in the first nine chapters, which make up the core of a one-term course. The remaining three chapters are devoted to physical applications, from which a selection can be made, and are intended mainly for selfstudy or reference.

Among major improvements, there are thirty new figures; and many of the old ones have been redrawn. Certain sections have been divided up in order to emphasize specific topics, and a number of new sections have been devoted exclusively to examples. Sections that can be skipped or postponed without disruption are more clearly identified in order to make more time for material that is absolutely essential in a first course, or for selected applications later on. Throughout the book, exercise sets occur more often than in earlier editions. As a result, the number of exercises in any given set is generally smaller, thus making it more convenient for an instructor in assigning homework.

As for other improvements in this edition, we mention that the introductory material on mappings in Chap. 2 has been simplified and now includes mapping properties of the exponential function. There has been some rearrangement of material in Chap. 3 on elementary functions, in order to make the flow of topics more natural. Specifically, the sections on logarithms now directly follow the one on the exponential function; and the sections on trigonometric and hyberbolic functions are now closer to the ones on their inverses. Encouraged by comments from users of the book in the past several years, we have brought some important material out of the exercises and into the text. Examples of this are the treatment of isolated zeros of analytic functions in Chap. 6 and the discussion of integration along indented paths in Chap. 7.

The first objective of the book is to develop those parts of the theory which are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. Special emphasis is given to the use of conformal mapping in solving boundary value problems that arise in studies of heat conduction, electrostatic potential, and fluid flow. Hence the book may be considered as a companion volume to the authors' "Fourier Series and Boundary Value Problems" and Ruel V. Churchill's "Operational Mathematics," where other classical methods for solving boundary value problems in partial differential equations are developed. The latter book also contains further applications of residues in connection with Laplace transforms.

This book has been used for many years in a three-hour course given each term at The University of Michigan. The classes have consisted mainly of seniors and graduate students majoring in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence, a first course in ordinary differential equations, and sometimes a term of advanced calculus. In order to accommodate as wide a range of readers as possible, there are footnotes referring to texts that give proofs and discussions of the more delicate results from calculus that are occasionally needed. Some of the material in the book need not be covered in lectures and can be left for students to read on their own. If mapping by elementary functions and applications of conformal mapping are desired earlier in the course, one can skip to Chapters 8, 9, and 10 immediately after Chapter 3 on elementary functions.

Most of the basic results are stated as theorems or corollaries, followed by examples and exercises illustrating those results. A bibliography of other books, many of which are more advanced, is provided in Appendix 1. A table of conformal transformations useful in applications appears in Appendix 2.

In the preparation of this edition, continual interest and support has been provided by a number of people, mar y of whom are family, colleagues, and students. They include Jacqueline R. Brown Ronald P. Morash, Margret H. Hoft, Sandra M. Weber, Joyce A. Moss, as well as Robert E. Ross and Michelle D. Munn of the editorial staff at McGraw-Hill Higher Education.

James Ward Brown

In this edition, the main changes appear in the first nine chapters, which make up the core of a one-term course. The remaining three chapters are devoted to physical applications, from which a selection can be made, and are intended mainly for selfstudy or reference.

Among major improvements, there are thirty new figures; and many of the old ones have been redrawn. Certain sections have been divided up in order to emphasize specific topics, and a number of new sections have been devoted exclusively to examples. Sections that can be skipped or postponed without disruption are more clearly identified in order to make more time for material that is absolutely essential in a first course, or for selected applications later on. Throughout the book, exercise sets occur more often than in earlier editions. As a result, the number of exercises in any given set is generally smaller, thus making it more convenient for an instructor in assigning homework.

As for other improvements in this edition, we mention that the introductory material on mappings in Chap. 2 has been simplified and now includes mapping properties of the exponential function. There has been some rearrangement of material in Chap. 3 on elementary functions, in order to make the flow of topics more natural. Specifically, the sections on logarithms now directly follow the one on the exponential function; and the sections on trigonometric and hyberbolic functions are now closer to the ones on their inverses. Encouraged by comments from users of the book in the past several years, we have brought some important material out of the exercises and into the text. Examples of this are the treatment of isolated zeros of analytic functions in Chap. 6 and the discussion of integration along indented paths in Chap. 7.

The first objective of the book is to develop those parts of the theory which are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. Special emphasis is given to the use of conformal mapping in solving boundary value problems that arise in studies of heat conduction, electrostatic potential, and fluid flow. Hence the book may be considered as a companion volume to the authors' "Fourier Series and Boundary Value Problems" and Ruel V. Churchill's "Operational Mathematics," where other classical methods for solving boundary value problems in partial differential equations are developed. The latter book also contains further applications of residues in connection with Laplace transforms.

This book has been used for many years in a three-hour course given each term at The University of Michigan. The classes have consisted mainly of seniors and graduate students majoring in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence, a first course in ordinary differential equations, and sometimes a term of advanced calculus. In order to accommodate as wide a range of readers as possible, there are footnotes referring to texts that give proofs and discussions of the more delicate results from calculus that are occasionally needed. Some of the material in the book need not be covered in lectures and can be left for students to read on their own. If mapping by elementary functions and applications of conformal mapping are desired earlier in the course, one can skip to Chapters 8, 9, and 10 immediately after Chapter 3 on elementary functions.

Most of the basic results are stated as theorems or corollaries, followed by examples and exercises illustrating those results. A bibliography of other books, many of which are more advanced, is provided in Appendix 1. A table of conformal transformations useful in applications appears in Appendix 2.

In the preparation of this edition, continual interest and support has been provided by a number of people, mar y of whom are family, colleagues, and students. They include Jacqueline R. Brown Ronald P. Morash, Margret H. Hoft, Sandra M. Weber, Joyce A. Moss, as well as Robert E. Ross and Michelle D. Munn of the editorial staff at McGraw-Hill Higher Education.

James Ward Brown

Preface

1 Complex Numbers

Sums and Products

Basic Algebraic Properties

Further Properties

Moduli

Complex Conjugates

Exponential Form

Products and Quotients in Exponential Form

Roots of Complex Numbers

Examples

Regions in the Complex Plane

2 Analytic Functions

Functions of a Complex Variable

Mappings

Mappings by the Exponential Function

Limits

Theorems on Limits

Limits Involving the Point at Infinity

Continuity

Derivatives

Differentiation Formulas

Cauchy-Riemann Equations

Sufficient Conditions for Differentiability

Polar Coordinates

Analytic Functions

Examples

Harmonic Functions

Uniquely Determined Analytic Functions

Reflection Principle

3 Elementary Functions

The Exponential Function

The Logarithmic Function

Branches mid Derivatives of Logarithms

Some Identities Involving Logarithms

Complex Exponents

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

Examples

Upper Bounds for Moduli of Contour Integrals

Antiderivatives

Examples

Cauchy-Goursat Theorem

Proof of the Theorem

Simply and Multiply Connected Domains

Cauchy Integral Formula

Derivatives of Analytic Functions

Liouville's Theorem and the Fundamental Theorem of Algebra

Maximum Modulus Principle

5 Series

Convergence of Sequences

Convergence of Series

Taylor Series

Examples

Laurent Series

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 Series6 Residues and Poles

6 Residues

Cauchy's Residue Theorem

Using a Single Residue

The Three Types of Isolated Singular Points

Residues at Poles

Examples

Zeros of Analytic Functions

Zeros and Poles

Behavior of f Near Isolated Singular Points

7 Applications of Residues

Evaluation of Improper Integrals

Example

Improper Integrals from Fourier Analysis

Jordan's Lemma

Indented Paths

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

Examples

8 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

The Transformation w=sin z

Mappings by z2 and Branches of z1/2

Square Roots of Polynomials

Riemann Surfaces

Surfaces for Related Functions

9 Conformal Mapping

Preservation of Angles

Scale Factors

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

Potential in a Cylindrical Space

Two-Dimensional Fluid Flow

The Stream Function

Flows Around a Corner 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

Related Boundary Value Problems

Schwarz Integral Formula

Dirichlet Problem for a Half Plane

Neumann Problems

Appendixes

Bibliography

Table of Transformations of Regions

Index

1 Complex Numbers

Sums and Products

Basic Algebraic Properties

Further Properties

Moduli

Complex Conjugates

Exponential Form

Products and Quotients in Exponential Form

Roots of Complex Numbers

Examples

Regions in the Complex Plane

2 Analytic Functions

Functions of a Complex Variable

Mappings

Mappings by the Exponential Function

Limits

Theorems on Limits

Limits Involving the Point at Infinity

Continuity

Derivatives

Differentiation Formulas

Cauchy-Riemann Equations

Sufficient Conditions for Differentiability

Polar Coordinates

Analytic Functions

Examples

Harmonic Functions

Uniquely Determined Analytic Functions

Reflection Principle

3 Elementary Functions

The Exponential Function

The Logarithmic Function

Branches mid Derivatives of Logarithms

Some Identities Involving Logarithms

Complex Exponents

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

Examples

Upper Bounds for Moduli of Contour Integrals

Antiderivatives

Examples

Cauchy-Goursat Theorem

Proof of the Theorem

Simply and Multiply Connected Domains

Cauchy Integral Formula

Derivatives of Analytic Functions

Liouville's Theorem and the Fundamental Theorem of Algebra

Maximum Modulus Principle

5 Series

Convergence of Sequences

Convergence of Series

Taylor Series

Examples

Laurent Series

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 Series6 Residues and Poles

6 Residues

Cauchy's Residue Theorem

Using a Single Residue

The Three Types of Isolated Singular Points

Residues at Poles

Examples

Zeros of Analytic Functions

Zeros and Poles

Behavior of f Near Isolated Singular Points

7 Applications of Residues

Evaluation of Improper Integrals

Example

Improper Integrals from Fourier Analysis

Jordan's Lemma

Indented Paths

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

Examples

8 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

The Transformation w=sin z

Mappings by z2 and Branches of z1/2

Square Roots of Polynomials

Riemann Surfaces

Surfaces for Related Functions

9 Conformal Mapping

Preservation of Angles

Scale Factors

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

Potential in a Cylindrical Space

Two-Dimensional Fluid Flow

The Stream Function

Flows Around a Corner 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

Related Boundary Value Problems

Schwarz Integral Formula

Dirichlet Problem for a Half Plane

Neumann Problems

Appendixes

Bibliography

Table of Transformations of Regions

Index

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