内容简介:
本书初版于20世纪40年代，是经典的本科数学教材之一，对复变函数的教学影响深远，被美国加州理工学院、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。.

本书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。..

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。...

目录:
Preface .

1 Complex Numbers

Sums and Products

Basic Algebraic Properties

Further Properties

Vectors and Moduli

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 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 and Derivatives of Logarithms

Some Identities Involving Logarithms

Complex Exponents

Trigonometric Functions

Hyperbolic Functions

Inverse Trigonometric and Hyperbolic Functions

4 Integrals

Derivatives of Funcfons w(t)

Definite Integrals of Functions w(t)

Contours

Contour Integrals

Some Examples

Examples with 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

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

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

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 I.emma

Indented Paths

An Indentation Around a Branch Point

Integration Along a Branch Cut

Definite Integrals Involving Sines and Cosines

Argument Principle

Roucht'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 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

Related Boundary Value Problems

Schwarz Integral Formula

Dirichlet Problem for a Half Plane

Neumann Problems

Appendixes

Bibliography

Table of Transformations of Regions

Index ...

前言:
This book is a revision of the seventh edition, which was published in 2004. 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 new 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. .

The first objective of the book is to develop those parts of the theory that are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. With regard to residues, special emphasis is given to their use in evaluating real improper integrals, finding inverse Laplace transforms, and locating zeros of functions. As for conformal mapping, considerable attention is paid to its use in solving boundary value problems that arise in studies of heat conduction and fluid flow. Hence the book may be considered as a companion volume to the authors' text "Fourier Series and Boundary Value Problems," where another classical method for solving boundary value problems in partial differential equations is developed.

The first nine chapters of this book have for many years formed the basis of a three-hour course given each term at The University of Michigan. The classes have consisted mainly of seniors and graduate students concentrating in mathematics, engineering, or one of the physical sciences. Before taking the course, the students have completed at least a three-term calculus sequence and a first course in ordinary differential equations. Much of the material in the book need not be covered in the lectures and can be left for self-study or used for reference. If mapping by elementary functions is desired earlier in the course, one can skip to Chap. 8 immediately after Chap. 3 on elementary functions.

In order to accommodate as wide a range of readers as possible, there are footnotes referring to other texts that give proofs and discussions of the more delicate results from calculus and advanced calculus that are occasionally needed. A bibliography of other books on complex variables, many of which are more advanced, is provided in Appendix 1. A table of conformal transformations that are useful in applications appears in Appendix 2. ..

The main changes in this edition appear in the first nine chapters. Many of those changes have been suggested by users of the last edition. Some readers have urged that sections which can be skipped or postponed without disruption be more clearly identified. The statements of Taylor's theorem and Laurent's theorem, for example, now appear in sections that are separate from the sections containing their proofs. Another significant change involves the extended form of the Cauchy integral formula for derivatives. The treatment of that extension has been completely rewritten, and its immediate consequences are now more focused and appear together in a single section.

Other improvements that seemed necessary include more details in arguments involving mathematical induction, a greater emphasis on rules for using complex exponents, some discussion of residues at infinity, and a clearer exposition of real improper integrals and their Cauchy principal values. In addition, some rearrangement of material was called for. For instance, the discussion of upper bounds of. moduli of integrals is now entirely in one section, and there is a separate section devoted to the definition and illustration of isolated singular points. Exercise sets occur more frequently than in earlier editions and, as a result, concentrate more directly on the material at hand.

Finally, there is an Instructor's Solutions Manual (ISBN: 978-0-07-333730-2; MHID: 0-07-333730-7) that is available upon request to instructors who adopt the book. It contains solutions of selected exercises in Chapters 1 through 7, covering the material through residues.

In the preparation of this edition, continual interest and support has been provided by a variety of people, especially the staff at McGraw-Hill and my wife Jacqueline Read Brown. ...

James Ward Brown

更多详情