(特价书)复变函数及应用(英文影印版.第8版)

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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