復(fù)變函數(shù)及應(yīng)用

前言

　　This book is a revision of the seventh edition， which was published in 2004. Thatedition has served, just as the earlier ones did, as a textbook for a oneterm 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, thefirst 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 areprominent 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, findinginverse Laplace transforms, and locating zeros of functions. As for conformal mapping, considerable attention is paid to its use in solving boundary value problemsthat arise in studies of heat conduction and fluid flow. Hence the book may beconsidered as a companion volume to the authors text "Fourier Series and Boundary Value Problems," where another classical method for solving boundary valueproblems in partial differential equations is developed.　　The first nine chapters of this book have for many years formed the basis of athreehour course given each term at The University of Michigan. The classes haveconsisted mainly of seniors and graduate students concentrating in mathematics,engineering, or one of the physical sciences. Before taking the course, the studentshave completed at least a threeterm calculus sequence and a first course in ordinarydifferential equations. Much of the material in the book need not be covered in thelectures and can be left for selfstudy or used for reference. If mapping by elementaryfunctions is desired earlier in the course, one can skip to Chap. 8 immediately afterChap. 3 on elementary functions.

內(nèi)容概要

本書初版于20世紀(jì)40年代，是經(jīng)典的本科數(shù)學(xué)教材之一，對復(fù)變函數(shù)的教學(xué)影響深遠(yuǎn)，被美國加州理工學(xué)院、加州大學(xué)伯克利分校、佐治亞理工學(xué)院、普度大學(xué)、達(dá)特茅斯學(xué)院、南加州大學(xué)等眾多名校采用。    本書闡述了復(fù)變函數(shù)的理論及應(yīng)用，還介紹了留數(shù)及保形映射理論在物理、流體及熱傳導(dǎo)等邊值問題中的應(yīng)用。    新版對原有內(nèi)容進(jìn)行了重新組織，增加了更現(xiàn)代的示例和應(yīng)用，更加方便教學(xué)。

作者簡介

　　James Ward Brown密歇根大學(xué)迪爾本分校數(shù)學(xué)系教授，美國數(shù)學(xué)學(xué)會會員。1964年于密歇根大學(xué)獲得數(shù)學(xué)博士學(xué)位。他曾經(jīng)主持研究美國國家自然科學(xué)基金項目，獲得過密歇根大學(xué)杰出教師獎，并被列入美國名人錄。　　Ruel V.Churchill已故密歇根大學(xué)知名教授。早在60多年前，就開始編寫一系列經(jīng)典教材。除本書外，還與James Ward Brown合著《Fourier Series and Boundary Value Problems》。

書籍目錄

Preface1 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 Functions 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  5 Series6 Residues and Poles7 Applications of Residues8 Mapping by Elementary Functions9 Conformal Mapping10 Applications of Conformal Mapping11 The Schwarz-Chrstoffer Transformation12 Integral Formulas of the Poisson TypeAppendixesIndex

章節(jié)摘錄

　　The first objective of.the book is to develop those parts of the theory that areprominent in applications of the subject. The second objective is to furnish an intro-duction to applications of residues and conformal mapping. With regard to residues，special emphasis is given to their use in evaluating real improper integrals, findinginverse Laplace transforms, and locating zeros of functions. As for conformal map-ping, considerable attention is paid to its use in solving boundary value problemsthat arise in studies of heat conduction and fluid flow. Hence the book may beconsidered as a companion volume to the authors text "Fourier Series and Bound-ary Value Problems," where another classical method for solving boundary valueproblems in partial differential equations is developed.　　The first nine chapters of this book have for many years formed the basis of athree-hour course given each term at The University of Michigan. The classes haveconsisted mainly of seniors and graduate students concentrating in mathematics,engineering, or one of the physical sciences. Before taking the course, the studentshave completed at least a three-term calculus sequence and a first course in ordinarydifferential equations. Much of the material in the book need not be covered in thelectures and can be left for self-study or used for reference.

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