复变量

出版时间：1970-1  出版社：世界图书出版公司  作者：阿布娄韦提兹  页数：647  
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内容概要

　　《复变量(第2版)》是Cambridge《应用数学系列丛书》之一，内容相当精辟，巧妙地展示了复变量在数学科学中的核心地位以及其在工程和物理科学应用中的关键性作用。复变量的引入不仅增加数学理论本身的完美性，更重要的是提供了一种解决一些数学疑难问题的途径，甚至可以说是解决有些问题的唯一途径。　　《复变量(第2版)》的内容分为两大部分。第一部分是整个课程的引入，包括：解析函数，积分，级数和残数积分等初等理论以及一些过渡性方法：复平面的普通微分方程、数值方法等。第二部分包括保形映射，渐近扩张以及Riemann-Hilbert问题。每章节都提供了大量的应用、图例以及练习，这些可以帮助读者加深对复变量的基本概念和基本定理的理解。新版本做了全新的改进，是研究生以及分析方向本科生的理想教程。

书籍目录

Sections denoted with an asterisk (*) can be either omitted or readindependently.PrefacePartⅠ Fundamentals and Techniques of Complex Function Theory1 Complex Numbers and Elementary Functions1.1 Complex Numbers and Their Properties1.2 Elementary Functions and Stereographic Projections1.2.1 Elementary Functions1.2.2 Stereographic Projections1.3 Limits， Continuity， and Complex Differentiation1.4 Elementary Applications to Ordinary Differential Equations2 Analytic Functions and Integration2.1 Analytic Functions2.1.1 The Cauchy-Riemann Equations2.1.2 Ideal Fluid Flow2.2 Multivalued Functions*2.3 More Complicated Multivalued Functions and Riemann Surfaces2.4 Complex Integration2.5 Cauchys Theorem2.6 Cauchys Integral Formula， Its a Generalization and Consequences2.6.1 Cauchys Integral Formula and Its Derivatives*2.6.2 Liouville， Morera， and Maximum-Modulus Theorems*2.6.3 Generalized Cauchy Formula and a Derivatives*2.7 Theoretical Developments3 Sequences， Series， and Singularities of Complex Functions3.1 Definitions and Basic Properties of Complex Sequences，Series3.2 Taylor Series3.3 Laurent Series*3.4 Theoretical Results for Sequences and Series3.5 Singularities of Complex Functions3.5.1 Analytic Continuation and Natural Barriers*3.6 Infinite Products and Mittag-Leffler Expansions*3.7 Differential Equations in the Complex Plane： Painleve Equations*3.8 Computational Methods*3.8.1 Laurent Series*3.8.2 Differential Equations4 Residue Calculus and Applications of Contour Integration4.1 Cauchy Residue Theorem4.2 Evaluation of Certain Definite Integrals4.3 Principal Value Integrals and Integrals with Branch Points4.3.1 Principal Value Integrals4.3.2 Integrals with Branch Points4.4 The Argument Principle， Rouches Theorem*4.5 Fourier and Laplace Transforms*4.6 Applications of Transforms to Differential EquationsPartⅡ Applications of Complex Function Theory5 Conformal Mappings and Applications5.1 Introduction5.2 Conformal Transformations5.3 Critical Points and Inverse Mappings5.4 Physical Applications*5.5 Theoretical Considerations - Mapping Theorems5.6 The Schwarz-Christoffel Transformation5.7 Bilinear Transformations*5.8 Mappings Involving Circular Arcs5.9 Other Considerations5.9.1 Rational Functions of the Second Degree5.9.2 The Modulus of a Quadrilateral*5.9.3 Computational Issues6 Asymptotic Evaluation of Integrals6.1 Introduction6.1.1 Fundamental Concepts6.1.2 Elementary Examples6.2 Laplace Type Integrals6.2.1 Integration by Parts6.2.2 Watsons Lemma6.2.3 Laplaces Method6.3 Fourier Type Integrals6.3.1 Integration by Parts6.3.2 Analog of Watsons Lcmma6.3.3 The Stationary Phase Method6.4 The Method of Steepest Descent6.4.1 Laplaces Method for Complex Contours6.5 Applications6.6 The Stokes Phenomenon*6.6.1 Smoothing of Stokes Discontinuities6.7 Related Techniques*6.7.1 WKB Method*6.7.2 The Mellin Transform Method7 Riemann-Hiibert Problems7.1 Introduction7.2 Cauchy Type Integrals7.3 Scalar Riemann-Hilbert Problems7.3.1 Closed Contours7.3.2 Open Contours7.3.3 Singular Integral Equations7.4 Applications of Scalar Riemann-Hilbert Problems7.4.1 Riemann-Hilbert Problems on the Real Axis7.4.2 The Fourier Transform7.4.3 The Radon Transform*7.5 Matrix Riemann-Hilbert Problems7.5.1 The Riemann-Hilbert Problem for Rational Matrices7.5.2 Inhomogeneous Riemann-Hilbert Problems and Singular Equations7.5.3 The Riemann-Hilbert Problem for Triangular Matrices7.5.4 Some Results on Zero Indices7.6 The DBAR Problem7.6.1 Generalized Analytic Functions*7.7 Applications of Matrix Riemann-Hilbert Problems and ProblemsAppendix A Answers to Odd-Numbered ExercisesBibliographyIndex

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