復變量

內容概要

　　《復變量(第2版)》是Cambridge《應用數學系列叢書》之一，內容相當精辟，巧妙地展示了復變量在數學科學中的核心地位以及其在工程和物理科學應用中的關鍵性作用。復變量的引入不僅增加數學理論本身的完美性，更重要的是提供了一種解決一些數學疑難問題的途徑，甚至可以說是解決有些問題的唯一途徑?！　　稄妥兞?第2版)》的內容分為兩大部分。第一部分是整個課程的引入，包括：解析函數，積分，級數和殘數積分等初等理論以及一些過渡性方法：復平面的普通微分方程、數值方法等。第二部分包括保形映射，漸近擴張以及Riemann-Hilbert問題。每章節(jié)都提供了大量的應用、圖例以及練習，這些可以幫助讀者加深對復變量的基本概念和基本定理的理解。新版本做了全新的改進，是研究生以及分析方向本科生的理想教程。

書籍目錄

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

圖書封面

圖書標簽Tags

無

評論、評分、閱讀與下載

---

    復變量 PDF格式下載

---