数学物理方法(第四版)[十二五国家级规划教材]

姚端正,武汉大学二级教授,博士导师,享受国务院政府特殊津贴,首届国家级教学名师奖的获得者,湖北省有突出贡献的中青年专家,全国高等院校数学物理研究会副理事长。姚端正教授长期从事非线性光学和数学物理领域的科研工作, 近年来她主要从事半导体限制结构(如量子阱,量子点)以及相干瞬态光学效应等光学非线性的理论研究。此外,还从事光纤中非线性光传输特性的解析研究。先后主持或承担湖北省自然科学基金、国家自然科学基金以及国防项目等科研项目10项,在PHYSICAL REVIEW A,Applied PhysicsLetters,Applied Physics B等国内外重要学术刊物上发表论文80余篇,以第一完成人获国家教育部科技进步二等奖一项。已指导硕、博士研究生30余名,目前正指导博士研究生4名。 目录 第一篇复变函数论 第一章解析函数 1.1复数及其运算 习题1.1 1.2复变函数 习题1.2 1.3微商及解析函数 习题1.3 1.4初等解析函数 习题1.4 1.5解析函数的几何性质 习题1.5 本章小结 第二章解析函数积分 2.1复变函数的积分 习题2.1 2.2柯西定理 习题2.2 2.3柯西积分公式 习题2.3 本章小结 第三章复变函数级数 3.1复级数 3.2幂级数 习题3.2 3.3泰勒级数 习题3.3 3.4洛朗级数 习题3.4 3.5单值函数的孤立奇点 习题3.5 本章小结 第四章解析延拓Γ函数B函数 4.1解析延拓 习题4.1 4.2Γ函数 习题4.2 *4.3B函数 习题4.3 本章小结 第五章留数理论 5.1留数定理 习题5.1 5.2利用留数理论计算实积分 习题5.2 5.3物理问题中的几个积分 习题5.3 *5.4多值函数的积分 习题5.4 本章小结 第二篇数学物理方程 第六章定解问题 6.1引言 6.2三类数理方程的导出 习题6.2 6.3定解条件 习题6.3 本章小结 第七章行波法 7.1无界弦的自由振动达朗贝尔公式 习题7.1 7.2无界弦的强迫振动 习题7.2 *7.3三维无界空间的自由振动泊松公式 习题7.3 *7.4三维无界空间的受迫振动推迟势 本章小结 第八章分离变量法 8.1有界弦的自由振动 习题8.1 8.2非齐次方程纯强迫振动 习题8.2 8.3非齐次边界条件的处理 习题8.3 8.4正交曲线坐标系中的分离变量法 习题8.4 本章小结 第九章积分变换法 9.1傅里叶变换 习题9.1 9.2傅里叶变换法 习题9.2 9.3拉普拉斯变换 习题9.3 9.4拉普拉斯变换法 习题9.4 本章小结 第十章格林函数法 10.1δ函数 习题10.1 10.2边值问题的格林函数法 习题10.2 10.3稳恒问题的格林函数 习题10.3 10.4电像法与狄氏格林函数 习题10.4 *10.5含时问题的格林函数法 习题10.5 本章小结 第三篇特殊函数 第十一章勒让德多项式 11.1勒让德多项式 习题11.1 11.2勒让德多项式的性质 习题11.2 11.3连带勒让德函数与球函数 习题11.3 本章小结 第十二章贝塞尔函数 12.1贝塞尔函数 习题12.1 12.2贝塞尔函数的性质 习题12.2 *12.3其他柱函数 习题12.3 本章小结 第十三章施图姆-刘维尔理论 13.1施图姆-刘维尔本征值问题 习题13.1 *13.2高斯方程和库默尔方程 本篇主要特殊函数性质小结 *第四篇近似方法及现代内容 第十四章变分法 14.1泛函和泛函的极值 习题14.1 14.2用变分法解数理方程 习题14.2 本章小结 第十五章非线性方程 15.1非线性方程的某些初等解法 习题15.1 15.2孤波和孤子 习题15.2 15.3解析近似法之正则摄动法 习题15.3 15.4数值解法之分步傅里叶变换法 本章小结 第十六章积分方程 16.1积分方程的几种解法 习题16.1 16.2施密特-希尔伯特理论 习题16.2 16.3维纳-霍普夫方法 习题16.3 本章小结 第十七章小波变换 17.1小波变换的由来 17.2小波变换 习题参考答案 参考文献 附录 I.矢量微分算子与拉普拉斯算符 II.傅里叶变换简表 III.拉普拉斯变换简表 索引 Contents Part One Theory of Complex Variable Function Chapter 1 Analytic Function 1.1 Complex numbers and their operations Problem 1.1 1.2 Function of a complex variable Problem 1.2 1.3 Derivative and analytic function Problem 1.3 1.4 Elementary analytic functions Problem 1.4 1.5 GeometricProperties of analytic functions Problem 1.5 Summary for chapter 1 Chapter 2 The Integral of the Analytic Function 2.1 The integral of the variable function Problem 2.1 2.2 The Cauchy theorem Problem 2.2 2.3 Cauchy integral formula Problem 2.3 Summary for chapter 2 Chapter 3 Series of Complex Variable Function 3.1 Complex series 3.2 Power series Problem 3.2 3.3 Taylor series Problem 3.3 3.4 Laurent series Problem 3.4 3.5 The isolated singularity of the single valued function Problem 3.5 Summary for chapter 3 Chapter 4 Analytic ContinuationΓ FunctionB Function 4.1 Analytic continuation Problem 4.1 4.2 Γ function Problem 4.2 *4.3 B function Problem 4.3 Summary for chapter 4 Chapter 5 The Residue Theory 5.1 The Residue theorem Problem 5.1 5.2 Real integral calculated by the Residue theorem Problem 5.2 5.3 Several integrals in physicalProblems Problem 5.3 *5.4 Integrals of the multi valued function Problem 5.4 Summary for chapter 5 Part TwoEquations of Mathematical Physics Chapter 6 Complete Mathematical Models 6.1 Introduction 6.2 Derivation for three type of equations of mathematical physics Problem 6.2 6.3 Boundary conditions and initial conditons Problem 6.3 Summary for chapter 6 Chapter 7 Method of Traveling Waves 7.1 Solution for the free vibration of the unbounded string D’Alembert formula Problem 7.1 7.2 Solution for the pure forced vibration of the unbounded string Problem 7.2 *7.3 The free vibration in three dimensional unbounded space Poisson formula Problem 7.3 *7.4 The forced vibration in three dimensional unbounded space Retarded potentials Summary for chapter 7 Chapter 8 The Method of Separation of Variables 8.1 The free vibration of the bounded string Problem 8.1 8.2 The non homogeneous equation the pure forced vibration Problem 8.2 8.3 The treatment of the non homogeneous boundary conditions Problem 8.3 8.4 The separation of variables in the orthogonal curvilinear coordinates Problem 8.4 Summary for chapter 8 Chapter 9 I ntegral Variable Method 9.1 Fourier transforms Problem 9.1 9.2 Fourier transform method Problem 9.2 9.3 Laplace transforms Problem 9.3 9.4 Laplace transform method Problem 9.4 Summary for chapter 9 Chapter 10 Green’s Function Method 10.1 δ function Problem 10.1 10.2 Boundary valueProblems solved by Green’s function method Problem 10.2 10.3 Green’s function for stableProblem Problem 10.3 10.4 Electro image method and Dirichlet Green’s function Problem 10.4 *10.5 Time related mathematicalProblems solved by Green’s function method Problem 10.5 Summary for chapter 10 Part ThreeSpecial Functions Chapter 11 Legendre Polynomials 11.1 Legendre polynomials Problem 11.1 11.2 TheProperty of Legendre polynomials Problem 11.2 11.3 Associated Legendre funtion and spherical function Problem 11.3 Summary for chapter 11 Chapter 12 Bessel functions 12.1 Bessel functions Problem 12.1 12.2 TheProperty of Bessel functions Problem 12.2 *12.3 Other Column Functions Problem 12.3 Summary for chapter 12 Chapter 13 Sturm Liouville Theory 13.1 Sturm Liouville EigenvalueProblem Problem 13.1 *13.2 Gauss equation and the Kummer equation Summary for theProperties of the principal special functions in part three *Part FourApproximation Method and Modern Content Chapter 14 The Variational Method 14.1 Functionals and functional extreme Problem 14.1 14.2 Solving equations of mathematical physics by the variational method Problem 14.2 Summary for chapter 14 Chapter 15 Nonlinear Equations 15.1 Some Primary solution method for nonlinear equations Problem 15.1 15.2 Isolated wave and soliton Problem 15.2 15.3 The regular perturbation method in analytical approximation methods Problem 15.3 15.4 A step by step Fourier transform method for numericalapproximate methods Summary for chapter 15 Chapter 16 Integral Equations 16.1 Several methods for solving integral equations Problem 16.1 16.2 Schimidt Hilbert theory Problem 16.2 16.3 Wiener Hopf method Problem 16.3 Summary for chapter 16 Chapter 17 Wavelet transform 17.1 The origin of wavelet transform 17.2 Wavelet transform Reference keys to theProblems Bibliography and references Appendix I.Vector Differential Operator and Laplace Operator II.Summary table of Fourier transform III.Summary table of Laplace transform Index