第1章复变函数与解析函数
1.1复数及其基本运算(complex numbers and operations)
1.1.1复数的基本概念(concepts of complex numbers)
1.1.2复数的表示方法(algebraic and geometric structure of complex numbers)
1.1.3复数的基本运算(operation of complex numbers)
1.1.4基于MATLAB的复数运算(complex number operations based on MATLAB)
1.2复变函数(complex variable functions)
1.2.1复变函数的概念(concepts and properties of complex variable function)
1.2.2区域的相关概念(concepts of domain)
1.2.3复变函数的极限和连续(limit and continuity of complex variable function)
1.3导数及解析函数(derivative and analytic function)
1.3.1导数(derivative)
1.3.2函数可导的充分必要条件(sufficient conditions for derivability)
1.3.3解析函数(analytic function)
1.3.4初等解析函数及性质(elementary analytic function and properties)
1.3.5运用MATLAB工具使复变函数可视化(visualization of complex function based on MATLAB)
1.4解析函数的应用(application of analytic function)
1.4.1解析函数在平面静电场中的应用(application of analytic function in the plane electrostatic field)
1.4.2保角变换及其几何解释(conformal mapping and its geometric interpretations)
1.4.3解析函数在系统稳态响应问题求解中的应用(application of analytic function in oscillation system)
第1章习题
第2章解析函数积分
2.1复变函数的积分(integral of complex variable function)
2.1.1复变函数积分的基本概念(concepts of complex integral)
2.1.2复变函数积分的性质(properties of complex integral)
2.1.3复变函数积分实例(examples of complex integral)
2.2柯西定理(Cauchy theorem)
2.2.1单连通区域情形的柯西定理(Cauchy theorem in simply connected domains)
2.2.2不定积分和原函数(indefinite integral and antiderivative)
2.2.3复连通区域的柯西定理(Cauchy theorem in multiply connected domains)
2.2.4复变函数积分的MATLAB运算(calculation of complex integral based on MATLAB)
2.3柯西公式及推论(Cauchy formula and extension)
2.3.1单连通区域的柯西积分公式(Cauchy formula in simply connected domain)
2.3.2复连通区域的柯西积分公式(Cauchy formula in multiply connected domain)
2.3.3无界区域中的柯西积分公式(Cauchy formula for unbounded domain)
2.3.4柯西公式推论(extension of Cauchy formula)
2.4柯西定理及柯西公式应用实例(application examples of Cauchy theorem and Cauchy formula)
第2章习题
第3章复变函数级数
3.1复数项级数(complex number series)
3.1.1复数项级数的概念(concepts of complex number series)
3.1.2复数项级数的性质(properties of complex number series)
3.1.3复变函数项级数(series of complex functions)
3.2幂级数(power series)
3.2.1幂级数概念(concepts of power series)
3.2.2收敛半径与收敛圆(radius of convergence and circle of convergence)
3.2.3幂级数的性质(properties of power series)
3.3泰勒级数(Taylor series)
3.3.1解析函数的泰勒展开式(Taylor expansion of analytic function)
3.3.2泰勒级数的收敛半径(radius of convergence of Taylor series)
3.3.3将函数展开成泰勒级数的实例(examples of Taylor series expansion)
3.4洛朗级数(Laurent series)
3.4.1洛朗级数定义(definition of Laurent series)
3.4.2洛朗级数的收敛性(convergence of Laurent series)
3.4.3洛朗级数展开实例(examples of Laurent series expansion)
3.5单值函数的孤立奇点(isolated singular points of singlevalued functions)
3.6基于MATLAB的幂级数展开(power series expansion based on MATLAB)
第3章习题
第4章留数定理及其应用
4.1留数定理(residue theorem)
4.1.1闭合回路积分与留数的关系(loop integral and residue)
4.1.2留数的计算(calculation of residue)
4.1.3基于MATLAB的留数计算(residue calculation based on MATLAB)
4.2利用留数定理计算实积分(application of residue theorem for calculation of real integral)
4.2.1类型Ⅰ实积分计算(type Ⅰ real integral)
4.2.2类型Ⅱ实积分计算(type Ⅱ real integral)
4.2.3类型Ⅲ实积分计算(type Ⅲ real integral)
4.3其他类型的实积分计算(calculation of other real integral)
4.4基于MATLAB的回路积分计算(loop integral calculation based on MATLAB)
第4章习题
第5章傅里叶级数
5.1周期函数的傅里叶展开(Fourier expansion of periodic function)
5.1.1傅里叶级数的定义(definition of Fourier series)
5.1.2傅里叶级数的实际意义(practical meaning of Fourier series)
5.1.3傅里叶级数的收敛性(convergence of Fourier series)
5.2奇函数及偶函数的傅里叶展开(Fourier expansion of odd and even function)
5.3定义在有界区间上函数的傅里叶展开(Fourier expansion of functions defined on an interval)
5.4复数形式的傅里叶级数(Fourier series in complex form)
5.5基于MATLAB的傅里叶级数可视化(visualization of Fourier series based on MATLAB)
第5章习题
第6章数学建模——数学物理定解问题
6.1基本概念(basic concepts)
6.2典型的数理方程(typical mathematical physics equation)
6.2.1波动方程(wave equation)
6.2.2热传导方程(heat conduction equation)
6.2.3泊松方程(Poisson equation)
6.3定解条件(definite solution condition)
6.3.1初始条件(initial condition)
6.3.2边界条件(boundary condition)
6.3.3数学物理定解问题的适定性(well posed problems in mathematical physics)
6.4二阶线性偏微分方程的分类和特征(classification and characteristics of second order linear partial differential equations)
6.4.1二阶线性偏微分方程的分类(classification of second order linear partial differential equations)
6.4.2二阶线性偏微分方程解的特征(characteristics of solutions of second order linear partial differential equations)
6.5行波法与达朗贝尔公式(traveling wave method and d’Alembert formula)
6.5.1一维波动方程的达朗贝尔公式(d’Alembert formula for one dimensional wave equation)
6.5.2达朗贝尔公式的物理意义(physical meaning of d’Alembert formula)
6.5.3达朗贝尔公式应用的MATLAB实现(application of d’Alembert formula based on MATLAB)
第6章习题
第7章分离变量法
7.1分离变量法的理论(theory of variable separation)
7.1.1分离变量法简介(introduction of variable separation method)
7.1.2偏微分方程可实施变量分离的条件(conditions for variable separation of PDEs)
7.1.3边界条件可实施变量分离的条件(conditions for variable separation of boundary conditions)
7.2直角坐标系中的分离变量法(variable separation method in rectangular coordinate system)
7.2.1分离变量法的求解步骤(steps of variable separation method)
7.2.2解的物理意义(physical meaning of solution)
7.2.3三维情况下的直角坐标分离变量(variable separation for 3D problem in rectangular coordinate)
7.3非齐次边界条件齐次化(homogenization of nonhomogeneous boundary conditions)
7.3.1非齐次边界条件齐次化的一般方法(general method)
7.3.2非齐次边界条件齐次化的特殊方法(special method)
7.4非齐次方程(inhomogeneous equation)
7.5泊松方程(Poisson equation)
7.6基于MATLAB的数学物理方程数值求解(numerical solution of mathematical physics equation based on MATLAB)
7.6.1有限元法介绍(introduction of the finite element method)
7.6.2MATLAB PDE工具箱(MATLAB PDE toolbox)
第7章习题
第8章二阶常微分方程的级数解法和本征值问题
8.1柱坐标系和球坐标系下的分离变量法(variable separation in spherical and cylindrical coordinate system)
8.1.1三种常用的正交坐标系(three types of coordinates)
8.1.2拉普拉斯方程的分离变量(Laplace equation)
8.1.3三维波动方程的分离变量(3D wave equation)
8.1.4三维输运方程/热传导方程的分离变量(3D transport equation/heat conduct equation)
8.1.5亥姆霍兹方程的分离变量(Helmholtz equation)
8.2常点邻域的级数解法(power series solution around ordinary points)
8.3施图姆刘维尔本征值问题(SturmLiouville eigenvalue problem)
8.3.1施图姆刘维尔型方程及本征值问题(SturmLiouville equation and eigenvalue problem)
8.3.2施图姆刘维尔本征值问题的性质及广义傅里叶级数(characteristics of SturmLiouville eigenvalue problem and generalized Fourier series)
第8章习题
第9章特殊函数
9.1勒让德多项式(Legendre polynomials)
9.1.1勒让德方程及其级数解(Legendre equation and power series solution)
9.1.2本征值问题(eigenvalue problem)
9.1.3勒让德多项式的表达式(Legendre polynomials)
9.1.4勒让德多项式的性质(characteristics of Legendre polynomials)
9.1.5勒让德多项式的MATLAB可视化(visualization of Legendre polynomials based on MATLAB)
9.1.6广义傅里叶级数(generalized Fourier series)
9.1.7轴对称定解问题(axisymmetric problems in spherical coordinate)
9.1.8勒让德多项式的生成函数(generating function of Legendre polynomial)
9.1.9勒让德多项式的递推公式(recurrence formula of Legendre polynomials)
9.2贝塞尔函数(Bessel function)
9.2.1三类柱函数(three types of cylindrical functions)
9.2.2贝塞尔函数和诺伊曼函数的MATLAB可视化(visualization of Bessel function and Neumann function based on MATLAB)
9.2.3贝塞尔函数的基本性质(characteristics of Bessel functions)
9.2.4贝塞尔方程本征值问题(Bessel equation eigenvalue problem)
9.2.5傅里叶贝塞尔级数(FourierBessel series)
9.3虚宗量贝塞尔函数(Bessel function of imaginary argument)
9.3.1虚宗量贝塞尔方程的解(solution of modified Bessel equation)
9.3.2虚宗量贝塞尔函数和虚宗量汉克尔函数的MATLAB可视化(visualization of modified Bessel function and modified Hankel function baese on MATLAB)
9.3.3虚宗量贝塞尔函数和虚宗量汉克尔函数的性质(characteristics of modified Bessel function and modified Hankel function)
9.4特殊函数的应用实例(application examples of special functions)
9.4.1拉普拉斯方程定解问题(Laplace equation problems)
9.4.2阶跃光纤的分析(analysis of step optical fibre)
9.4.3表面等离激元(plasmonics)
第9章习题
第10章数理方程的其他方法
10.1保角变换法(conformal mapping)
10.1.1常用的保角变换函数(analytic functions for conformal mapping)
10.1.2应用举例(examples)
10.2有限差分法(finite difference method)
10.2.1差分的基本概念(concepts)
10.2.2二维拉普拉斯方程的差分方程(difference equation of 2D Laplace equation)
10.2.3边界上的差分方程(difference equation on boundary grids)
10.2.4二维静态电磁场差分方程的迭代法求解(iterative method solution for 2D static electromagnetic field difference equation)
10.3有限元法(finite element method)
10.3.1有限元法的基本原理(principle of FEM)
10.3.2有限元法求解案例(examples)
第10章习题
习题答案
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