第1章 複變函數與解析函數 1.1 複數及其基本運算(complex numbers and operations) 1.1.1 複數的基本概念(concepts of complex numbers) 1.1.2 複數的表示方法(algebraic and geometric structure of complex numbers) 1.1.3 複數的基本運算(operation of complex numbers) 1.1.4 基於MATLAB的複數運算(complex number operations based on MATLAB) 1.2 複變函數(complex variable functions) 1.2.1 複變函數的概念(concepts and properties of complex variable function) 1.2.2 區域的相關概念(concepts of domain) 1.2.3 複變函數的極限和連續(limit and continuity of complex variable function) 1.3 導數及解析函數(derivative and analytic function) 1.3.1 導數(derivative) 1.3.2 函數可導的充分必要條件(sufficient conditions for derivability) 1.3.3 解析函數(analytic function) 1.3.4 初等解析函數及性質(elementary analytic function and properties) 1.3.5 運用MATLAB工具使複變函數可視化(visualization of complex function based on MATLAB) 1.4 解析函數的應用(application of analytic function) 1.4.1 解析函數在平面靜電場中的應用(application of analytic function in the plane electrostatic field) 1.4.2 保角變換及其幾何解釋(conformal mapping and its geometric interpretations) 1.4.3 解析函數在系統穩態響應問題求解中的應用(application of analytic function in oscillation system) 第1章習題 第2章 解析函數積分 2.1 複變函數的積分(integral of complex variable function) 2.1.1 複變函數積分的基本概念(concepts of complex integral) 2.1.2 複變函數積分的性質(properties of complex integral) 2.1.3 複變函數積分實例(examples of complex integral) 2.2 柯西定理(Cauchy theorem) 2.2.1 單連通區域情形的柯西定理(Cauchy theorem in simply connected domains) 2.2.2 不定積分和原函數(indefinite integral and antiderivative) 2.2.3 復連通區域的柯西定理(Cauchy theorem in multiply connected domains) 2.2.4 複變函數積分的MATLAB運算(calculation of complex integral based on MATLAB) 2.3 柯西公式及推論(Cauchy formula and extension) 2.3.1 單連通區域的柯西積分公式(Cauchy formula in simply connected domain) 2.3.2 復連通區域的柯西積分公式(Cauchy formula in multiply connected domain) 2.3.3 無界區域中的柯西積分公式(Cauchy formula for unbounded domain) 2.3.4 柯西公式推論(extension of Cauchy formula) 2.4 柯西定理及柯西公式應用實例(application examples of Cauchy theorem and Cauchy formula) 第2章習題 第3章 複變函數級數 3.1 複數項級數(complex number series) 3.1.1 複數項級數的概念(concepts of complex number series) 3.1.2 複數項級數的性質(properties of complex number series) 3.1.3 複變函數項級數(series of complex functions) 3.2 冪級數(power series) 3.2.1 冪級數概念(concepts of power series)
3.3.3 將函數展開成泰勒級數的實例(examples of Taylor series expansion) 3.4 洛朗級數(Laurent series) 3.4.1 洛朗級數定義(definition of Laurent series) 3.4.2 洛朗級數的收斂性(convergence of Laurent series) 3.4.3 洛朗級數展開實例(examples of Laurent series expansion) 3.5 單值函數的孤立奇點(isolated singular points of single-valued functions) 3.6 基於MATLAB的冪級數展開(power series expansion based on MATLAB) 第3章習題 第4章 留數定理及其應用 4.1 留數定理(residue theorem) 4.1.1 閉合迴路積分與留數的關係(loop integral and residue) 4.1.2 留數的計算(calculation of residue) 4.1.3 基於MATLAB的留數計算(residue calculation based on MATLAB) 4.2 利用留數定理計算實積分(application of residue theorem for calculation of real integral) 4.2.1 類型Ⅰ實積分計算(type Ⅰ real integral) 4.2.2 類型Ⅱ實積分計算(type Ⅱ real integral) 4.2.3 類型Ⅲ實積分計算(type Ⅲ real integral) 4.3 其他類型的實積分計算(calculation of other real integral) 4.4 基於MATLAB的迴路積分計算(loop integral calculation based on MATLAB) 第4章習題 第5章 傅里葉級數 5.1 周期函數的傅里葉展開(Fourier expansion of periodic function) 5.1.1 傅里葉級數的定義(definition of Fourier series) 5.1.2 傅里葉級數的實際意義(practical meaning of Fourier series) 5.1.3 傅里葉級數的收斂性(convergence of Fourier series) 5.2 奇函數及偶函數的傅里葉展開(Fourier expansion of odd and even function) 5.3 定義在有界區間上函數的傅里葉展開(Fourier expansion of functions defined on an interval) 5.4 複數形式的傅里葉級數(Fourier series in complex form) 5.5 基於MATLAB的傅里葉級數可視化(visualization of Fourier series based on MATLAB) 第5章習題 第6章 數學建模——數學物理定解問題 6.1 基本概念(basic concepts) 6.2 典型的數理方程(typical mathematical physics equation) 6.2.1 波動方程(wave equation) 6.2.2 熱傳導方程(heat-conduction equation) 6.2.3 泊松方程(Poisson equation) 6.3 定解條件(definite solution condition) 6.3.1 初始條件(initial condition) 6.3.2 邊界條件(boundary condition) 6.3.3 數學物理定解問題的適定性(well-posed problems in mathematical physics) 6.4 二階線性偏微分方程的分類和特徵(classification and characteristics of second-order linear partial differential equations) 6.4.1 二階線性偏微分方程的分類(classification of second-order linear partial differential equations) 6.4.2 二階線性偏微分方程解的特徵(characteristics of solutions of second-order linear partial differential equations) 6.5 行波法與達朗貝爾公式(traveling wave method and d』Alembert formula) 6.5.1 一維波動方程的達朗貝爾公式(d』Alembert formula for one dimensional wave equation) 6.5.2 達朗貝爾公式的物理意義(physical meaning of d』Alembert formula)
7.1.1 分離變數法簡介(introduction of variable separation method) 7.1.2 偏微分方程可實施變數分離的條件(conditions for variable separation of PDEs) 7.1.3 邊界條件可實施變數分離的條件(conditions for variable separation of boundary conditions) 7.2 直角坐標系中的分離變數法(variable separation method in rectangular coordinate system) 7.2.1 分離變數法的求解步驟(steps of variable separation method) 7.2.2 解的物理意義(physical meaning of solution) 7.2.3 三維情況下的直角坐標分離變數(variable separation for 3D problem in rectangular coordinate) 7.3 非齊次邊界條件齊次化(homogenization of non-homogeneous boundary conditions) 7.3.1 非齊次邊界條件齊次化的一般方法(general method) 7.3.2 非齊次邊界條件齊次化的特殊方法(special method) 7.4 非齊次方程(inhomogeneous equation) 7.5 泊松方程(Poisson equation) 7.6 基於MATLAB的數學物理方程數值求解(numerical solution of mathematical physics equation based on MATLAB) 7.6.1 有限元法介紹(introduction of the finite element method) 7.6.2 MATLAB PDE工具箱(MATLAB PDE toolbox) 第7章習題 第8章 二階常微分方程的級數解法和本征值問題 8.1 柱坐標系和球坐標系下的分離變數法(variable separation in spherical and cylindrical coordinate system) 8.1.1 三種常用的正交坐標系(three types of coordinates) 8.1.2 拉普拉斯方程的分離變數(Laplace equation) 8.1.3 三維波動方程的分離變數(3D wave equation) 8.1.4 三維輸運方程/熱傳導方程的分離變數(3D transport equation heat-conduct equation) 8.1.5 亥姆霍茲方程的分離變數(Helmholtz equation) 8.2 常點鄰域的級數解法(power series solution around ordinary points) 8.3 施圖姆-劉維爾本征值問題(Sturm-Liouville eigenvalue problem) 8.3.1 施圖姆-劉維爾型方程及本征值問題(Sturm-Liouville equation and eigenvalue problem) 8.3.2 施圖姆-劉維爾本征值問題的性質及廣義傅里葉級數(characteristics of Sturm-Liouville eigenvalue problem and generalized Fourier series) 第8章習題 第9章 特殊函數 9.1 勒讓德多項式(Legendre polynomials) 9.1.1 勒讓德方程及其級數解(Legendre equation and power series solution) 9.1.2 本征值問題(eigenvalue problem) 9.1.3 勒讓德多項式的表達式(Legendre polynomials) 9.1.4 勒讓德多項式的性質(characteristics of Legendre polynomials) 9.1.5 勒讓德多項式的MATLAB可視化(visualization of Legendre polynomials based on MATLAB) 9.1.6 廣義傅里葉級數(generalized Fourier series) 9.1.7 軸對稱定解問題(axisymmetric problems in spherical coordinate) 9.1.8 勒讓德多項式的生成函數(generating function of Legendre polynomial) 9.1.9 勒讓德多項式的遞推公式(recurrence formula of Legendre polynomials) 9.2 貝塞爾函數(Bessel function) 9.2.1 三類柱函數(three types of cylindrical functions) 9.2.2 貝塞爾函數和諾伊曼函數的MATLAB可視化(visualization of Bessel function and Neumann f 9.4.1 拉普拉斯方程定解問題(Laplace equation problems) 9.4.2 階躍光纖的分析(analysis of step optical fibre) 9.4.3 表面等離激元(plasmonics) 第9章習題 第10章 數理方程的其他方法 10.1 保角變換法(conformal mapping) 10.1.1 常用的保角變換函數(analytic functions for conformal mapping) 10.1.2 應用舉例(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 二維拉普拉斯方程的差分方程(difference equation of 2D Laplace equation) 10.2.3 邊界上的差分方程(difference equation on boundary grids) 10.2.4 二維靜態電磁場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章習題 習題答案 參考文獻
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