第1章復(fù)變函數(shù)與解析函數(shù) 1.1復(fù)數(shù)及其基本運(yùn)算(complex numbers and operations) 1.1.1復(fù)數(shù)的基本概念(concepts of complex numbers) 1.1.2復(fù)數(shù)的表示方法(algebraic and geometric structure of complex numbers) 1.1.3復(fù)數(shù)的基本運(yùn)算(operation of complex numbers) 1.1.4基于MATLAB的復(fù)數(shù)運(yùn)算(complex number operations based on MATLAB) 1.2復(fù)變函數(shù)(complex variable functions) 1.2.1復(fù)變函數(shù)的概念(concepts and properties of complex variable function) 1.2.2區(qū)域的相關(guān)概念(concepts of domain) 1.2.3復(fù)變函數(shù)的極限和連續(xù)(limit and continuity of complex variable function) 1.3導(dǎo)數(shù)及解析函數(shù)(derivative and analytic function) 1.3.1導(dǎo)數(shù)(derivative) 1.3.2函數(shù)可導(dǎo)的充分必要條件(sufficient conditions for derivability) 1.3.3解析函數(shù)(analytic function) 1.3.4初等解析函數(shù)及性質(zhì)(elementary analytic function and properties) 1.3.5運(yùn)用MATLAB工具使復(fù)變函數(shù)可視化(visualization of complex function based on MATLAB) 1.4解析函數(shù)的應(yīng)用(application of analytic function) 1.4.1解析函數(shù)在平面靜電場(chǎng)中的應(yīng)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(shù)在系統(tǒng)穩(wěn)態(tài)響應(yīng)問題求解中的應(yīng)用(application of analytic function in oscillation system) 第1章習(xí)題 第2章解析函數(shù)積分 2.1復(fù)變函數(shù)的積分(integral of complex variable function) 2.1.1復(fù)變函數(shù)積分的基本概念(concepts of complex integral) 2.1.2復(fù)變函數(shù)積分的性質(zhì)(properties of complex integral) 2.1.3復(fù)變函數(shù)積分實(shí)例(examples of complex integral) 2.2柯西定理(Cauchy theorem) 2.2.1單連通區(qū)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(shù)(indefinite integral and antiderivative) 2.2.3復(fù)連通區(qū)域的柯西定理(Cauchy theorem in multiply connected domains) 2.2.4復(fù)變函數(shù)積分的MATLAB運(yùn)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(qū)域的柯西積分公式(Cauchy formula in simply connected domain) 2.3.2復(fù)連通區(qū)域的柯西積分公式(Cauchy formula in multiply connected domain) 2.3.3無界區(qū)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(yīng)用實(shí)例(application examples of Cauchy theorem and Cauchy formula) 第2章習(xí)題 第3章復(fù)變函數(shù)級(jí)數(shù) 3.1復(fù)數(shù)項(xiàng)級(jí)數(shù)(complex number series) 3.1.1復(fù)數(shù)項(xiàng)級(jí)數(shù)的概念(concepts of complex number series) 3.1.2復(fù)數(shù)項(xiàng)級(jí)數(shù)的性質(zhì)(properties of complex number series) 3.1.3復(fù)變函數(shù)項(xiàng)級(jí)數(shù)(series of complex functions) 3.2冪級(jí)數(shù)(power series) 3.2.1冪級(jí)數(shù)概念(concepts of power series) 3.2.2收斂半徑與收斂圓(radius of convergence and circle of convergence) 3.2.3冪級(jí)數(shù)的性質(zhì)(properties of power series) 3.3泰勒級(jí)數(shù)(Taylor series) 3.3.1解析函數(shù)的泰勒展開式(Taylor expansion of analytic function) 3.3.2泰勒級(jí)數(shù)的收斂半徑(radius of convergence of Taylor series) 3.3.3將函數(shù)展開成泰勒級(jí)數(shù)的實(shí)例(examples of Taylor series expansion) 3.4洛朗級(jí)數(shù)(Laurent series) 3.4.1洛朗級(jí)數(shù)定義(definition of Laurent series) 3.4.2洛朗級(jí)數(shù)的收斂性(convergence of Laurent series) 3.4.3洛朗級(jí)數(shù)展開實(shí)例(examples of Laurent series expansion) 3.5單值函數(shù)的孤立奇點(diǎn)(isolated singular points of singlevalued functions) 3.6基于MATLAB的冪級(jí)數(shù)展開(power series expansion based on MATLAB) 第3章習(xí)題 第4章留數(shù)定理及其應(yīng)用 4.1留數(shù)定理(residue theorem) 4.1.1閉合回路積分與留數(shù)的關(guān)系(loop integral and residue) 4.1.2留數(shù)的計(jì)算(calculation of residue) 4.1.3基于MATLAB的留數(shù)計(jì)算(residue calculation based on MATLAB) 4.2利用留數(shù)定理計(jì)算實(shí)積分(application of residue theorem for calculation of real integral) 4.2.1類型Ⅰ實(shí)積分計(jì)算(type Ⅰ real integral) 4.2.2類型Ⅱ?qū)嵎e分計(jì)算(type Ⅱ real integral) 4.2.3類型Ⅲ實(shí)積分計(jì)算(type Ⅲ real integral) 4.3其他類型的實(shí)積分計(jì)算(calculation of other real integral) 4.4基于MATLAB的回路積分計(jì)算(loop integral calculation based on MATLAB) 第4章習(xí)題 第5章傅里葉級(jí)數(shù) 5.1周期函數(shù)的傅里葉展開(Fourier expansion of periodic function) 5.1.1傅里葉級(jí)數(shù)的定義(definition of Fourier series) 5.1.2傅里葉級(jí)數(shù)的實(shí)際意義(practical meaning of Fourier series) 5.1.3傅里葉級(jí)數(shù)的收斂性(convergence of Fourier series) 5.2奇函數(shù)及偶函數(shù)的傅里葉展開(Fourier expansion of odd and even function) 5.3定義在有界區(qū)間上函數(shù)的傅里葉展開(Fourier expansion of functions defined on an interval) 5.4復(fù)數(shù)形式的傅里葉級(jí)數(shù)(Fourier series in complex form) 5.5基于MATLAB的傅里葉級(jí)數(shù)可視化(visualization of Fourier series based on MATLAB) 第5章習(xí)題 第6章數(shù)學(xué)建模——數(shù)學(xué)物理定解問題 6.1基本概念(basic concepts) 6.2典型的數(shù)理方程(typical mathematical physics equation) 6.2.1波動(dòng)方程(wave equation) 6.2.2熱傳導(dǎo)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(shù)學(xué)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征(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征(characteristics of solutions of second order linear partial differential equations) 6.5行波法與達(dá)朗貝爾公式(traveling wave method and d’Alembert formula) 6.5.1一維波動(dòng)方程的達(dá)朗貝爾公式(d’Alembert formula for one dimensional wave equation) 6.5.2達(dá)朗貝爾公式的物理意義(physical meaning of d’Alembert formula) 6.5.3達(dá)朗貝爾公式應(yīng)用的MATLAB實(shí)現(xiàn)(application of d’Alembert formula based on MATLAB) 第6章習(xí)題 第7章分離變量法 7.1分離變量法的理論(theory of variable separation) 7.1.1分離變量法簡(jiǎn)介(introduction of variable separation method) 7.1.2偏微分方程可實(shí)施變量分離的條件(conditions for variable separation of PDEs) 7.1.3邊界條件可實(shí)施變量分離的條件(conditions for variable separation of boundary conditions) 7.2直角坐標(biāo)系中的分離變量法(variable separation method in rectangular coordinate system) 7.2.1分離變量法的求解步驟(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(biāo)分離變量(variable separation for 3D problem in rectangular coordinate) 7.3非齊次邊界條件齊次化(homogenization of nonhomogeneous 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基于MATLAB的數(shù)學(xué)物理方程數(shù)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) 7.6.2MATLAB PDE工具箱(MATLAB PDE toolbox) 第7章習(xí)題 第8章二階常微分方程的級(jí)數(shù)解法和本征值問題 8.1柱坐標(biāo)系和球坐標(biāo)系下的分離變量法(variable separation in spherical and cylindrical coordinate system) 8.1.1三種常用的正交坐標(biāo)系(three types of coordinates) 8.1.2拉普拉斯方程的分離變量(Laplace equation) 8.1.3三維波動(dòng)方程的分離變量(3D wave equation) 8.1.4三維輸運(yùn)方程/熱傳導(dǎo)方程的分離變量(3D transport equation/heat conduct equation) 8.1.5亥姆霍茲方程的分離變量(Helmholtz equation) 8.2常點(diǎn)鄰域的級(jí)數(shù)解法(power series solution around ordinary points) 8.3施圖姆劉維爾本征值問題(SturmLiouville eigenvalue problem) 8.3.1施圖姆劉維爾型方程及本征值問題(SturmLiouville equation and eigenvalue problem) 8.3.2施圖姆劉維爾本征值問題的性質(zhì)及廣義傅里葉級(jí)數(shù)(characteristics of SturmLiouville eigenvalue problem and generalized Fourier series) 第8章習(xí)題 第9章特殊函數(shù) 9.1勒讓德多項(xiàng)式(Legendre polynomials) 9.1.1勒讓德方程及其級(jí)數(shù)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(xiàng)式的表達(dá)式(Legendre polynomials) 9.1.4勒讓德多項(xiàng)式的性質(zhì)(characteristics of Legendre polynomials) 9.1.5勒讓德多項(xiàng)式的MATLAB可視化(visualization of Legendre polynomials based on MATLAB) 9.1.6廣義傅里葉級(jí)數(shù)(generalized Fourier series) 9.1.7軸對(duì)稱定解問題(axisymmetric problems in spherical coordinate) 9.1.8勒讓德多項(xiàng)式的生成函數(shù)(generating function of Legendre polynomial) 9.1.9勒讓德多項(xiàng)式的遞推公式(recurrence formula of Legendre polynomials) 9.2貝塞爾函數(shù)(Bessel function) 9.2.1三類柱函數(shù)(three types of cylindrical functions) 9.2.2貝塞爾函數(shù)和諾伊曼函數(shù)的MATLAB可視化(visualization of Bessel function and Neumann function based on MATLAB) 9.2.3貝塞爾函數(shù)的基本性質(zhì)(characteristics of Bessel functions) 9.2.4貝塞爾方程本征值問題(Bessel equation eigenvalue problem) 9.2.5傅里葉貝塞爾級(jí)數(shù)(FourierBessel series) 9.3虛宗量貝塞爾函數(shù)(Bessel function of imaginary argument) 9.3.1虛宗量貝塞爾方程的解(solution of modified Bessel equation) 9.3.2虛宗量貝塞爾函數(shù)和虛宗量漢克爾函數(shù)的MATLAB可視化(visualization of modified Bessel function and modified Hankel function baese on MATLAB) 9.3.3虛宗量貝塞爾函數(shù)和虛宗量漢克爾函數(shù)的性質(zhì)(characteristics of modified Bessel function and modified Hankel function) 9.4特殊函數(shù)的應(yīng)用實(shí)例(application examples of special functions) 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(xí)題 第10章數(shù)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(shù)(analytic functions for conformal mapping) 10.1.2應(yīng)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(tài)電磁場(chǎng)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(xí)題 習(xí)題答案 參考文獻(xiàn)
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