目錄�おお�
優(yōu)質(zhì)篇復變函數(shù)論

第1章復變函數(shù)與解析函數(shù)

1.1復數(shù)及其基本運算(complex numbers and operations)

1.1.1復數(shù)的基本概念(concepts of complex numbers)

1.1.2復數(shù)的表示方法(algebraic and geometric structure of complex numbers)

1.1.3復數(shù)的基本運算(operation of complex numbers)

1.1.4基于MATLAB的復數(shù)運算(complex number operations based on MATLAB)

1.2復變函數(shù)(complex variable functions)

1.2.1復變函數(shù)的概念(concepts and properties of complex variable function)

1.2.2區(qū)域的相關概念(concepts of domain)

1.2.3復變函數(shù)的極限和連續(xù)(limit and continuity of complex variable function)

1.3導數(shù)及解析函數(shù)(derivative and analytic function)

1.3.1導數(shù)(derivative)

1.3.2函數(shù)可導的充分必要條件(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運用MATLAB工具使復變函數(shù)可視化(visualization of complex function
based on MATLAB)

1.4解析函數(shù)的應用(application of analytic function)

1.4.1解析函數(shù)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(shù)在系統(tǒng)穩(wěn)態(tài)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(shù)積分

2.1復變函數(shù)的積分(integral of complex variable function)

2.1.1復變函數(shù)積分的基本概念(concepts of complex integral)

2.1.2復變函數(shù)積分的性質(zhì)(properties of complex integral)

2.1.3復變函數(shù)積分實例(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復連通區(qū)域的柯西定理(Cauchy theorem in multiply connected domains)

2.2.4復變函數(shù)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(qū)域的柯西積分公式(Cauchy formula in simply connected domain)

2.3.2復連通區(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用實例(application examples of Cauchy theorem
and Cauchy formula)

第2章習題

第3章復變函數(shù)級數(shù)

3.1復數(shù)項級數(shù)(complex number series)

3.1.1復數(shù)項級數(shù)的概念(concepts of complex number series)

3.1.2復數(shù)項級數(shù)的性質(zhì)(properties of complex number series)

3.1.3復變函數(shù)項級數(shù)(series of complex functions)

3.2冪級數(shù)(power series)

3.2.1冪級數(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冪級數(shù)的性質(zhì)(properties of power series)

3.3泰勒級數(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)展開成泰勒級數(shù)的實例(examples of Taylor series expansion)

3.4洛朗級數(shù)(Laurent series)

3.4.1洛朗級數(shù)定義(definition of Laurent series)

3.4.2洛朗級數(shù)的收斂性(convergence of Laurent series)

3.4.3洛朗級數(shù)展開實例(examples of Laurent series expansion)

3.5單值函數(shù)的孤立奇點(isolated singular points of single�瞯alued functions)

3.6基于MATLAB的冪級數(shù)展開(power series expansion based on MATLAB)

第3章習題

第4章留數(shù)定理及其應用

4.1留數(shù)定理(residue theorem)

4.1.1閉合回路積分與留數(shù)的關系(loop integral and residue)

4.1.2留數(shù)的計算(calculation of residue)

4.1.3基于MATLAB的留數(shù)計算(residue calculation based on MATLAB)

4.2利用留數(shù)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?qū)嵎e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基于MATLAB的回路積分計算(loop integral calculation based on MATLAB)

第4章習題

第二篇數(shù)學物理方程及求解方法

第5章傅里葉級數(shù)

5.1周期函數(shù)的傅里葉展開(Fourier expansion of periodic function)

5.1.1傅里葉級數(shù)的定義(definition of Fourier series)

5.1.2傅里葉級數(shù)的實際意義(practical meaning of Fourier series)

5.1.3傅里葉級數(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復數(shù)形式的傅里葉級數(shù)(Fourier series in complex form)

5.5使用MATLAB完成傅里葉級數(shù)的可視化(visualization of Fourier series based
on MATLAB)

第5章習題

第6章數(shù)學建模——數(shù)學物理定解問題

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波動方程(wave equation)

6.2.2熱傳導方程(heat�瞔onduction 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)學物理定解問題的適定性(well�瞤osed 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�瞣rder linear partial differential equations)

6.4.1二階線性偏微分方程的分類(classification of second�瞣rder 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�瞣rder
linear partial differential equations)

6.5行波法與達朗貝爾公式(traveling wave method and D�餉lembert formula)

6.5.1一維波動方程的達朗貝爾公式(D�餉lembert formula for one
dimensional wave equation)

6.5.2達朗貝爾公式的物理意義(physical meaning of D�餉lembert formula)

6.5.3達朗貝爾公式應用的MATLAB實現(xiàn)(application of D�餉lembert
formula based on MATLAB)

第6章習題

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分離變量法簡介(introduction of variable separation method)

7.1.2偏微分方程可實施變量分離的條件(conditions for variable
separation of PDEs)

7.1.3邊界條件可實施變量分離的條件(conditions for variable separation
of boundary conditions)

7.2直角坐標系中的分離變量法(variable separation method in rectangular
coordinate