工程數學——複變函數與數學物理方法

《工程數學——複變函數與數學物理方法》是2017年北京郵電大學出版社出版的圖書。作者是石霞，默會霞，錢江，楊建奎。本書可作為理工科院校非數學專業本科生的雙語教材，也可供有關工程技術人員參考。

基本介紹

• 中文名：工程數學——複變函數與數學物理方法
• 作者：石霞、默會霞、錢江、楊建奎
• 出版社：北京郵電大學出版社
• ISBN：9787563552641

內容簡介

《工程數學：複變函數與數學物理方法（英文版）/普通高等教育“十三五”規劃教材》是一本用於同名課程雙語教學的英文教材，編者按照國家教育部對本課程的基本要求，參考了多本有關的英文教材，結合多年的教學實踐編著而成。《工程數學：複變函數與數學物理方法（英文版）/普通高等教育“十三五”規劃教材》包含複變函數和數學物理方法兩部分。複變函數部分的內容包括：複數與複變函數的基本概念、複變函數的導數和積分、解析函式的定義和性質、解析函式的冪級數表示、留數定理及其套用等。數學物理方法部分的內容包括：三類典型方程的導出和定解問題的定義、二階線性偏微分方程的分類和化簡以及達朗貝爾公式求解波動方程、分離變數法求解定解問題和特徵值問題、貝塞爾函式和勒讓德多項式、傅立葉變換和拉普拉斯變換。

　　《工程數學：複變函數與數學物理方法（英文版）/普通高等教育“十三五”規劃教材》可作為理工科院校非數學專業本科生的雙語教材，也可供有關工程技術人員參考。

圖書目錄

Part Ⅰ Functions of a Complex Variable

Chapter 1 Complex Numbers and Complex Functions

1．1 Complex number and its operations

1．1．1 Complex number and its expression

1．1．2 The operations of complex numbers

1．1．3 Regions in the complex plane

Exercises 1．1

1．2 Functions of a complex variable

1．2．1 Definition of function of a complex variable

1．2．2 Complex mappings

Exercises 1．2

1．3 Limit and continuity of a complex function

1．3．1 Limit of a complex function

1．3．2 Continuity of a complex function

Exercises 1．3

Chapter 2 Analytic Functions

2．1 Derivatives of complex functions

2．1．1 Derivatives

2．1．2 Some properties of derivatives

2．1．3 A necessary condition on differentiability

2．1．4 Sufficient conditions on differentiability

Exercises 2．1

2．2 Analytic functions

2．2．1 Analytic functions

2．2．2 Harmonic functions

Exercises 2．2

2．3 Elementary functions

2．3．1 Exponential functions

2．3．2 Logarithmic functions

2．3．3 Complex exponents

2．3．4 Trigonometric functions

2．3．5 Hyperbolic functions

2．3．6 Inverse trigonometric and hyperbolic functions

Exercises 2．3

Chapter 3 Integral of Complex Function

3．1 Derivatives and definite integrals of functions w（t）

3．1．1 Derivatives of functions w（t）

3．1．2 Definite integrals of functions w（t）

Exercises 3．1

3．2 Contour integral

3．2．1 Contour

3．2．2 Definition of contour integra

3．2．3 Antiderivatives

Exercises 3．2

3．3 Cauchy integral theorem

3．3．1 CauchyGoursat theorem

3．3．2 Simply and multiply connected domains

Exercises 3．3

3．4 Cauchy integral formula and derivatives of analytic functions

3．4．1 Cauchy integral formula

3．4．2 Higherorder derivatives formula of analytic functions

Exercises 3．4

Chapter 4 Complex Series

4．1 Complex series and its convergence

4．1．1 Complex sequences and its convergence

4．1．2 Complex series and its convergence

Exercises 4．1

4．2 Power series

4．2．1 The definition of power series

4．2．2 The convergence of power series

4．2．3 The operations of power series

Exercises 4．2

4．3 Taylor series

4．3．1 Taylor's theorem

4．3．2 Taylor expansions of analytic functions

Exercises 4．3

4．4 Laurent series

4．4．1 Laurent's theorem

4．4．2 Laurent series expansion of analytic functions

Exercises 4．4

Chapter 5 Residues and Its Application

5．1 Three types of isolated singular points

Exercises 5．1

5．2 Residues and Cauchy's residue theorem

Exercises 5．2

5．3 Application of residues on definite integrals

5．3．1 Improper integrals

5．3．2 Improper integrals involving sines and cosines

5．3．3 Integrals on ［0，2穡？ involving sines and cosines

Exercises 5．3

Part Ⅱ Mathematical Methods for Physics

Chapter 6 Equations of Mathematical Physics and

Problems for Defining Solutions

6．1 Basic concept and definition

6．1．1 Basic concept

6．1．2 Linear operator and linear composition

6．1．3 Calculation rule of operator

6．2 Three typical

Partial differential equations and problems for defining solutions

6．2．1 Wave equations and physical derivations

6．2．2 Heat （conduction） equations and physical derivations

6．2．3 Laplace equations and physical derivations

6．3 Wellposed problem

6．3．1 Initial conditions

6．3．2 Boundary conditions

Chapter 7 Classification and Simplification for Linear Second Order PDEs

7．1 Classification of linear second order

Partial differential equations with two

variables

Exercises 7．1

7．2 Simplification to standard forms

Exercises 7．2

Chapter 8 Integral Method on Characteristics

8．1 D'Alembert formula for one dimensional infinite string oscillation

Exercises 8．1

8．2 Small oscillations of semiinfinite string with rigidly fixed or free ends， method

of prolongation

Exercises 8．2

8．3 Integral method on characteristics for other second order PDEs， some examples162

Exercises 8．3

Chapter 9 The Method of Separation of Variables on Finite Region

9．1 Separation of variables for （1 1）dimensional homogeneous equations

9．1．1 Separation of variables for wave equation on finite region

9．1．2 Separation of variables for heat equation on finite region

Exercises 9．1

9．2 Separation of variables for 2dimensional Laplace equations

9．2．1 Laplace equation with rectangular boundary

9．2．2 Laplace equation with circular boundary

Exercises 9．2

9．3 Nonhomogeneous equations and nonhomogeneous boundary conditions

Exercises 9．3

9．4 SturmLiouville eigenvalue problem

Exercises 9．4

Chapter 10 Special Functions

10．1 Bessel function

10．1．1 Introduction to the Bessel equation

10．1．2 The solution of the Bessel equation

10．1．3 The recurrence formula of the Bessel function

10．1．4 The properties of the Bessel function

10．1．5 Application of Bessel function

Exercises 10．1

10．2 Legendre polynomial

10．2．1 Introduction of the Legendre equation

10．2．2 The solution of the Legendre equation

10．2．3 The properties of the Legendre polynomial and recurrence formula

10．2．4 Application of Legendre polynomial

Exercises 10．2

Chapter 11 Integral Transformations

11．1 Fourier integral transformation

11．1．1 Definition of Fourier integral transformation

11．1．2 The properties of Fourier integral transformation

11．1．3 Convolution and its Fourier transformation

11．1．4 Application of Fourier integral transformation

Exercises 11．1

11．2 Laplace integral transformation

11．2．1 Definition of Laplace transformation

11．2．2 Properties of Laplace transformation

11．2．3 Convolution and its Laplace transformation

11．2．4 Application of Laplace integral transformation

Exercises 11．2

References