Table of Contents

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穑？ 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