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Part Ⅰ Functions of a Complex Variable 3

Chapter 1 Complex Numbers and Complex Functions 3

1.1 Complex number and its operations 3

1.1.1 Complex number and its expression 3

1.1.2 The operations of complex numbers 6

1.1.3 Regions in the complex plane 13

Exercises 1.1 14

1.2 Functions of a complex variable 15

1.2.1 Definition of function of a complex variable 15

1.2.2 Complex mappings 17

Exercises 1.2 20

1.3 Limit and continuity of a complex function 21

1.3.1 Limit of a complex function 21

1.3.2 Continuity of a complex function 26

Exercises 1.3 27

Chapter 2 Analytic Functions 29

2.1 Derivatives of complex functions 29

2.1.1 Derivatives 29

2.1.2 Some properties of derivatives 31

2.1.3 A necessary condition on differentiability 31

2.1.4 Sufficient conditions on differentiability 34

Exercises 2.1 36

2.2 Analytic functions 37

2.2.1 Analytic functions 37

2.2.2 Harmonic functions 39

Exercises 2.2 41

2.3 Elementary functions 41

2.3.1 Exponential functions 41

2.3.2 Logarithmic functions 42

2.3.3 Complex exponents 45

2.3.4 Trigonometric functions 46

2.3.5 Hyperbolic functions 48

2.3.6 Inverse trigonometric and hyperbolic functions 49

Exercises 2.3 50

Chapter 3 Integral of Complex Function 52

3.1 Derivatives and definite integrals of functionsw（t） 52

3.1.1 Derivatives of functionsw（t） 52

3.1.2 Definite integrals of functionsw（t） 53

Exercises 3.1 56

3.2 Contour integral 56

3.2.1 Contour 56

3.2.2 Definition of contour integral 58

3.2.3 Antiderivatives 66

Exercises 3.2 73

3.3 Cauchy integral theorem 75

3.3.1 Cauchy-Goursat theorem 75

3.3.2 Simply and multiply connected domains 76

Exercises 3.3 80

3.4 Cauchy integral formula and derivatives of analytic functions 81

3.4.1 Cauchy integral formula 81

3.4.2 Higher-order derivatives formula of analytic functions 84

Exercises 3.4 87

Chapter 4 Complex Series 89

4.1 Complex series and its convergence 89

4.1.1 Complex sequences and its convergence 89

4.1.2 Complex series and its convergence 90

Exercises 4.1 93

4.2 Power series 93

4.2.1 The definition of power series 93

4.2.2 The convergence of power series 95

4.2.3 The operations of power series 97

Exercises 4.2 97

4.3 Taylor series 98

4.3.1 Taylor's theorem 98

4.3.2 Taylor expansions of analytic functions 100

Exercises 4.3 104

4.4 Laurent series 105

4.4.1 Laurent's theorem 105

4.4.2 Laurent series expansion of analytic functions 109

Exercises 4.4 111

Chapter 5 Residues and Its Application 113

5.1 Three types of isolated singular points 113

Exercises 5.1 118

5.2 Residues and Cauchy's residue theorem 118

Exercises 5.2 123

5.3 Application of residues on definite integrals 123

5.3.1 Improper integrals 124

5.3.2 Improper integrals involving sines and cosines 125

5.3.3 Integrals on［0,2π］involving sines and cosines 128

Exercises 5.3 130

Part Ⅱ Mathematical Methods for Physics 135

Chapter 6 Equations of Mathematical Physics and Problems for Defining Solutions 135

6.1 Basic concept and definition 135

6.1.1 Basic concept 136

6.1.2 Linear operator and linear composition 138

6.1.3 Calculation rule of operator 140

6.2 Three typical partial differential equations and problems for defining solutions 141

6.2.1 Wave equations and physical derivations 141

6.2.2 Heat（conduction）equations and physical derivations 143

6.2.3 Laplace equations and physical derivations 144

6.3 Well-posed problem 145

6.3.1 Initial conditions 146

6.3.2 Boundary conditions 146

Chapter 7 Classification and Simplification for Linear Second Order PDEs 148

7.1 Classification of linear second order partial differential equations with two variables 148

Exercises 7.1 149

7.2 Simplification to standard forms 149

Exercises 7.2 156

Chapter 8 Integral Method on Characteristics 158

8.1 D'Alembert formula for one dimensional infinite string oscillation 158

Exercises 8.1 160

8.2 Small oscillations of semi-infinite string with rigidly fixed or free ends,method of prolongation 160

Exercises 8.2 162

8.3 Integral method on characteristics for other second order PDEs,some examples 162

Exercises 8.3 165

Chapter 9 The Method of Separation of Variables on Finite Region 166

9.1 Separation of variables for（1＋1）-dimensional homogeneous equations 167

9.1.1 Separation of variables for wave equation on finite region 167

9.1.2 Separation of variables for heat equation on finite region 170

Exercises 9.1 172

9.2 Separation of variables for 2-dimensional Laplace equations 174

9.2.1 Laplace equation with rectangular boundary 174

9.2.2 Laplace equation with circular boundary 177

Exercises 9.2 180

9.3 Nonhomogeneous equations and nonhomogeneous boundary conditions 181

Exercises 9.3 192

9.4 Sturm-Liouville eigenvalue problem 192

Exercises 9.4 198

Chapter 10 Special Functions 199

10.1 Bessel function 199

10.1.1 Introduction to the Bessel equation 199

10.1.2 The solution of the Bessel equation 201

10.1.3 The recurrence formula of the Bessel function 204

10.1.4 The properties of the Bessel function 207

10.1.5 Application of Bessel function 210

Exercises 10.1 213

10.2 Legendre polynomial 214

10.2.1 Introduction of the Legendre equation 214

10.2.2 The solution of the Legendre equation 216

10.2.3 The properties of the Legendre polynomial and recurrence formula 218

10.2.4 Application of Legendre polynomial 221

Exercises 10.2 223

Chapter 11 Integral Transformations 224

11.1 Fourier integral transformation 224

11.1.1 Definition of Fourier integral transformation 225

11.1.2 The properties of Fourier integral transformation 228

11.1.3 Convolution and its Fourier transformation 230

1 1.1.4 Application of Fourier integral transformation 231

Exercises 11.1 235

11.2 Laplace integral transformation 236

11.2.1 Definition of Laplace transformation 236

11.2.2 Properties of Laplace transformation 238

11.2.3 Convolution and its Laplace transformation 241

11.2.4 Application of Laplace integral transformation 242

Exercises 11.2 244

References 245