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ngineering Mathematics—Functions of a Complex Variable and Mathematical Methods for Physics

Contents

Contents


Part ¢ñ Functions of a Complex Variable

Chapter 1 Complex Numbers and Complex Functions3

1.1 Complex number and its operations3

1.1.1 Complex number and its expression3

1.1.2 The operations of complex numbers6

1.1.3 Regions in the complex plane13

Exercises 1.114

1.2 Functions of a complex variable15

1.2.1 Definition of function of a complex variable15

1.2.2 Complex mappings17

Exercises 1.220

1.3 Limit and continuity of a complex function21

1.3.1 Limit of a complex function21

1.3.2 Continuity of a complex function26

Exercises 1.327

Chapter 2 Analytic Functions29

2.1 Derivatives of complex functions29

2.1.1 Derivatives29

2.1.2 Some properties of derivatives31

2.1.3 A necessary condition on differentiability31

2.1.4 Sufficient conditions on differentiability34

Exercises 2.136

2.2 Analytic functions37

2.2.1 Analytic functions37

2.2.2 Harmonic functions39

Exercises 2.241

2.3 Elementary functions41

2.3.1 Exponential functions41

2.3.2 Logarithmic functions42

2.3.3 Complex exponents45

2.3.4 Trigonometric functions46

2.3.5 Hyperbolic functions48

2.3.6 Inverse trigonometric and hyperbolic functions49

Exercises 2.350

Chapter 3 Integral of Complex Function52

3.1 Derivatives and definite integrals of functions w(t)52

3.1.1 Derivatives of functions w(t)52

3.1.2 Definite integrals of functions w(t)53

Exercises 3.156

3.2 Contour integral56

3.2.1 Contour56

3.2.2 Definition of contour integral58

3.2.3 Antiderivatives66

Exercises 3.273

3.3 Cauchy integral theorem75

3.3.1 Cauchyª²Goursat theorem 75

3.3.2 Simply and multiply connected domains76

Exercises 3.380

3.4 Cauchy integral formula and derivatives of analytic functions81

3.4.1 Cauchy integral formula81

3.4.2 Higherª²order derivatives formula of analytic functions84

Exercises 3.487

Chapter 4 Complex Series89

4.1 Complex series and its convergence89

4.1.1 Complex sequences and its convergence89

4.1.2 Complex series and its convergence90

Exercises 4.193

4.2 Power series93

4.2.1 The definition of power series93

4.2.2 The convergence of power series95

4.2.3 The operations of power series97

Exercises 4.297

4.3 Taylor series98

4.3.1 Taylor's theorem98

4.3.2 Taylor expansions of analytic functions100

Exercises 4.3104

4.4 Laurent series105

4.4.1 Laurent's theorem105

4.4.2 Laurent series expansion of analytic functions109

Exercises 4.4111

Chapter 5 Residues and Its Application113

5.1 Three types of isolated singular points113

Exercises 5.1118

5.2 Residues and Cauchy's residue theorem118

Exercises 5.2123

5.3 Application of residues on definite integrals123

5.3.1 Improper integrals124

5.3.2 Improper integrals involving sines and cosines125

5.3.3 Integrals on £Û0,2π£Ý involving sines and cosines128

Exercises 5.3130

Part ¢ò Mathematical Methods for Physics
Chapter 6 Equations of Mathematical Physics and
Problems for Defining Solutions135

6.1 Basic concept and definition135

6.1.1 Basic concept 136

6.1.2 Linear operator and linear composition138

6.1.3 Calculation rule of operator140

6.2 Three typical partial differential equations and problems for defining solutions141

6.2.1 Wave equations and physical derivations141

6.2.2 Heat (conduction) equations and physical derivations143

6.2.3 Laplace equations and physical derivations144

6.3 Wellª²posed problem145

6.3.1 Initial conditions146

6.3.2 Boundary conditions146

Chapter 7 Classification and Simplification for Linear Second Order PDEs148

7.1 Classification of linear second order partial differential equations with two
variables148

Exercises 7.1149

7.2 Simplification to standard forms149

Exercises 7.2156

Chapter 8 Integral Method on Characteristics158

8.1 D'Alembert formula for one dimensional infinite string oscillation158

Exercises 8.1160

8.2 Small oscillations of semiª²infinite string with rigidly fixed or free ends, method
of prolongation160

Exercises 8.2162

8.3 Integral method on characteristics for other second order PDEs, some examples162

Exercises 8.3165

Chapter 9 The Method of Separation of Variables on Finite Region166

9.1 Separation of variables for (1 1)ª²dimensional homogeneous equations167

9.1.1 Separation of variables for wave equation on finite region167

9.1.2 Separation of variables for heat equation on finite region170

Exercises 9.1 172

9.2 Separation of variables for 2ª²dimensional Laplace equations174

9.2.1 Laplace equation with rectangular boundary174

9.2.2 Laplace equation with circular boundary177

Exercises 9.2180

9.3 Nonhomogeneous equations and nonhomogeneous boundary conditions181

Exercises 9.3192

9.4 Sturmª²Liouville eigenvalue problem192

Exercises 9.4198

Chapter 10 Special Functions199

10.1 Bessel function199

10.1.1 Introduction to the Bessel equation199

10.1.2 The solution of the Bessel equation201

10.1.3 The recurrence formula of the Bessel function204

10.1.4 The properties of the Bessel function207

10.1.5 Application of Bessel function210

Exercises 10.1213

10.2 Legendre polynomial214

10.2.1 Introduction of the Legendre equation214

10.2.2 The solution of the Legendre equation216

10.2.3 The properties of the Legendre polynomial and recurrence formula218

10.2.4 Application of Legendre polynomial 221

Exercises 10.2223

Chapter 11 Integral Transformations224

11.1 Fourier integral transformation224

11.1.1 Definition of Fourier integral transformation225

11.1.2 The properties of Fourier integral transformation228

11.1.3 Convolution and its Fourier transformation230

11.1.4 Application of Fourier integral transformation231

Exercises 11.1235

11.2 Laplace integral transformation236

11.2.1 Definition of Laplace transformation236

11.2.2 Properties of Laplace transformation238

11.2.3 Convolution and its Laplace transformation241

11.2.4 Application of Laplace integral transformation242

Exercises 11.2244

References245