Table of Contents

Chapter l Complex Numbers and Functions of a Complex Variable 1
1.1 Complex numbers and its four fundamental operations 1
1.2 Geometric representation of complex numbers 3
1.3 Complex conjugates 4
1.4 Powers and roots 6
1.5 Riemann sphere and infinity 7
1.6 Complex number sets 8
1.7 Functions of a complex variable 9
Exercise 1 13
Chapter 2 Analytic Functions 17
2.1 The concept of analytic function 17
2.2 Necessary and sufficient conditions of analytic functions 20
2.3 Elementary functions 24
Exercise 2 32
Chapter 3 Complexlntegrals 36
3.1 The concept ofcomplexintegral 36
3.2 Cauchy integral theorem 41
3.3 Cauchy integral formula 47
3.4 Analytic functions and harmonic functions 55
Exercise 3 59
Chapter Series 63
4.1 Series of complex numbers and series of complex functions 63
4.2 Power series 72
4.3 Taylor series 75
4.4 Laurent series 80
Exercise 4 88
Chapter 5 Residues 91
5.1 Isolated singularities 91
5.2 Residues 98
5.3 Application of residues in evaluating definite and improper integrals 105
Exercise 5 117
Chapter 6 Conformal Mappings 120
6.1 The concept of conformal mapping 120
6.2 Fractional linear transformations 123
6.3 Condition of uniqueness 131
6.4 Some important fractional linear transformations 132
6.5 Mapping by some elementary functions 135
Exercise 6 141
Chapter 7 Fourier Transform 143
7.1 Fourier integral and Fourier integral theorem 143
7.2 Fourier transform and inverse Fourier transform 149
7.3 Unit impulse functions 153
7.4 Generalized Fourier transform 161
7.5 The properties of Fourier transform 161
7.6 Convolution 168
Exercise 7 177
Chapter 8 Laplace Transform 181
8.1 The concept of Laplace transform 181
8.2 The properties of Laplace transform 190
8.3 Inverse Laplace transform 202
8.4 Application of Laplace transform 205
Exercise 8 209
Answers to Selected Exercises 215
Exercise 1 215
Exercise 2 215
Exercise 3 218
Exercise 4 220
Exercise 5 222
Exercise 6 224
Exercise 7 225
Exercise 8 226
Bibliography 230
Appendix 231
Appendix A Table of Fourier transform 231
Appendix B Table of Laplace transform 239
Index 248

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