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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