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| Chapter 1 Complex Numbers and Functions of a Complex Variable 1.1 Complex numbers and its four fundamental operations 1.2 Geometric representation of complex numbers 1.3 Complex conjugates 1.4 Powers and roots 1.5 Riemann sphere and infinity 1.6 Complex number sets 1.7 Functions of a complex variable Exercise 1 Chapter 2 Analytic Functions 2.1 The concept of analytic function 2.2 Necessary and sufficient conditions of analytic functions 2.3 Elementary functions Exercise 2 Chapter 3 Complex Integrals 3.1 The concept of complex integral 3.2 Cauchy integral theorem 3.3 Cauchy integral formula 3.4 Analytic functions and harmonic functions Exercise 3 Chapter 4 Series 4.1 Series of complex numbers and series of complex functions" 4.2 Power series 4.3 Taylor series 4.4 Laurent series Exercise 4 Chapter 5 Residues 5.1 Isolated singularities 5.2 Residues 5.3 Application of residues in evaluating definite and improper integrals Exercise 5 Chapter 6 Conformal Mappings 6.1 The concept of conformal mapping 6.2 Fractional linear transformations 6.3 Condition of uniqueness 6.4 Some important fractional linear transformations 6.5 Mapping by some elementary functions Exercise 6 Chapter 7 Fourier Transform 7.1 Fourier integral and Fourier integral theorem 7.2 Fourier transform and inverse Fourier transform 7.3 Unit impulse functions 7.4 Generalized Fourier transform 7.5 The properties of Fourier transform 7.6 Convolution Exercise 7 Chapter 8 Laplace Transform 8.1 The concept of Laplace transform 8.2 The properties of Laplace transform 8.3 Inverse Laplace transform 8.4 Application of Laplace transform Exercise 8 Answers to Selected Exercises Exercise 1 Exercise 2 Exercise 3 Exercise 4 Exercise 5 Exercise 6 Exercise 7 Exercise 8 Bibliography Appendix Index |
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