A Quick Introduction To Complex Analysis

by Kalyan Chakraborty, Shigeru Kanemitsu, Takako Kuzumaki

Overview

The aim of the book is to give a smooth analytic continuation from calculus to complex analysis by way of plenty of practical examples and worked-out exercises. The scope ranges from applications in calculus to complex analysis in two different levels.If the reader is in a hurry, he can browse the quickest introduction to complex analysis at the beginning of Chapter 1, which explains the very basics of the theory in an extremely user-friendly way. Those who want to do self-study on complex analysis can concentrate on Chapter 1 in which the two mainstreams of the theory — the power series method due to Weierstrass and the integration method due to Cauchy — are presented in a very concrete way with rich examples. Readers who want to learn more about applied calculus can refer to Chapter 2, where numerous practical applications are provided. They will master the art of problem solving by following the step by step guidance given in the worked-out examples.This book helps the reader to acquire fundamental skills of understanding complex analysis and its applications. It also gives a smooth introduction to Fourier analysis as well as a quick prelude to thermodynamics and fluid mechanics, information theory, and control theory. One of the main features of the book is that it presents different approaches to the same topic that aids the reader to gain a deeper understanding of the subject.

Product Details

ISBN-13:	9789813108516
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/03/2016
Pages:	208
Product dimensions:	6.00(w) x 8.90(h) x 0.60(d)

Table of Contents

Preface vii

1 A Quick Introduction to Complex Analysis with Applications 1

1.1 The quickest introduction to complex analysis 1

1.2 Complex number system 7

1.3 Power series and Euler's identity 10

1.3.1 Power series 10

1.3.2 Euler's identity 12

1.3.3 Laurent expansion, residues 18

1.4 Residue calculus 20

1.4.1 Cauchy integral formula and its consequences 27

1.5 Review on vector-valued functions 28

1.5.1 Differentiation 28

1.5.2 Fluid mechanics 30

1.5.3 Thermodynamic intermission 31

1.6 Cauchy-Riemann equation 33

1.7 Inverse functions 40

1.8 Around Jensen's formula 45

1.9 Residue calculus again 47

1.10 Partial fraction expansion 60

1.10.1 Partial fraction expansions for rational functions 60

1.11 Second-order systems and the Laplace transform 61

1.11.1 Examples of second-order systems 61

1.11.2 The Laplace transform method 62

1.11.3 Partial fraction expansion for the cotangent function and some of its applications 67

1.12 Robust controller for servo systems 73

1.12.1 Prerequisites 74

1.12.2 A generalized Nevanlinna-Pick interpolation problem 75

1.12.3 Proof of Theorem 1.23 80

1.12.4 Another statement 82

1.13 Paley-Wiener theorem 83

1.14 Bernstein polynomials 84

1.14.1 Time-limited polynomial extrapolation 84

1.15 Some far-reaching principles in mathematics 85

2 Applicable Real and Complex Functions 89

2.1 Preliminaries 89

2.2 Algebra of complex numbers 92

2.2.1 Algebraic preliminaries and embeddings 92

2.2.2 Complex number system 96

2.3 Power series again 107

2.3.1 Limes principals 107

2.3.2 Radius of convergence 109

2.3.3 Function series 110

2.4 Improper integrals* 114

2.5 Differentiation* 120

2.6 Computation of definite integrals* 121

2.6.1 Line integrals and Green's formula* 124

2.7 Cauchy integral theorem 125

2.8 Cauchy integral formula 130

2.9 Taylor expansions and extremal values 132

2.9.1 Taylor expansions for real functions* 132

2.9.2 Taylor expansion 134

2.9.3 Extremal values* 135

2.10 Laurent expansions 141

2.11 Differential equations 144

2.11.1 The logarithm function* 144

2.11.2 Autonomous DE 145

2.11.3 First-order reaction 145

2.11.4 System of reactions 148

2.11.5 Other reactions 150

2.11.6 Poisson distribution 151

2.11.7 Decay of radio-active substances 152

2.12 Inverse functions 154

2.12.1 Inverse trigonometric functions* 154

2.12.2 A. Gaudi and the catenary curve 156

2.13 Rudiments of the Fourier transform 158

2.14 Paley-Wiener theorem and signal transmission 163

2.14.1 Fourier analysis again 164

2.14.2 Restoration of signals 167

Appendix 169

A.1 Integration* 169

A.1.1 Integration by parts and change of variables 172

A.1.2 Multiple integrals* 175

A.2 Answers and hi it Is 179

Bibliography 187

Index 191