A QUICK INTRODUCTION TO COMPLEX ANALYSIS
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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.Contents:A Quick Introduction to Complex Analysis with Applications:The Quickest Introduction to Complex AnalysisComplex Number SystemPower Series and Euler's IdentityResidue CalculusReview on Vector-Valued FunctionsCauchy–Riemann EquationInverse FunctionsAround Jensen's FormulaResidue Calculus AgainPartial Fraction ExpansionSecond-Order Systems and the Laplace TransformRobust Controller for Servo SystemsPaley–Wiener TheoremBernstein PolynomialsSome Far-Reaching Principles in MathematicsApplicable Real and Complex Functions:PreliminariesAlgebra of Complex NumbersPower Series AgainImproper IntegralsDifferentiationDifferential Calculus of One and Several VariablesComputation of Definite IntegralsCauchy Integral TheoremCauchy Integral FormulaTaylor Expansions and Extremal ValuesComplex Power SeriesLaurent ExpansionsDifferential EquationsInverse FunctionsRudiments of the Fourier TransformPaley–Wiener Theorem and Signal TransmissionAppendices:IntegrationAnswers and HintsReadership: Advanced undergraduate mathematics, physics and engineering students; researchers in the field of complex analysis; also suitable for self-study.Key Features:With so many existing excellent books on complex analysis, the raison d'être of the present book is that it is extremely reader-friendly while keeping rigor of serious mathematics and in-depth analysis of practical applications to various subjects including thermodynamics, fluid dynamics, control theory, information theory etc.The method of residues is treated rather thoroughly and the reader can master the skills without too much painMany examples and worked-out exercises will help the reader to master the fundamental skillsIt gives some different approaches to the same topic and so just browsing through them is beneficial