A Quick Introduction to Complex Analysis

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 Analysis
  • Complex Number System
  • Power Series and Euler's Identity
  • Residue Calculus
  • Review on Vector-Valued Functions
  • Cauchy–Riemann Equation
  • Inverse Functions
  • Around Jensen's Formula
  • Residue Calculus Again
  • Partial Fraction Expansion
  • Second-Order Systems and the Laplace Transform
  • Robust Controller for Servo Systems
  • Paley–Wiener Theorem
  • Bernstein Polynomials
  • Some Far-Reaching Principles in Mathematics

• Applicable Real and Complex Functions:

  • Preliminaries
  • Algebra of Complex Numbers
  • Power Series Again
  • Improper Integrals
  • Differentiation
  • Differential Calculus of One and Several Variables
  • Computation of Definite Integrals
  • Cauchy Integral Theorem
  • Cauchy Integral Formula
  • Taylor Expansions and Extremal Values
  • Complex Power Series
  • Laurent Expansions
  • Differential Equations
  • Inverse Functions
  • Rudiments of the Fourier Transform
  • Paley–Wiener Theorem and Signal Transmission

• Appendices:

  • Integration
  • Answers and Hints

Readership: 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 pain
• Many examples and worked-out exercises will help the reader to master the fundamental skills
• It gives some different approaches to the same topic and so just browsing through them is beneficial