A Friendly Approach to Complex Analysis

✍ Scribed by Sara Maad Sasane, Amol Sasane

• Publisher: World Scientific
• Year: 2023
• Tongue: English
• Leaves: 219
• Edition: 2
• Category: Library

✦ Synopsis

The book constitutes a basic, concise, yet rigorous first course in complex analysis, for undergraduate students who have studied multivariable calculus and linear algebra. The textbook should be particularly useful for students of joint programmes with mathematics, as well as engineering students seeking rigour. The aim of the book is to cover the bare bones of the subject with minimal prerequisites. The core content of the book is the three main pillars of complex analysis: the Cauchy-Riemann equations, the Cauchy Integral Theorem, and Taylor and Laurent series. Each section contains several problems, which are not drill exercises, but are meant to reinforce the fundamental concepts. Detailed solutions to all the 243 exercises appear at the end of the book, making the book ideal for self-study. There are many figures illustrating the text.

The second edition corrects errors from the first edition, and includes 89 new exercises, some of which cover auxiliary topics that were omitted in the first edition. Two new appendices have been added, one containing a detailed rigorous proof of the Cauchy Integral Theorem, and another providing background in real analysis needed to make the book self-contained.

✦ Table of Contents

Contents
Overview
What is complex analysis?
Why study complex analysis?
Chapter 1. Complex numbers and their geometry
1.1. The field of complex numbers
1.2. Geometric representation of complex numbers
1.3. Topology of C
1.4. The exponential function and kith
Chapter 2. Complex differentiability
2.1. Complex differentiability
2.2. Cauchy-Riemann equations
2.3. Geometric meaning of the complex derivative
Chapter 3. Cauchy Integral Theorem
3.1. Definition of the contour integral
3.2. Properties of the contour integral
3.3. Fundamental Theorem of Contour Integration
3.4. The Cauchy Integral Theorem
3.5. Existence of a primitive
3.6. The Cauchy Integral Formula
3.7. Holomorphic functions are infinitely complex differentiable
3.8. Liouville’s Theorem, Fundamental Theorem of Algebra
3.9. Morera’s Theorem
3.10. Appendix
Chapter 4. Taylor and Laurent series
4.1. Series
4.2. Power series
4.3. Taylor series
4.4. Classification of zeroes
4.5. The Identity Theorem
4.6. The Maximum Modulus Theorem
4.7. Laurent series
4.8. Classification of singularities
4.9. Residue Theorem
Chapter 5. Harmonic functions
5.1. What is a harmonic function?
5.2. Link between harmonic and holomorphic functions
5.3. Consequences of the two way traffic
Solutions
Solutions to the exercises from the Introduction
Solutions to the exercises from Chapter 1
Solutions to the exercises from Chapter 2
Solutions to the exercises from Chapter 3
Solutions to the exercises from Chapter 4
Solutions to the exercises from Chapter 5
Some real analysis background
Notes
Bibliography
Index