An Introduction to Complex Analysis and the Laplace Transform

✍ Scribed by Vladimir Eiderman

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Tongue: English
• Leaves: 399
• Series: Textbooks in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

The aim of this comparatively short textbook is a sufficiently full exposition of the fundamentals of the theory of functions of a complex variable to prepare the student for various applications. Several important applications in physics and engineering are considered in the book.

This thorough presentation includes all theorems (with a few exceptions) presented with proofs. No previous exposure to complex numbers is assumed. The textbook can be used in one-semester or two-semester courses.

In one respect this book is larger than usual, namely in the number of detailed solutions of typical problems. This, together with various problems, makes the book useful both for self- study and for the instructor as well.

A specific point of the book is the inclusion of the Laplace transform. These two topics are closely related. Concepts in complex analysis are needed to formulate and prove basic theorems in Laplace transforms, such as the inverse Laplace transform formula. Methods of complex analysis provide solutions for problems involving Laplace transforms.

Complex numbers lend clarity and completion to some areas of classical analysis. These numbers found important applications not only in the mathematical theory, but in the mathematical descriptions of processes in physics and engineering.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Author
Introduction
1. Complex Numbers and Their Arithmetic
1.1. Complex Numbers
1.2. Operations with Complex Numbers
2. Functions of a Complex Variable
2.1. The Complex Plane
2.1.1. Curves in the complex plane
2.1.2. Domains
2.2. Sequences of Complex Numbers and Their Limits
2.3. Functions of a Complex Variable; Limits and Continuity
3. Differentiation of Functions of a Complex Variable
3.1. The Derivative. Cauchy-Riemann Conditions
3.1.1. The derivative and the differential
3.1.2. Cauchy-Riemann conditions
3.1.3. Analytic functions
3.2. The Connection between Analytic and Harmonic Functions
3.3. The Geometric Meaning of the Derivative. Conformal Mappings
3.3.1. The geometric meaning of the argument of the derivative
3.3.2. The geometric meaning of the modulus of the derivative
3.3.3. Conformal mappings
4. Conformal Mappings
4.1. Linear and Mobius Transformations
4.1.1. Linear functions
4.1.2. Mobius transformations
4.2. The Power Function. The Concept of Riemann Surface
4.3. Exponential and Logarithmic Functions
4.3.1. Exponential function
4.3.2. The logarithmic function
4.4. Power, Trigonometric, and Other Functions
4.4.1. The general power function
4.4.2. The trigonometric functions
4.4.3. Inverse trig functions
4.4.4. The Zhukovsky function
4.5. General Properties of Conformal Mappings
5. Integration
5.1. Definition of the Contour Integral
5.1.1. Properties of the contour integral
5.2. Cauchy-Goursat Theorem
5.3. Indefinite Integral
5.4. The Cauchy Integral Formula
6. Series
6.1. Definitions
6.2. Function Series
6.3. Power Series
6.4. Power Series Expansion
6.5. Uniqueness Property
6.6. Analytic Continuations
6.7. Laurent Series
7. Residue Theory
7.1. Isolated Singularities
7.2. Residues
7.3. Computing Integrals with Residues
7.3.1. Integrals over closed curves
7.3.2. Real integrals of the form 2R0R(cos°; sin°) d°; where R is a rational function of cos °and sin °
7.3.3. Improper integrals
7.4. Logarithmic Residues and the Argument Principle
8. Applications
8.1. The Schwarz-Christo el Transformation
8.2. Hydrodynamics. Simply-connected Domains
8.2.1. Complex potential of a vector field
8.2.2. Simply-connected domains
8.3. Sources and Sinks. Flow around Obstacles
8.3.1. Sources and sinks
8.3.2. Vortices
8.3.3. Flow around obstacles
8.3.4. The Zhukovsky airfoils
8.3.5. Lifting force
8.4. Other Interpretations of Vector Fields
8.4.1. Electrostatics
8.4.2. Heat ow
8.4.3. Remarks on boundary value problems
9. The Laplace Transform
9.1. The Laplace Transform
9.2. Properties of the Laplace Transformation
9.3. Applications to Differential Equations
9.3.1. Linear ODEs
9.3.2. Finding the original function from its transform
9.3.3. Differential equations with piecewise defined right hand sides
9.3.4. Application of the convolution operation to solving differential equations
9.3.5. Systems of differential equations
Solutions, Hints, and Answers to Selected Problems
Appendix
Bibliography
Index

✦ Subjects

Complex Analysis; Complex Variable; Functions; Complex Numbers;