Complex Analysis with Applications

by Richard A. Silverman

Overview

This basic book on functions of a complex variable represents the irreducible minimum of what every scientist and engineer should know about this important subject. From a preliminary discussion of complex numbers and functions to key topics such as the Cauchy theory, power series, and residues, distinguished mathematical writer Richard Silverman presents the fundamentals of complex analysis in a concise manner designed not to overwhelm the beginner. The author's lively style and simplicity of approach enable the reader to grasp essential topics without being distracted by secondary issues.
Contents include: Complex Numbers; Some Special Mapping; Limits in the Complex Plane; Multiple-Valued Functions' Complex Functions; Taylor Series; Differentiation in the Complex Plane; Laurent Series; Integration in the Complex Plane; Applications of Residues; Complex Series; Mapping of Polygonal Domains; Power Series; and Some Physical Applications.
Abundant exercise material and examples, as well as section-by-section comments at the end of each chapter make this book especially valuable to students and anyone encountering complex analysis for the first time.

Product Details

ISBN-13:	9780486647623
Publisher:	Dover Publications
Publication date:	10/18/2010
Series:	Dover Books on Mathematics
Edition description:	Dover ed
Pages:	304
Product dimensions:	5.38(w) x 8.50(h) x (d)

About the Author

Richard A. Silverman: Dover's Trusted Advisor
Richard Silverman was the primary reviewer of our mathematics books for well over 25 years starting in the 1970s. And, as one of the preeminent translators of scientific Russian, his work also appears in our catalog in the form of his translations of essential works by many of the greatest names in Russian mathematics and physics of the twentieth century. These titles include (but are by no means limited to): Special Functions and Their Applications (Lebedev); Methods of Quantum Field Theory in Statistical Physics (Abrikosov, et al); An Introduction to the Theory of Linear Spaces, Linear Algebra, and Elementary Real and Complex Analysis (all three by Shilov); and many more.

During the Silverman years, the Dover math program attained and deepened its reach and depth to a level that would not have been possible without his valuable contributions.

Table of Contents

Preface
1. Complex Numbers
1.1. Basic Concepts
1.2. The Complex Plane
1.3. The Modulus and Argument
1.4. Inversion
Comments
Problems
2. Limits in the Complex Plane
2.1. The Principle of Nested Rectangles
2.2. Limit Points
2.3. Convergent Complex Sequences
2.4. The Riemann Sphere and the Extended Complex Plane
Comments
Problems
3. Complex Functions
3.1. Basic Concepts
3.2. Curves and Domains
3.3. Continuity of a Complex Function
3.4. Uniform Continuity
Comments
Problems
4. Differentiation in the Complex Plane
4.1. The Derivative of a Complex Function
4.2. The Cauchy-Riemann Equations
4.3. Conformal Mapping
Comments
Problems
5. Integration in the Complex Plane
5.1. The Integral of a Complex Function
5.2. Basic Properties of the Integral
5.3. Integrals along Polygonal Curves
5.4. Cauchy's Integral Theorem
5.5. Indefinite Complex Integrals
5.6. Cauchy's Integral Formula
5.7. Infinite Differentiability of Analytic Functions
5.8. Harmonic Functions
Comments
Problems
6. Complex Series
6.1. Convergence vs. Divergence
6.2. Absolute vs. Conditional Convergence
6.3. Uniform Convergence
Comments
Problems
7. Power Series
7.1. Basic Theory
7.2. Determination of the Radius of Convergence
Comments
Problems
8. Some Special Mappings
8.1. The Exponential and Related Functions
8.2. Fractional Linear Transformations
Comments
Problems
9. Multiple-Valued Functions
9.1. Domains of Univalence
9.2. Branches and Branch Points
9.3. Riemann Surfaces
Comments
Problems
10. Taylor Series
10.1. The Taylor Expansion of an Analytic Function
10.2. Uniqueness Theorems
10.3. The Maximum Modulus Principle and Its Implications
Comments
Problems
11. Laurent Series
11.1. The Laurent Expansion of an Analytic Function
11.2. Isolated Singular Points
11.3. Residues
Comments
Problems
12. Applications of Residues
12.1. Logarithmic Residues and the Argument Principle
12.2. Rouché's Theorem and Its Implications
12.3. Evaluation of Improper Real Integrals
12.4. Integrals Involving Multiple-Valued Functions
Comments
Problems
13. Further Theory
13.1. More on Harmonic Functions
13.2. The Dirichlet Problem
13.3. More Conformal Mapping
13.4. Analytic Continuation
13.5. The Symmetry Principle
Comments
Problems
14. Mapping of Polygonal Domains
14.1. The Schwarz-Christoffel Transformation
14.2. Examples
Comments
Examples
15. Some Physical Applications
15.1. Fluid Dynamics
15.2. Examples
15.3. Electrostatics
Comments
Problems
Selcted Hints and Answers
Bibliography
Index