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- ISBN-10:
- 0486647625
- ISBN-13:
- 9780486647623
- Pub. Date:
- Publisher:

## 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.

## Related collections and offers

## Product Details

ISBN-13: | 9780486647623 |
---|---|

Publisher: | Dover Publications |

Publication date: | 10/18/2010 |

Series: | Dover Books on Mathematics Series |

Edition description: | Dover ed |

Pages: | 304 |

Product dimensions: | 5.37(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

Preface1. 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