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Basic Complex Analysis

Basic Complex Analysis - 3rd edition

ISBN13: 978-0716728771

Cover of Basic Complex Analysis 3RD 99 (ISBN 978-0716728771)
ISBN13: 978-0716728771
ISBN10: 071672877X
Cover type: Hardback
Edition/Copyright: 3RD 99
Publisher: W.H. Freeman
Published: 1999
International: No
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Basic Complex Analysis - 3RD 99 edition

ISBN13: 978-0716728771

Jerrold E. Marsden and Michael J. Hoffman

ISBN13: 978-0716728771
ISBN10: 071672877X
Cover type: Hardback
Edition/Copyright: 3RD 99
Publisher: W.H. Freeman

Published: 1999
International: No
Summary
  • Generous in the number of examples, exercises, and applications provided
  • Applications include electric potentials, heat conduction, and hydrodynamics-studied with the aid of harmonic functions, conformal mappings, Laplace transforms, asymptotic expansions, and Gamma and Bessel functions
  • Intuitive approach enables application-oriented students to skip the more technical parts without sacrificing an understanding of the main theoretical points
  • Highly readable text motivates students and encourages self-study

Table of Contents

Preface

1. Analytic Functions

Introduction to Complex Numbers
Properties of Complex Numbers
Some Elementary Functions
Continuous Functions
Analytic Functions
Differentiation of the Elementary Functions
Review Exercises for Chapter 1

2. Cauchy's Theorem

Contour Integrals
Supplement: Riemann Sums
Cauchy's Theorem: Intuitive Version
Cauchy's Theorem: Precise Version
Supplement A: Integrals Along
Continuous Curves
Supplement B: Relationship of Cauchy's Theorem to the Jordan Curve Theorem
Cauchy's Integral Formula
Maximum Modulus Theorem and Harmonic Function
Review Exercises for Chapter 2

3. Series Representation of Analytic Functions

Convergent Series of Analytic Functions
Power Series and Taylor's Theorem
Laurent's Series and Classification of Singularities
Review Exercises for Chapter 3

4. Calculus of Residues

Calculation of Residues
The Residue Theorem
Evaluation of Definite Integrals
Evaluation of Infinite Series and
Partial-Fraction Expansions
Review Exercises for Chapter 4

5. Conformal Mappings

Basic Theory of Conformal Mappings
Fractional Linear and Schwarz-Christoffel Transformations
Applications of Conformal Mapping to Laplace's Equation, Heat
Conduction, Electrostatics, and Hydrodynamics
Review Exercises for Chapter 5

6. Further Development of the Theory

Analytic Continuation and Elementary Riemann Surfaces
Rouché's Theorem and the Principle of the Argument
Mapping Properties of Analytic
Functions
Supplement A: Normal Families and The Riemann Mapping Theorem
Supplement B: The Dynamics of Complex Analytic Mappings
Review Exercises for Chapter 6

7. Asymptotic Methods

Infinite Products and the Gamma Function
Asymptotic Expansions and the Method of Steepest Descent
Supplement: Bounded Variation and the Proof of the Stationary Phase Formula
Stirling's Formula and Bessel
Functions
Review Exercises for Chapter 7

8. The Laplace Transform and Applications

Basic Properties of LaPlace Transforms
The Complex Inversion Formula
Application of Laplace Transforms to Ordinary Differential Equations
Supplement: The Fourier Transform and the Wave Equation
Review Exercises for Chapter 8

Index

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