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by Jerrold E. Marsden and Michael J. Hoffman

Edition: 3RD 99Copyright: 1999

Publisher: W.H. Freeman

Published: 1999

International: No

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- 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

**Preface**

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

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

Convergent Series of Analytic Functions

Power Series and Taylor's Theorem

Laurent's Series and Classification of Singularities

Review Exercises for Chapter 3

Calculation of Residues

The Residue Theorem

Evaluation of Definite Integrals

Evaluation of Infinite Series and

Partial-Fraction Expansions

Review Exercises for Chapter 4

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

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

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

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**

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
#### 1. Analytic Functions

#### 2. Cauchy's Theorem

#### 3. Series Representation of Analytic Functions

#### 4. Calculus of Residues

#### 5. Conformal Mappings

#### 6. Further Development of the Theory

#### 7. Asymptotic Methods

#### 8. The Laplace Transform and Applications

**Preface**

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

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

Convergent Series of Analytic Functions

Power Series and Taylor's Theorem

Laurent's Series and Classification of Singularities

Review Exercises for Chapter 3

Calculation of Residues

The Residue Theorem

Evaluation of Definite Integrals

Evaluation of Infinite Series and

Partial-Fraction Expansions

Review Exercises for Chapter 4

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

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

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

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**

Publisher Info

Publisher: W.H. Freeman

Published: 1999

International: No

Published: 1999

International: No

- 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

**Preface**

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

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

Convergent Series of Analytic Functions

Power Series and Taylor's Theorem

Laurent's Series and Classification of Singularities

Review Exercises for Chapter 3

Calculation of Residues

The Residue Theorem

Evaluation of Definite Integrals

Evaluation of Infinite Series and

Partial-Fraction Expansions

Review Exercises for Chapter 4

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

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

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

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**