# Book:Jerrold E. Marsden/Basic Complex Analysis/Third Edition

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## Jerrold E. Marsden and Michael J. Hoffman:

## Jerrold E. Marsden and Michael J. Hoffman: *Basic Complex Analysis (2nd Edition)*

Published $\text {1999}$

### Subject Matter

### Contents

- Preface

- 1 Analytic Functions
- 1.1 Introduction to Complex Numbers
- 1.2 Properties of Complex Numbers
- 1.3 Some Elementary Functions
- 1.4 Continuous Functions
- 1.5 Basic Properties of Analytic Functions
- 1.6 Differentiation of Elementary Functions

- 2 Cauchy's Theorem
- 2.1 Contour Integrals
- 2.2 Cauchy's Theorem – A First Look
- 2.3 A Closer Look at Cauchy's Theorem
- 2.4 Cauchy's Integral Formula
- 2.5 Maximum Modulus Theorem and Harmonic Functions

- 3 Series Representations of Analytic Functions
- 3.1 Convergent Series of Analytic Functions
- 3.2 Power Series and Taylor's Theorem
- 3.3 Laurent Series and Classification of Singularities

- 4 Calculus of Residues
- 4.1 Calculations of Residues
- 4.2 Residue Theorem
- 4.3 Evaluation of Definite Integrals
- 4.4 Evaluation of Infinite Series and Partial-Fraction Expansions

- 5 Conformal Mappings
- 5.1 Basic Theory of Conformal Mappings
- 5.2 Fractional Linear and Schwarz-Christoffel Transformations
- 5.3 Applications of Conformal Mappings to Laplace's Equation, Heat Conduction, Electrostatics and Hydrodynamics

- 6 Further Development of the Theory
- 6.1 Analytic Continuation and Elementary Riemann Surfaces
- 6.2 RouchÃ©'s Theorem and Principle of the Argument
- 6.3 Mapping Properties of Analytic Functions

- 7 Asymptotic Methods
- 7.1 Infinite Products and the Gamma Function
- 7.2 Asymptotic Expansions and the Method of Steepest Descent
- 7.3 Stirling's Formula and Bessel Functions

- 8 Laplace Transform and Applications
- 8.1 Basic Properties of Laplace Transforms
- 8.2 Complex Inversion Formula
- 8.3 Application of Laplace Transforms to Ordinary Differential Equations

- Answers to Odd-Numbered Exercises

- Index