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

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

Published $\text {1999}$

Subject Matter

• Complex Analysis

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 Rouché'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