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- Basic Complex Analysis

# Basic Complex Analysis

## Third Edition| ©1999 Jerrold E. Marsden, California Institute of Technology; Michael J. Hoffman, California State University, Los Angeles

*Basic Complex Analysis*skillfully combines a clear exposition of core theory with a rich variety of applications. Designed for undergraduates in mathematics, the physical sciences, and engineering who have completed two years of calculus and are taking complex analysis for the

*Basic Complex Analysis*skillfully combines a clear exposition of core theory with a rich variety of applications. Designed for undergraduates in mathematics, the physical sciences, and engineering who have completed two years of calculus and are taking complex analysis for the first time.

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*Basic Complex Analysis*skillfully combines a clear exposition of core theory with a rich variety of applications. Designed for undergraduates in mathematics, the physical sciences, and engineering who have completed two years of calculus and are taking complex analysis for the first time.

Features

**Only first year calculus required**--all necessary linear algebra is explained

Incorporates wide range of

**physical applications**,**dozens of graphics**, and a large number of**exercises****Boxes**highlight important definitions and formulas

**Notes to the student**offer further help on exceptionally difficult topics

New to This Edition

**
Basic Complex Analysis**

Third Edition| ©1999

Jerrold E. Marsden, California Institute of Technology; Michael J. Hoffman, California State University, Los Angeles

# Digital Options

**Basic Complex Analysis**

Third Edition| 1999

Jerrold E. Marsden, California Institute of Technology; Michael J. Hoffman, California State University, Los Angeles

## Table of Contents

**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 the 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

3.2 Power Series and Taylor's Theorem

3.3 Laurent Series and Classification of Singularities

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 Representation of Analytic Functions**

3.1 Convergent Series of Analytic Functions3.2 Power Series and Taylor's Theorem

3.3 Laurent Series and Classification of Singularities

**4. Calculus of Residues**

4.1 Calculation of Residues

4.2 Residue Theorem

4.3 Evaluation of Definite Integrals

4.4 Evaluation of Infinite Series and Partial-Fraction Expansions

5.2 Fractional Linear and Schwarz-Christoffel Transformations

5.3 Applications of Conformal Mappings to Laplace's Equation, Heat Conduction, Electrostatics, and Hydrodynamics

6.2 Rouche Theorem and Principle of the Argument

6.3 Mapping Properties of Analytic Functions

7.2 Asymptotic Expansions and the Method of Steepest Descent

7.3 Stirlings Formula and Bessel Functions

**5. Conformal Mappings**

5.1 Basic Theory of Conformal Mappings5.2 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 Surfaces6.2 Rouche Theorem and Principle of the Argument

6.3 Mapping Properties of Analytic Functions

**7. Asymptotic Methods**

7.1 Infinite Products and the Gamma Function7.2 Asymptotic Expansions and the Method of Steepest Descent

7.3 Stirlings 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**

**Basic Complex Analysis**

Third Edition| 1999

Jerrold E. Marsden, California Institute of Technology; Michael J. Hoffman, California State University, Los Angeles

## Authors

### Jerrold E. Marsden

### Michael J. Hoffman

**Basic Complex Analysis**

Third Edition| 1999

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