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. Cauchys Theorem
2.1 Contour Integrals
2.2 Cauchys Theorem-A First Look
2.3 A Closer Look at Cauchys Theorem
2.4 Cauchys Integral Formula
2.5 Maximum Modulus Theorem and Harmonic Functions

3. Series Representation of Analytic Functions
3.1 Convergent Series of Analytic Functions
3.2 Power Series and Taylors 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. Conformal Mappings
5.1 Basic Theory of Conformal Mappings
5.2 Fractional Linear and Schwarz-Christoffel Transformations
5.3 Applications of Conformal Mappings to Laplaces Equation, Heat Conduction, Electrostatics, and Hydrodynamics


6. Further Development of the Theory
6.1 Analytic Continuation and Elementary Riemann Surfaces
6.2 Rouche Theorem and Principle of the Argument
6.3 Mapping Properties of Analytic Functions

7. Asymptotic Methods
7.1 Infinite Products and the Gamma Function
7.2 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