Table of Contents

Chapter 1- Complex Numbers
1.1 The Origin of Complex Numbers
1.2 The Algebra of Complex Numbers
1.3 The Geometry of Complex Numbers
1.4 The Geometry of Complex Numbers, Continued
1.5 The Algebra of Complex Numbers, Revised
1.6 The Topology of Complex Numbers

Chapter 2- Complex Functions
2.1 Functions of a Complex Variable
2.2 Transformations and Linear Mappings
2.3 The Mappings w=z n and w = z 1/n
2.4 Limits and Continuity
2.5 Branches and Functions
2.6 The Reciprocal Transformation w=1/z

Chapter 3- Analytic and Harmonic Functions
3.1 Differentiable and Analytic Functions
3.2 The Cauchy-Riemann Equations
3.3 Harmonic Functions

Chapter 4- Sequences, Julia and Mandelbrot Sets, and Power Series
4.1 Sequences and Series
4.2 Julia and Mandelbrot Sets
4.3 Geometric Series and Convergence Theorems
4.4 Power Series Functions

Chapter 5- Elementary Functions
5.1 The Complex Exponential Function
5.2 The Complex Logarithm
5.3 Complex Exponents
5.4 Trigonometric and Hyperbolic Functions
5.5 Inverse Trigonometric and Hyperbolic Functions

Chapter 6- Complex Integration
6.1 Complex Integrals
6.2 Contours and Contour Integrals
6.3 The Cauchy-Goursat Theorem
6.4 The Fundamental Theorems of Integration
6.5 Integral Representations for Analytic Functions
6.6 The Theorems of Morera and Liouville and Some Applications

Chapter 7- Taylor and Laurent Series
7.1 Uniform Convergence
7.2 Taylor Series Representations
7.3 Laurent Series Representations
7.4 Singularities, Zeros, and Poles
7.5 Applications of Taylor and Laurent Series

Chapter 8- Residue Theory
8.1 The Residue Theorem
8.2 Trigonometric Integrals
8.3 Improper Integrals of Rational Functions
8.4 Improper Integrals Involving Trigonometric Functions
8.5 Indented Contour Integrals
8.6 Integrands with Branch Points
8.7 The Argument Principle and Rouche's Theorem

Chapter 9- z-Transforms and Applications
9.1 The z-Transform
9.2 Second Order Homogenous Difference Equations
9.3 Digital Signal Features

Chapter 10- Conformal Mapping
10.1 Basic Properties of Conformal Mappings
10.2 Bilinear Transformations
10.3 Mappings Involving Elementary Functions
10.4 Mapping be Trigonometric Functions

Chapter 11- Applications of Harmonic Functions
11.1 Preliminaries
11.2 Invariance of Laplace's Equation and the Dirichlet Problem
11.3 Poisson's Integral Formula for the Upper Half Plane
11.4 Two-Dimensional Mathematical Models
11.5 Steady State Temperatures
11.6 Two-Dimensional Electrostatics
11.7 Two-Dimensional Fluid Flow
11.8 The Joukowski Airfoil
11.9 The Schwarz-Christoffel Transformation
11.10 Image of a Fluid Flow
11.11 Sources and Sinks

Chapter 12- Fourier Series and the Laplace Transform
12.1 Fourier Series
12.2 The Dirichlet Problem for the Unit Disk
12.3 Vibrations in Mechanical Systems
12.4 The Fourier Transform
12.5 The Laplace Transform
12.6 Laplace Transforms of Derivatives and Integrals
12.7 Shifting Theorems and the Step Function
12.8 Multiplication and Division by t
12.9 Inverting the Laplace Transform
12.10 Convolution 

Answers

Index