Chapter 1. Complex Numbers

1. The Origin of Complex Numbers
2. The Algebra of Complex Numbers
3. The Geometry of Complex Numbers
4. The Geometry of Complex Numbers, Continued
5. The Algebra of Complex Numbers, Revisited
6. The Topology of Complex Numbers

 

Chapter 2. Complex Functions

1. Complex Functions and Linear Mappings
2. The Mappings and
3. Complex Limits and Continuity
4. Branches of Complex Functions
5. The Reciprocal Transformation

 

Chapter 3. Analytic and Harmonic Functions

1. Differentiable and Analytic Functions
2. The Cauchy-Riemann Equations
3. Harmonic Functions

 

Chapter 4. Sequences, Series, and Julia and Mandelbrot Sets

1. Complex Sequences and Series
2. Julia and Mandelbrot Sets
3. Geometric Series and Convergence Theorems
4. Power Series Functions

 

Chapter 5. Elementary Functions

1. The Complex Exponential Function
2. The Complex Logarithm Function
3. Complex Exponents and Powers
4. Trigonometric and Hyperbolic Functions
5. Inverse Trigonometric and Hyperbolic Functions

 

Chapter 6. Complex Integration

1. Complex Integrals
2. Contours and Contour Integrals
3. The Cauchy-Goursat Theorem
4. The Fundamental Theorem of Integration
5. Integral Representations for Analytic Functions
6. The Theorems of Morera and Liouville and Applications

 

Chapter 7. Taylor and Laurent Series

1. Uniform Convergence
2. Taylor Series Representations
3. Laurent Series Representations
4. Singularities, Zeros and Poles
5. Applications of Taylor and Laurent Series

 

Chapter 8. Residue Theory

1. The Residue Theorem
2. Trigonometric Integrals
3. Improper Integrals of Rational Functions
4. Improper Integrals Involving Trigonometric Functions
5. Indented Contour Integrals
6. Integrands with Branch Points
7. The Argument Principle and Rouche's Theorem

  

Chapter 9. The z-Transforms and Applications

1. The z-transform
2. Second-Order Homogeneous Difference Equations
3. Digital Signal Filters

 

Chapter 10. Conformal Mapping

1. Basic Properties of Conformal Mappings
2. Mobius Transformations - Bilinear Transformations
3. Mappings Involving Elementary Functions
4. Mappings by Trigonometric Functions

 

Chapter 11. Applications of Harmonic Functions

1. Preliminaries
2. Invariance of Laplace's Equation and the Dirichlet Problem
3. Poisson's Integral Formula for the Upper Half Plane
4. Two-Dimensional Mathematical Models
5. Steady State Temperatures
6. Two-Dimensional Electrostatics
7. Two-Dimensional Fluid Flow
8. The Joukowski Airfoil
9. The Schwarz-Christoffel Transformation
10. Image of a Fluid Flow
11. Sources and Sinks

 

Chapter 12. Fourier Series and the Laplace Transform

1. Fourier Series
2. The Dirichlet Problem for the Unit Disk
3. Vibrations in Mechanical Systems
4. The Fourier Transform
5. The Laplace Transform
6. Laplace Transforms of Derivatives and Integrals
7. Shifting Theorems and the Step Function
8. Multiplication and Division by t
9. Inverting the Laplace Transform
10. Convolution

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