Table of Contents

Preamble

• Introduction
• License
• Acknowledgements
• Reader's Comments

Chapter 1

• A Brief History
• Terminology and Notation
• Geometric Interpretation
• The Principal Argument
• Roots of Complex Numbers
• Topology of the Complex Plane

Chapter 2

• Complex Functions
• Limits
• Riemann Sphere
• Continuity
• Complex Differentiation
• The Logarithmic Function
• Riemann surfaces: Examples

Chapter 3

• Mappings
• The Transformation $1/z$
• Analytic Landscapes
• Domain Coloring
• The Complex Power Function

Chapter 4

• Curves in the Complex Plane
• Complex Integration
• Integrals of Functions with Branch Cuts

Chapter 5

• Series
• Taylor Series
• Laurent Series
• Classification of Singularities
• The Mandelbrot Set
• The Julia Set

Chapter 6

• Conformal Mapping
• Linear Rational Transformations
• Applications of Conformal Mappings
• Complex Potential: Basic examples
• Uniform Flow Around a Circle
• Joukowsky Airfoil

Further Reading

• Bibliography