Contents

Introduction

Part 1. Complex Made Simple

Chapter 0. Differentiability and the Cauchy-Riemann Equations

Chapter 1. Power Series

Chapter 2. Preliminary Results on Holomorphic Functions

Chapter 3. Elementary Results on Holomorphic Functions

Chapter 4. Logarithms, Winding Numbers and Cauchy's Theorem

Chapter 5. Counting Zeroes and the Open Mapping Theorem

Chapter 6. Euler's Formula for sin(z)

6.0. Motivation

6.1. Proof by the Residue Theorem

6.2. Estimating Sums Using Integrals

6.3. Proof Using Liouville's Theorem

Chapter 7. Inverses of Holomorphic Maps

Chapter 8. Conformal Mappings

8.0. Meromorphic Functions and the Riemann Sphere