### Introduction

In this Section we introduce Cauchy’s theorem which allows us to simplify the calculation of certain contour integrals. A second result, known as Cauchy’s integral formula, allows us to evaluate some integrals of the form ${\∮}_{C}\frac{f\left(z\right)}{z-{z}_{0}}\phantom{\rule{0.3em}{0ex}}dz$ where ${z}_{0}$ lies inside $C$ .

#### Prerequisites

- be familiar with the basic ideas of functions of a complex variable
- be familiar with line integrals

#### Learning Outcomes

- state and use Cauchy’s theorem
- state and use Cauchy’s integral formula

#### Contents

1 Cauchy’s theorem1.1 Simply-connected regions

2 Cauchy’s integral formula

2.1 The derivative of an analytic function