### Introduction

In this Section we consider two important features of complex functions. The Cauchy-Riemann equations provide a necessary and sufficient condition for a function $f\left(z\right)$ to be analytic in some region of the complex plane; this allows us to find ${f}^{\prime}\left(z\right)$ in that region by the rules of the previous Section.

A mapping between the $z$ -plane and the $w$ -plane is said to be conformal if the angle between two intersecting curves in the $z$ -plane is equal to the angle between their mappings in the $w$ -plane. Such a mapping has widespread uses in solving problems in fluid flow and electromagnetics, for example, where the given problem geometry is somewhat complicated.

#### Prerequisites

- understand the idea of a complex function and its derivative

#### Learning Outcomes

- use the Cauchy-Riemann equations to obtain the derivative of complex functions
- appreciate the idea of a conformal mapping

#### Contents

1 The Cauchy-Riemann equations1.1 Analytic functions and harmonic functions

2 Conformal mapping

2.1 Function as mapping

2.2 Inversion