### Introduction

In this introduction to functions of a complex variable we shall show how the operations of taking a limit and of finding a derivative, which we are familiar with for functions of a real variable, extend in a natural way to the complex plane. In fact the
**
notation
**
used for functions of a complex variable and for complex operations is almost identical to that used for functions of a real variable. In effect, the
**
real variable x
**
is simply replaced by the
**
complex variable
**
$z$
. However, it is the
interpretation
of functions of a complex variable and of complex operations that differs significantly from the real case. In effect, a function of a complex variable is equivalent to
**
two
**
functions of a real variable and our standard interpretation of a function of a real variable as being a curve on an
$xy$
plane no longer holds.
There are many situations in engineering which are described quite naturally by specifying two harmonic functions of a real variable: a harmonic function is one satisfying the two-dimensional Laplace equation:

$\phantom{\rule{2em}{0ex}}\frac{{\partial}^{2}f}{\partial {x}^{2}}+\frac{{\partial}^{2}f}{\partial {y}^{2}}=0.$

Fluids and heat flow in two dimensions are particular examples. It turns out that knowledge of functions of a complex variable can significantly ease the calculations involved in this area.

#### Prerequisites

- understand how to use the polar and exponential forms of a complex number
- be familiar with trigonometric relations, hyperbolic and logarithmic functions
- be able to form a partial derivative
- be able to take a limit

#### Learning Outcomes

- explain the meaning of the term analytic function

#### Contents

1 Complex functions2 The limit of a function

2.1 Definition of continuity

3 Differentiating functions of a complex variable