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:

$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}=0.$

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