### Introduction

In this Section we introduce a third way of expressing a complex number: the exponential form. We shall discover, through the use of the complex number notation, the intimate connection between the exponential function and the trigonometric functions. We shall also see, using the exponential form, that certain calculations, particularly multiplication and division of complex numbers, are even easier than when expressed in polar form.

The exponential form of a complex number is in widespread use in engineering and science.

#### Prerequisites

- be able to convert from degrees to radians
- understand how to use the Cartesian and polar forms of a complex number
- be familiar with the hyperbolic functions $coshx$ and $sinhx$

#### Learning Outcomes

- explain the relations between the exponential function ${\text{e}}^{x}$ and the trigonometric functions $cosx$ , $sinx$
- interchange between Cartesian, polar and exponential forms of a complex number
- explain the relation between hyperbolic and trigonometric functions

#### Contents

1 Series expansions for exponential and trigonometric functions2 The exponential form

3 Hyperbolic and trigonometric functions

4 Engineering Example 1

4.1 Feedback applied to an amplifier