### Introduction

In this Section we show how a Fourier series can be expressed more concisely if we introduce the complex number $\text{i}$ where ${i}^{2}=-1$ . By utilising the Euler relation:

$\phantom{\rule{2em}{0ex}}{e}^{\text{i}\theta}\equiv cos\theta +\text{i}sin\theta $

we can replace the trigonometric functions by complex exponential functions. By also combining the Fourier coefficients ${a}_{n}$ and ${b}_{n}$ into a complex coefficient ${c}_{n}$ through

$\phantom{\rule{2em}{0ex}}{c}_{n}=\frac{1}{2}\left({a}_{n}-\text{i}{b}_{n}\right)$

we find that, for a given periodic signal, both sets of constants can be found in one operation.

We also obtain Parseval’s theorem which has important applications in electrical engineering.

The complex formulation of a Fourier series is an important precursor of the Fourier transform which attempts to Fourier analyse non-periodic functions.

#### Prerequisites

- know how to obtain a Fourier series
- be competent working with the complex numbers
- be familiar with the relation between the exponential function and the trigonometric functions

#### Learning Outcomes

- express a periodic function in terms of its Fourier series in complex form
- understand Parseval’s theorem

#### Contents

1 Complex exponential form of a Fourier series2 Revision of the exponential form of a complex number

3 Properties of the complex Fourier coefficients

4 Parseval’s theorem

4.1 Proof of Parseval’s theorem.