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:

$e^{\text{ i}\theta }\equiv cos\theta +\text{ i}sin\theta$

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

$c_n=\frac{1}{2}\left(a_n-\text{ i}{b}_n\right)$

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