The Complex Form

Introduction

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

$e^{i θ} \equiv cos θ + i sin θ$

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

$c_{n} = \frac{1}{2} (a_{n} - i b_{n})$

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

1 Complex exponential form of a Fourier series

2 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.

The Complex Form

Introduction

Prerequisites

Learning Outcomes

Contents

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