### Introduction

In this Section we show how a periodic function can be expressed as a series of sines and cosines. We begin by obtaining some standard integrals involving sinusoids. We then assume that if $f\left(t\right)$ is a periodic function, of period $2\pi $ , then the Fourier series expansion takes the form:

$\phantom{\rule{2em}{0ex}}f\left(t\right)=\frac{{a}_{0}}{2}+{\sum}_{n=1}^{\infty}\left({a}_{n}cosnt+{b}_{n}sinnt\right)$

Our main purpose here is to show how the constants in this expansion, ${a}_{n}$ (for $n=0,1,2,3\dots $ and ${b}_{n}$ (for $n=1,2,3,\dots $ ), may be determined for any given function $f\left(t\right)$ .

#### Prerequisites

- know what a periodic function is
- be able to integrate functions involving sinusoids
- have knowledge of integration by parts

#### Learning Outcomes

- calculate Fourier coefficients of a function of period $2\pi $
- calculate Fourier coefficients of a function of general period

#### Contents

1 Introduction2 Functions of period $2\pi $

2.1 Orthogonality properties of sinusoids

3 Calculation of Fourier coefficients

4 Examples of Fourier series

5 Fourier series for functions of general period