### 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