### 1 Complex exponential form of a Fourier series

So far we have discussed the
**
trigonometric
**
form of a Fourier series i.e. we have represented functions of period
$T$
in the terms of sinusoids, and possibly a constant term, using

$\phantom{\rule{2em}{0ex}}f\left(t\right)=\frac{{a}_{0}}{2}+{\sum}_{n=1}^{\infty}\left\{{a}_{n}cos\left(\frac{2n\pi t}{T}\right)+{b}_{n}sin\left(\frac{2n\pi t}{T}\right)\right\}.$

If we use the angular frequency

$\phantom{\rule{2em}{0ex}}{\omega}_{0}=\frac{2\pi}{T}$

we obtain the more concise form

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

We have seen that the Fourier coefficients are calculated using the following integrals:

$\phantom{\rule{2em}{0ex}}{a}_{n}=\frac{2}{T}{\int}_{-\frac{T}{2}}^{\frac{T}{2}}f\left(t\right)cosn{\omega}_{0}t\phantom{\rule{1em}{0ex}}dt\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n=0,1,2,\dots $ (1)

$\phantom{\rule{2em}{0ex}}{b}_{n}=\frac{2}{T}{\int}_{-\frac{T}{2}}^{\frac{T}{2}}f\left(t\right)sinn{\omega}_{0}t\phantom{\rule{1em}{0ex}}dt\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n=1,2,\dots $ (2)

An alternative, more concise form, of a Fourier series is available using
**
complex quantities
**
. This form is quite widely used by engineers, for example in Circuit Theory and Control Theory, and leads naturally into the Fourier Transform which is the subject of
**
HELM booklet
**
24.