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

$f\left(t\right)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }\left\{a_{n}cos\left(\frac{2n\pi t}{T}\right)+b_{n}sin\left(\frac{2n\pi t}{T}\right)\right\}.$

If we use the angular frequency

${\omega }_0=\frac{2\pi }{T}$

we obtain the more concise form

$f\left(t\right)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }\left(a_{n}cosn{\omega }_{0}t+b_{n}sinn{\omega }_{0}t\right).$

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

$a_n=\frac{2}{T}{\int \; <!--nolimits\; -->}_{-\frac{T}{2}}^{\frac{T}{2}}f\left(t\right)cosn{\omega }_{0}tdtn=0,1,2,\dots$ (1)

$b_n=\frac{2}{T}{\int \; <!--nolimits\; -->}_{-\frac{T}{2}}^{\frac{T}{2}}f\left(t\right)sinn{\omega }_{0}tdtn=1,2,\dots$ (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.