So from the fourier series, we can simplify it further and use trig identities to get the following:
$$ f(t) = \frac{a_0}{2} + \sum^{\infty}_{n=1} \left(\frac{a_n}{2}+\frac{b_n}{2i}\right)e^{i n \omega t} + \left(\frac{a_n}{2}-\frac{b_n}{2i}\right)e^{-i n \omega t} $$
So how do you go from the above line to
$$ f(t) = \sum^{\infty}_{n=-\infty} C_n e^{in\omega t} $$
and
$$ f(t) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} F(\omega) e^{-i \omega t} d\omega $$
Thanks