Fourier transform

An initially identity that is useful: $x(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x(t)e^{-j2\pi ft}dt$

Suppose that we have some function, say $\beta (t)$, that is nonperiodic and finite in duration.
This means that $\beta (t)=0$ for some $T_{\alpha }<\left|t\right|$

Now let's make a periodic function $\gamma (t)$ by repeating $\beta (t)$ with a fundamental period $T_{\zeta }$.
The Fourier Series representation of $\gamma (t)$ is
$\gamma (t)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k{e}^{j2\pi fkt}$ where $f=\frac{1}{T_{\alpha }}$
and ${\alpha }_k=\frac{1}{T_{\alpha }}\underset{-\frac{T_{\alpha }}{2}}{\overset{\frac{T_{\alpha }}{2}}{\int }}\gamma (t)e^{-j2\pi kt}dt$