Fourier transform

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

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
$\gamma (t)=\sum _{k=-\infty }^{\infty }\alpha _{k}e^{j2\pi fkt}$ where $f={1 \over T_{\alpha }}$
and $\alpha _{k}={1 \over T_{\alpha }}\int _{-{T_{\alpha } \over 2}}^{T_{\alpha } \over 2}\gamma (t)e^{-j2\pi kt}\,dt$