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 $T_{\alpha }<\left|t\right|$