Asymptotic behaviours from Fourier transforms I have completely forgotten how one derives the asymptotic behavior in frequency space, given the asymptotic behavior of the function in real space (e.g. time). As an example example, it is often said that when $f(t)\sim t^\alpha$ for $t\to\infty$, then $f(\omega)\sim\omega^{-\alpha-1}$ for $\omega\to 0$. Aside from a dimensional analysis, how do you derive this result a bit more strictly?

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See Method of Steepest Descent : With $\ds{\int_{-\infty}^{\infty}t^{\alpha}\expo{\ic\omega t}\dd t}$ you get the ${\it saddle\ point}$ $\ds{t_{s} = \alpha\ic/\omega}$ and the ${\it asymptotic\ behavior}$
$$
\bbx{\pars{\root{2\pi}\,\ic^{\alpha - 1}\,\alpha^{\alpha + 1/2}\,\,\expo{-\alpha}}\ {1 \over \omega^{\alpha + 1}}}
$$
