Mellin inversion formula
= Mellin inversion formula
{c}
{wiki=Mellin_inversion_theorem}
Under standard decay and regularity hypotheses,
$$
f(x)=\frac1{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}
\mathcal Mf(s)x^{-s}\,ds.
$$
If $f$ is smooth and compactly supported in $(0,\infty)$, its Mellin transform is entire, decays faster than every power on vertical lines, and the formula holds for every real $\sigma$ and every $x>0$.