Mellin inversion theorem

In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.

If ${\displaystyle \varphi (s)}$ is analytic in the strip ${\displaystyle a<\Re (s)<b}$, and if it tends to zero uniformly as ${\displaystyle \Im (s)\to \pm \infty }$ for any real value c between a and b, with its integral along such a line converging absolutely, then if

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