Mellin inversion theorem

In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. If φ ( s ) {displaystyle varphi (s)} is analytic in the strip a < ℜ ( s ) < b {displaystyle a<Re (s)<b} ,and if it tends to zero uniformly as ℑ ( s ) → ± ∞ {displaystyle Im (s) o pm infty } for any real value c between a and b, with its integral along such a line converging absolutely, then if