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 is analytic in the strip , and if it tends to zero uniformly as for any real value c between a and b, with its integral along such a line converging absolutely, then if we have that Conversely, suppose is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral is absolutely convergent when . Then is recoverable via the inverse Mellin transform from its Mellin transform . These results can be obtained by relating the Mellin transform to the Fourier transform by a change of variables and then applying an appropriate version of the Fourier inversion theorem. The boundedness condition on can be strengthened if is continuous. If is analytic in the strip , and if , where K is a positive constant, then as defined by the inversion integral exists and is continuous; moreover the Mellin transform of is for at least . On the other hand, if we are willing to accept an original which is a generalized function, we may relax the boundedness condition on to simply make it of polynomial growth in any closed strip contained in the open strip . We may also define a Banach space version of this theorem. If we call by the weighted Lp space of complex valued functions on the positive reals such that where ν and p are fixed real numbers with , then if is in with , then belongs to with and Here functions, identical everywhere except on a set of measure zero, are identified. Since the two-sided Laplace transform can be defined as these theorems can be immediately applied to it also.