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. Method 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. Boundedness condition The boundedness condition on can be strengthen 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. See also Mellin transform Nachbin's theorem References External links Tables of Integral Transforms at EqWorld: The World of Mathematical Equations. Integral transforms Theorems in complex analysis Laplace transforms