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.[1]
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.