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 f(x) 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 f is recoverable via the inverse Mellin transform from its Mellin transform .
We may strengthen the boundedness condition on if f(x) is continuous. If is analytic in the strip , and if , where K is a positive constant, then f(x) as defined by the inversion integral exists and is continuous; moreover the Mellin transform of f is for at least .
On the other hand, if we are willing to accept an original f 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 .
where ν and p are fixed real numbers with p>1, then if f(x) 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.
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