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.
Method.
If formula_1 is analytic in the strip formula_2,
and if it tends to zero uniformly as formula_3 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 formula_2. Then "f" is recoverable via the inverse Mellin transform from its Mellin transform formula_8.
Boundedness condition.
We may strengthen the boundedness condition on formula_1 if 
"f"("x") is continuous. If formula_1 is analytic in the strip formula_2, and if formula_12, 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 formula_8 for at least formula_2.
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 
formula_8 to
simply make it of polynomial growth in any closed strip contained in the open strip formula_2.
We may also define a Banach space version of this theorem. If we call by
formula_17 the weighted Lp space of complex valued functions "f" on the positive reals such that
where ν and "p" are fixed real numbers with "p">1, then if "f"("x")
is in formula_17 with formula_20, then
formula_1 belongs to formula_22 with formula_23 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.