Get Mellin Inversion Theorem essential facts below. View Videos or join the Mellin Inversion Theorem discussion. Add Mellin Inversion Theorem to your PopFlock.com topic list for future reference or share this resource on social media.
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
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 f on the positive reals such that
where ? 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.