following Riemann's trick, here is what i did :

start with the contour integral :

the contour is the usual Hankel contour. consider :

- the Mittag-Leffler function admits the beautiful continuation :

or

now :

and the second integral could be thought of as:

or :

lets go back to the 1st integral, and expand the Mittag-leffler function :

now the problem becomes finding a function of the variable s -lets call it - such that:

define:

then :

and the problem becomes proving the existence of for all s, and of course, finding it !!

here is my strategy for finding :

for some function/distribution . clearly the relation above defines a Mellin pair. Using Mellin inversion theorem, along with Mellin convolution, one can find a function , such that:

where :

under Mellin transform .

that being said, i'm stuck on doing the following Mellin type integral :

any insights are more than welcome