analytic continuation of an integral

greetings. we have the following integral :

where

is the mittag-leffler function

the integral is well defined for

i was wondering if we can apply Riemann's trick, and replace this integral with a contour integral to obtain a meromorphic integral - one that is analytic almost everywhere in the complex plane- !?

namely, consider the contour integral :

where the contour starts and ends at +∞ and circles the origin once. using this contour along with the Mellin-Barnes integral rep. of the mittag-leffler function, can we start working the analytic continuation of the original integral ?

Re: analytic continuation of an integral

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