analytic continuation of an integral
greetings. we have the following integral :
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 :
and the second integral could be thought of as:
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:
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:
under Mellin transform .
that being said, i'm stuck on doing the following Mellin type integral :
any insights are more than welcome