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 ?