greetings. we have the following integral :

$\displaystyle I(s)=s\int_{0}^{\infty}\frac{E_{s}(x^{s})-1}{xe^{x}(e^{x}-1)}dx$


$\displaystyle E_{s}(x)$ is the mittag-leffler function

the integral is well defined for $\displaystyle \Re(s)>1$

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 :

$\displaystyle K(s)=-s\oint _{\gamma}\frac{E_{s}((-x)^{s})-1}{xe^{x}(e^{x}-1)}dx $

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 ?