greetings. we have the following integral :

$$I(s)=s\int_{0}^{\infty} \frac{dx}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)-\left(E_{s/2}(( x)^{s/2})-1\right)e^{-x}$$

where : $E_{\alpha}(z)$ is the mittag-leffler fuction

$\omega(x)=\frac{\theta(ix)-1}{2}$

$\theta(x)$ is the jacobi theta function

and $\zeta(s)$ is the Riemann zeta function

the integral behaves well for $Re(s)>1$ . i am trying to extend the domain of the integral to the whole complex plane except for some points. i have tried the identities:

$\omega(x^{-1})=-\frac{1}{2}+\frac{1}{2}x^{1/2}+x^{1/2}\omega(x)$

and

$E_{\alpha}(z^{-1})=1-E_{-\alpha}(z)$

to split the integration at 1 , and replace $x$ by $x^{-1}$ in the interval [0,1], but it seems i am lost !!! hence, the post .

edit: another representation of the integral above is - for a minute assume i'm correct about this equivalence - : $$I(s)=s\int_{0}^{\infty}\frac{E_{s}(x^{s})-1}{xe^{x}(e^{x}-1)}dx$$

i was wondering if we can apply Riemann's trick, and replace this integral with a contour integral to obtain a meromorphic integral !?