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greetings . we have the integral :

[tex] I(s)=\int_{0}^{\infty}\frac{s(E_{s}(x^{s})-1)-x}{x(e^{x}-1)}dx [/tex]

which is equivalent to

[tex] =I(s)=\frac{1}{4}\int_{0}^{\infty}\frac{\theta(ix)\left(sE_{s/2} ((\pi x)^{s/2})-s-2x^{1/2}\right)}{x}dx [/tex]

[itex]E_{\alpha}(z)[/itex] being the mittag-leffler function

and [itex] \theta(x) [/itex] is the jacobi theta function

the integral above behaves well for Re(s)>1 . i am trying to extend the domain of [itex]I(s)[/itex] to the whole complex plane except for some points. but i have no idea where to start !!

[tex] I(s)=\int_{0}^{\infty}\frac{s(E_{s}(x^{s})-1)-x}{x(e^{x}-1)}dx [/tex]

which is equivalent to

[tex] =I(s)=\frac{1}{4}\int_{0}^{\infty}\frac{\theta(ix)\left(sE_{s/2} ((\pi x)^{s/2})-s-2x^{1/2}\right)}{x}dx [/tex]

[itex]E_{\alpha}(z)[/itex] being the mittag-leffler function

and [itex] \theta(x) [/itex] is the jacobi theta function

the integral above behaves well for Re(s)>1 . i am trying to extend the domain of [itex]I(s)[/itex] to the whole complex plane except for some points. but i have no idea where to start !!

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