The polylogarithm can be expressed in terms of the integral of the Bose–Einstein distribution according to https://en.wikipedia.org/wiki/Polylogarithm

This converges for Re(s) > 0 and all z except for z real and ≥ 1.

I am interested in to know if the same name "the integral of the Bose–Einstein distribution" is still valid for recognizing that integral when z = exp(2*k*i), being i the imaginary unit and k a non-negative integer (k=1, 2, 3...)?

Is that definition for the "integral of the Bose–Einstein distribution" when z=exp(2*k*i) possible or has to be only when z has another domain?

Best regards,

Carlos López