# Definite integral of a function containing an exponential

I have to calculate analytically this integral: $${\rm J}\left(q\right) = \int_{0}^{\infty}{{\rm d}x \over x^{q}\left({\rm e}^{kx}-1\right)}$$ where $-1\le q\le N$ with: $N\in\mathbb{N}$ and $q\in\mathbb{N}$, $k\le 5\times10^{-5}$ I didn't find anything on the Gradshteyn Ryzhik and Mathematica isn't able to integrate it. Is it possible to make some approximation because the little value of $k$? Thanks in advance.


It isn't bounded for $q\ge 0$.

For $q<0$, it equals $$\zeta(1-q)\Gamma(1-q)k^q,$$ I believe.

• $k^{q - 1}$ instead of $k^{q}$. – Felix Marin Jan 18 '14 at 8:35

Mathematica is perfectly capable of evaluating this:

ConditionalExpression[k^(-1 + q) Gamma[1 - q] PolyLog[1 - q, 1], q < 0 && k > 0]

Since the integrate has an $x^{-q - 1}$ singularity at the origin, the integral diverges for $q \geq 0.$