Debye Functions

$$\int_0^x {t^n\,dt\over e^t-1} = x^n\left[{{1\over n} - {x\ov... ... + \sum_{k=1}^\infty {B_{2k}x^{2k}\over (2k+n)(2k!)}}\right],$$ (1)

where $\vert x\vert < 2\pi$ and $B_n$ are Bernoulli Numbers.

$$\int_x^\infty {t^n\,dt\over e^t-1}= \sum_{k=1}^\infty e^{-kx... ...{n(n-1)x^{n-2}\over k^3} + \ldots + {n!\over k^{n+1}}}\right],$$ (2)

where $x > 0$. The sum of these two integrals is

$$\int_0^\infty {t^n\,dt\over e^t-1}= n!\zeta (n+1),$$ (3)

where $\zeta(z)$ is the Riemann Zeta Function.

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Debye Functions.'' §27.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 998, 1972.