Proof of Integral: $\int_0^{\infty}\frac{dx x^2}{e^x - 1} = 2\zeta(3)$

[nicksauce]

An integral that seems to come up a lot in stat mech is

$$\int_0^{\infty}\frac{dx x^2}{e^x - 1} = 2\zeta(3)$$

Does anyone know how to prove this?

 

[arildno]

Hmm..the denominator looks, perhaps amenable to make some contour integral in the complex plane, and then utilize the residue theorem?

I'm not at all sure..years since I've done that sort of thing.

 

[jgens]

I'm not particularly familiar with the zeta function, but if you haven't read this, it might help . . .

http://mathworld.wolfram.com/RiemannZetaFunction.html

 

[Count Iblis]

x^2/[exp(x) - 1] =

x^2 exp(-x)/[1 - exp(-x)] =

Sum from n = 1 to infinity of x^2 exp(-nx)

Integrate this from zero to infinity and interchange summation and integration:

Sum from n = 1 to infinity Integral from zero to infinity
x^2 exp(-nx)dx =

Sum from n = 1 to infinity Integral from zero to infinity
1/n^3 t^2 exp(-t)dt =

2 Zeta(3)

 

[nicksauce]

F***ing brilliant! Thanks!