Integrating a Photon Gas: Contour Integration for the Grand Potential

[Parmenides]

When considering the grand potential for a photon gas, one encounters an integral of the form:
$$\Sigma = a\int_{0}^{\infty}x^2\ln\Big(1 - e^{-bx}\Big)dx$$
I have never had to integrate something like this before; I was told it is done via contour integration, but I have never used such a method and the examples typically given are not of this form. Could somebody please provide some assistance? I have tried to learn a bit myself, but I remain perplexed. What would be the contour enclosing such an integral, for example? Thanks.

UPDATE: I noticed that integration by parts puts the integral in a form of $$\int_{0}^{\infty}\frac{x^3dx}{e^{bx} - 1}$$ ignoring constants. I now recognize this as a familiar integral found in Stefan's Law, but it would still be nice to see someone's perspective of the contour method to evaluate it.

 

[RUber]

Looking at your updated integral, this is an analytic function away from the singularity (simple pole) at x=0. The contour method would have you draw an imaginary contour in the RE, IM plane which traces the imaginary axis from +infty to +epsilon*i then makes a small half circle cut to +epsilon and follows the real axis to +infty, then has a semicircular closure with radius +infty. This contour excludes the singularity at 0, so clearly is analytic on the interior of the contour, which means it will be zero.

Therefore, the evaluation of your integral simplifies to the evaluation of the residue at x=zero.
See the wikipedia pages for more thorough explanation.

http://en.wikipedia.org/wiki/Residue_theorem
http://en.wikipedia.org/wiki/Residue_(complex_analysis )

 

[acegikmoqsuwy]

I think that differentiating your original equations with respect to b on both sides might help.