I think the tenth (Cauchy residue) and eleventh (Fourier domain) are my favourites, and what I'd use if I had to produce a proof to save my life.

I didn't know you could use the Stirling Approximation though. Anyone who does any thermodynamics should be very familiar with this trick, but I think people rarely include the second order term with the suspicious sqrt(pi) factor, much less think about where it comes from.