Does anyone know how to compute analytically or numerically the following integral (for $T=10^4$K)?:
$$n_\gamma=\frac{1}{\hbar^3\pi^2c^3}\int\limits_{2.1789\cdot 10^{-18}}^{+\infty}\dfrac{E^2\mathrm{d}E}{e^{\frac{E}{kT}}-1}$$
I tried with R, MATLAB, Maxima, Maple and Wolfram but I failed. I also search an analytical solution during a least a whole week....
Thx in advance for your help.
Sometimes computers have issues dealing with precision. Also try a change of variables like u=E/kT. Then factor out all the constants. So you get something like u^2du/(e^u-1) in your integral. This will simplify things in your numerical calculation. Once you calculate this integral, then you can just multiply back in the constants at the end.
Sometimes, one just need to do a little work before the analytic software can handle the integration. Here is a suggestion. First write it in the form: $$ \frac{A}{\exp(Bx)-1}=\frac{A\exp(-Bx)}{1-\exp(-Bx)} . $$ Then expand the denominator $$ \frac{A\exp(-Bx)}{1-\exp(-Bx)} = A\exp(-Bx)\sum_n \exp(-nBx) . $$ Now you (or Maple, etc.) should be able to integrate every term after which you can resum the series.
