can the prime really not solve this integral?

[Manolo Sobrino]

(02-27-2014 12:21 PM)parisse Wrote:  You get a numeric answer with:
int(x^3/(exp(x)-1),x,0,inf)
then shift-enter.
x^3/(exp(x)-1) does not have an antiderivative than you can express with elementary function (special functions required, polylogs here). There is probably a trick than can give you the exact answer (pi^4/15) for the definite integral, any idea? On a voyage 200, you don't get the exact value by the way.

No, no, you don't use primitives for this, come on. You could use contour integration/Residue Theorem, but it's a bit tricky for this. See for instance:

http://en.wikipedia.org/wiki/Stefan_bolt...w#Appendix

The easy way is:

http://math.stackexchange.com/questions/...u3-eu-1-du

That integral appears when you integrate the Planck distribution to all frequencies in order to recover the Stefan-Boltzmann law, also in the Debye theory of specific heat. The functions defined by these integrals are called Debye functions. See Abramowitz-Stegun §27.1

http://people.math.sfu.ca/~cbm/aands/page_998.htm

(Now that I'm thinking about it, it shouldn't be too difficult to implement a large class of definite integrals by the Residue theorem. That would be come in handy.)