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Stefan-Boltzman's radiation law problem

  • #1
Hello everyone,
I cannot solve the following integral
integral from 0 to infinity of (x^3)/(exp(kx)-1) dx with k>0
as you may be able to guess, this integral is very famous one which can be used to deduce Stefan-Boltzman's radiation law
those of you who are experts may quickly recognize it and easily solve it
I've ever learnt the trick 4 years ago, now forget hehe
so I would be grateful if you could teach me that trick
any comments are welcome

regards
 

Answers and Replies

  • #2
dextercioby
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1.
HINT:Write the integrand as a product between the
[tex] x^{3} [/tex]
and the fraction containing the exponetial;then write the fraction as a sum of a geometic series with ratio [itex] e^{-kx} [/itex].
Take the sum in the exterior of the integral and then perform three times partial integration.U'll end up with:
[tex] I=6\sum_{k=1}^{+\infty} \frac{1}{k^{4}} [/tex]

which is
[tex] I=6\zeta(4) [/tex]

2.Search the integral in the tables (e.g.Abramowitz & Stegun,Gradsteyn & Rytzik ) to find
[tex] I=\Gamma(4)\zeta(4) [/tex]

Daniel.
 
  • #3
thank you
before my first posting, I had tried to use the geometrical series expansion, but I performed the summation inside the integral and I didn't realize that with limit of integration from 0 to infinity, I just could replace x by 0.

thanks anyway
 

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