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Stefan-Boltzman's radiation law problem

  1. Jan 11, 2005 #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
     
  2. jcsd
  3. Jan 11, 2005 #2

    dextercioby

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    Homework Helper

    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.
     
  4. Jan 11, 2005 #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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