Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof for integral

  1. Aug 2, 2009 #1

    nicksauce

    User Avatar
    Science Advisor
    Homework Helper

    An integral that seems to come up a lot in stat mech is

    [tex]
    \int_0^{\infty}\frac{dx x^2}{e^x - 1} = 2\zeta(3)
    [/tex]

    Does anyone know how to prove this?
     
  2. jcsd
  3. Aug 2, 2009 #2

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Hmm..the denominator looks, perhaps amenable to make some contour integral in the complex plane, and then utilize the residue theorem?

    I'm not at all sure..years since I've done that sort of thing.
     
  4. Aug 2, 2009 #3

    jgens

    User Avatar
    Gold Member

  5. Aug 2, 2009 #4
    x^2/[exp(x) - 1] =

    x^2 exp(-x)/[1 - exp(-x)] =

    Sum from n = 1 to infinity of x^2 exp(-nx)

    Integrate this from zero to infinity and interchange summation and integration:

    Sum from n = 1 to infinity Integral from zero to infinity
    x^2 exp(-nx)dx =

    Sum from n = 1 to infinity Integral from zero to infinity
    1/n^3 t^2 exp(-t)dt =

    2 Zeta(3)
     
  6. Aug 2, 2009 #5

    nicksauce

    User Avatar
    Science Advisor
    Homework Helper

    F***ing brilliant! Thanks!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof for integral
  1. Integral proof (Replies: 4)

  2. Proof of integrals (Replies: 18)

  3. Proof of an integral (Replies: 17)

  4. Integration proof (Replies: 5)

Loading...