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Deriving Planck's law

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data

    I need to find the Planck's law: [tex] R(\lambda)=\frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1} [/tex]

    2. Relevant equations

    3. The attempt at a solution

    I've done most of the derivation, but I got stuck with an integral: [tex] R(\lambda)=\frac{1}{4\pi^3 \hbar^3 c^2} \int^{\infty}_{0} {\frac{E^3}{\exp{{\frac{E}{kT}}}-1}}dE} [/tex]

    Basically, I need a formula for [tex] \int^{\infty}_{0} {\frac{x^x}{e^x-1}}dx} [/tex]

    Could anyone give me the formula or perhaps a link where I could find it myself or maybe just point me in the right direction somehow?

    Thank you.
     
  2. jcsd
  3. Nov 11, 2008 #2

    Hootenanny

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  4. Nov 11, 2008 #3
    Found it: [tex] \int^{\infty}_{0} {\frac{x^3}{e^x-1}}dx}=\frac{\pi^4}{15} [/tex]

    And evidently I don't really have to use it. :)

    Thanks for your help.
     
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