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How to solve this integral

  1. Dec 2, 2005 #1
    When integrating plank's fomula to obatin boltzman law,
    I need to integrate

    f(x) = x^3/(e^x-1) from 0 to infinity, the result is pi^4/15

    Does anybody have any idea on how to do this elegantly??
    Thank you.
    Jaap
     
  2. jcsd
  3. Dec 2, 2005 #2

    George Jones

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    This is a problem in Arfken. It involves the ploygamma function and the Riemann zeta function. I won't have time to think about it until later today.

    Regards,
    George
     
  4. Dec 2, 2005 #3

    saltydog

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    This is a problem previously addressed by Daniel:

    The integral is the Debye-Einstein integral:

    [tex]\mathcal{D}_3=\int_0^{\infty} \frac{x^3}{e^x-1}dx=\int_0^\infty \frac{x^3e^{-x}}{1-e^{-x}}dx[/tex]

    Since:

    [tex]\frac{1}{1-e^{-x}}=\sum_{n=0}^{\infty} \left(\frac{1}{e^x}\right)^n[/tex]

    then:

    [tex]\sum_{n=1}^{\infty}\int_0^{\infty} x^3 e^{-nx}dx=\Gamma(x)\zeta(4)[/tex]
     
    Last edited: Dec 2, 2005
  5. Dec 2, 2005 #4

    George Jones

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    Very nice. :smile:

    Picking a nit - there's a minor typo in the last line.

    I had hoped to have a go at this problem this afternoon after finishing my "real" work; now I guess I'll have to find something else to do.

    Regards,
    George
     
  6. Dec 2, 2005 #5

    saltydog

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    Thanks for pointing that out. Should it read:

    [tex]\sum_{n=1}^{\infty}\int_0^{\infty} x^3 e^{-nx}dx=\Gamma(4)\zeta(4)[/tex]

    And thus, would we have:

    [tex]\mathcal{D}_n=\Gamma(n+1)\zeta(n+1)\quad ?[/tex]

    I'm not sure and will need to look at it a bit. Well, . . . how about you Jaap?

    Edit:

    Yep, yep, I think we should re-phrase the question:

    Japp, kindly prove or disprove the following:

    [tex]\int_0^{\infty}\frac{x^n}{e^x-1}dx \:?=\:\Gamma(n+1)\zeta(n+1)[/tex]

    (and he also showed me how to put that question mark on top of the equal sign but I forgot)
     
    Last edited: Dec 2, 2005
  7. Dec 2, 2005 #6
    Thanks guys! let me chew on that one a bit. Note however, I'm an engineer not a mathematician. Nice to know there is a community out here to help, Makes me feel good.

    I'll let Y'all know if I have any questions

    Jaap
     
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