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How to evaluate this integral to get pi^2/6:

  1. Mar 16, 2012 #1
    [itex]\int_0^\infty \frac{u}{e^u - 1}[/itex]

    I know that this integral is [itex]\frac{\pi^2}{6}[/itex], just from having seen it before, but I'm not really sure if I can evaluate it directly to show that.

    I know that:

    [itex] \zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du [/itex]

    Does the value [itex]\frac{\pi^2}{6}[/itex] come from using other methods of showing the result for [itex]\zeta(2)[/itex] and solving the equation, or is that integral another way of evaluating [itex]\zeta(2)[/itex]?
  2. jcsd
  3. Mar 16, 2012 #2
    never mind ... my complex variable technique is rusty ...
    Last edited: Mar 16, 2012
  4. Mar 17, 2012 #3
    Anyone else have any input?
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