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

by hb1547
Tags: evaluate, integral, pi2 or 6
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hb1547
#1
Mar16-12, 01:39 AM
P: 35
[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]?
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sunjin09
#2
Mar16-12, 01:05 PM
P: 312
Quote Quote by hb1547 View Post
[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]?
never mind ... my complex variable technique is rusty ...
hb1547
#3
Mar17-12, 05:06 PM
P: 35
Anyone else have any input?


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