Contour Integration

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We were given the value of an integral but told to have a go finding it out if we got bored.... now im bored and annoyed at contour integration :D

The integral is from 0 to infinity of x^3 / (e^x - 1)

I have drawn a contour but im not sure if its right as i dont get the correct answer using it. The contour is a sort of 3/4 circle (top right quater missing) with the origin and positive y axis cut out.

Is there a better contour or am i just going wrong from here on in? (the latter is most likely the case) :)

(i dont really want a full written solution, just some pointers to get me through)

Thank you

[Xerxes314]

That seems like a bad idea, since the denominator has an infinite number of poles running up and down the imaginary axis. You prolly want to run down the positive real axis at +iε, curl around the one pole at the origin and then run back along −iε.

Xerxes

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Ahh, the cuts go further than i thought, didnt think about negative x, but if the branch/whatever it is goes for the whole imaginary axis does the contour look like a quater of a circle with the pointy bit indented?

[Xerxes314]

Oops, I retract what I said. There is no pole at zero. You can solve this by explicit integration into polylogs. The limits at zero and infinity are both finite. In fact, the limit at infinity goes to zero.

Xerxes

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Are polylogs what parrots use to perch on?
See my problem is we've been given 3 examples of contour integration and are expected to know how it works in all cases, such is life :)

I might go back to the lecturer on this one, unless you want to try teaching a new technique via the board :)

i think the problem is that the integral was from my statistical physics lecturer not my complex variable theory one, so im not sure if he knows what i've been taught.

[Xerxes314]

I'm arguing that this isn't a contour integral at all. The function is well defined all along the positive real axis. Check out formula (4) on Mathworld on Polylogarithms.

Xerxes

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Thanks for the link, in that case i'll definately go back to the lecturer and ask what the heck he was going on about :)
Thanks again