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An integral related to the zeta function at the point $2$ is given by

$$\zeta(2) = \int\nolimits_0^\infty \dfrac{t}{e^t - 1}\mathrm dt$$

How to calculate this integral?

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  • $\begingroup$ Well... you already did (compute this integral), didn't you? Since you know its value. Or do you want a proof that it is indeed zeta(2)? $\endgroup$ – Did Sep 14 '11 at 13:12
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    $\begingroup$ @Dan: ?? $ $ $ $ $\endgroup$ – Did Sep 14 '11 at 13:15
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    $\begingroup$ Didier, throw a needle! $\endgroup$ – Dan Brumleve Sep 14 '11 at 13:17
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    $\begingroup$ Is the fact that the function being integrated is the generating function of the Bernoulli numbers of interest here? $\endgroup$ – Michael Hardy Sep 14 '11 at 14:04
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    $\begingroup$ @Michael Hardy: Yes it is of interest. We can see this directly with the "notion" of a negative Bernoulli number. (Non rigorous, it might be possible to make it rigorous though) The generating series fact can be reworded as $$B_n =\lim_{x\rightarrow 0} \frac{d^n}{dx^n} \frac{x}{e^x-1}.$$ In some "sense" the integral of $\frac{t}{e^t-1}$ should be like $B_{-1}$ since we would want to take $n=-1$. Recall as well that $-n\zeta(1-n)=B_n$ for positive $n$. If we "define" the negative Bernoulli numbers by that formula as well, then $B_{-1}=\zeta(2)$. Voilà. $\endgroup$ – Eric Naslund Sep 14 '11 at 15:02
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The integrand can be expressed as a geometric series with first term $te^{-t}$ and common ratio $e^{-t}$. Integrate term-by-term (after justifying it, of course) and see if you don't recognize the result as $\zeta(2)$.

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Somewhat equivalent to Gerry's answer: let $t=-\log(1-u)$, giving the integral

$$-\int_0^1 \frac{\log(1-u)}{u}\mathrm du$$

Expand the logarithm as a series, swap summation and integration, and then you should be able to see something familiar...

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    $\begingroup$ Or notice that this integral is the dilogarithm evaluated at $1$. Wait, I guess you prove that $\text{Li}_2 (1)=\frac{\pi^2}{6}$ by expanding the integral anyway so nevermind.... $\endgroup$ – Eric Naslund Sep 14 '11 at 13:33
  • $\begingroup$ @Eric: I didn't want to bring it up. You're right, of course. ;) $\endgroup$ – J. M. is a poor mathematician Sep 14 '11 at 13:34

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