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In the course of solving a certain problem, I've had to evaluate integrals of the form:

$$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$

for several values of k. I've noticed that that, for k a positive integer other than 1, the result is seemingly always a dyadic rational multiple of $\zeta(k)$, which is not particularly surprising given some of the identities for $\zeta$ (k=7 is the first noninteger value).

However, I've been unable to find a nice way to evaluate this integral. I'm reasonably sure there's a way to change this expression into $\int \frac{x^{k-1}}{e^x+1} \mathrm{d}x$, but all the things I tried didn't work. Integration by parts also got too messy quickly, and Mathematica couldn't solve it (though it could calculate for a particular value of k very easily).

So I'm looking for a simple way to evaluate the above integral.

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  • $\begingroup$ The indefinite integral involves a polylogarithm, and when you plug in x=0 you get your zeta function. $\endgroup$ – GEdgar May 8 '11 at 2:36
  • $\begingroup$ At least for $k$ even you should be able to do this by contour integration. Are you familiar with complex analysis? $\endgroup$ – Qiaochu Yuan May 8 '11 at 2:38
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Just note that $$ \frac{1}{1 + \cosh x} = \frac{2e^{-x}}{(1 + e^{-x})^2} = 2 \frac{d}{dx} \frac{1}{1 + e^{-x}} = 2 \sum_{n = 1}^{\infty} (-1)^{n-1} n e^{-n x}.$$ Thus we have $$ \begin{eqnarray*}\int_{0}^{\infty} \frac{x^k}{1 + \cosh x} \, dx & = & 2 \sum_{n = 1}^{\infty} (-1)^{n-1} n \int_{0}^{\infty} x^{k} e^{-n x} \, dx \\ & = & 2 \sum_{n = 1}^{\infty} (-1)^{n-1} \frac{\Gamma(k+1)}{n^k} \\ & = & 2 (1 - 2^{1-k}) \zeta(k) \Gamma(k+1). \end{eqnarray*}$$ This formula works for all $k > -1$, where we understand that the Dirichlet eta function $\eta(s) = (1 - 2^{1-s})\zeta(s)$ is defined, by analytic continuation, for all $s \in \mathbb{C}$.

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  • $\begingroup$ A very nice solution. I wonder why Mathematica couldn't do it. $\endgroup$ – TheBlaarg May 8 '11 at 2:42
  • $\begingroup$ Very tidy. :) $\,$ $\endgroup$ – J. M. is a poor mathematician May 8 '11 at 2:56
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sos440 Added it to my collection $$\int\limits_0^\infty \frac{x^{s-1}}{e^x-1}dx=\zeta(s)\Gamma(s)$$

$$\int\limits_0^\infty \frac{x^{s-1}}{e^x+1}dx=\eta(s)\Gamma(s)$$

$$\int\limits_0^\infty \frac{x^{s}e^x}{(e^x-1)^2}dx=\zeta(s)\Gamma(s+1)$$

$$\int\limits_0^\infty \frac{x^{s}e^x}{(e^x+1)^2}dx=\eta(s)\Gamma(s+1)$$

$$\int\limits_0^1 \frac{x^s}{1-x}\log x dx = \sum\limits_{k=0}^\infty \frac{1}{(k+s)^2}$$ I'm guessing my memory is all right, but feel free to correct any mistakes!

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  • $\begingroup$ @sos440 Guess this wiki will be helpful for similar questions arising in the future. $\endgroup$ – Pedro Tamaroff Mar 21 '12 at 4:36
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    $\begingroup$ The last one is $\large\Psi\,'\left(s\right)$ where $\large\Psi\,'$ is the $\large\it\mbox{Trigamma Function}$. $\endgroup$ – Felix Marin Mar 1 '14 at 18:25
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    $\begingroup$ Those are very popular in Statistical Physics. In the Pathria Book appendix they are studied in full detail including asymptotic expansions, zeta function analytical continuation, etc... $\endgroup$ – Felix Marin Mar 1 '14 at 18:30

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