I need to evaluate $$\int_0^\infty \left(\frac{x}{\sinh(x)}\right)^3dx$$ I know that I need to use the residue theorem to solve it, but I don't understand how to choose contour properly.

Thank you for any help!

  • 1
    $\begingroup$ See heeehttps://math.stackexchange.com/questions/2136089/contour-integral-of-int-0-infty-fracx-sinh-x-operatornamedx?rq=1 Then by Cauchy you have $$ 0 = \int_{-\infty}^\infty \left(\frac{x}{\sinh(x)}\right)^3dx - \int_{\epsilon}^\infty\left(\frac{x+i\pi}{\sinh(x+i\pi)}\right)^3dx -\int_{C_\epsilon} \left(\frac{z}{\sinh(z)}\right)^3dz - \int_{-\infty}^{-\epsilon} \left(\frac{x+i\pi}{\sinh(x+i\pi)}\right)^3dx $$ where $C_\epsilon$ is counter-clockwise half semicircle under the pole at $i\pi.$ $\endgroup$ – Guy Fsone Dec 11 '17 at 19:46

You do not need to use the residue theorem.

By substituting $x=\log t$ we get

$$ \int_{1}^{+\infty}\frac{8t^2\log^3(t)}{(t^2-1)^3}\,dt = \int_{0}^{1}\frac{-8t^2\log^3(t)}{(1-t^2)^3}\,dt $$ where $$ \frac{t^2}{(1-t^2)^3}=\sum_{n\geq 1}\frac{n(n+1)}{2}\,t^{2n} $$ and $$ \int_{0}^{1}t^{2n}(-\log^3 t)\,dt =\frac{6}{(2n+1)^4}$$ lead to $$ \int_{0}^{+\infty}\left(\frac{x}{\sinh x}\right)^3\,dx = 24\sum_{n\geq 1}\frac{n(n+1)}{(2n+1)^4}=\color{red}{\frac{\pi^2(12-\pi^2)}{16}}. $$


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