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I'm studying presently studying the calculus of residues, and I'm stuck in this problem.

How do I evaluate the integral, $$ \int_{0}^{\infty} \frac{dx}{(x^2 + a^2) \cosh \pi x} $$?

My first thought about the contour for this integral was a rectangular contour, with a small indentation at $z=ai$ with the pole $ z = \frac{i}{2} $ inside the contour. But I'm unable to get the correct answer. Any help would be highly appreciated.

Vaibhav C M
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    Include what you tried. Otherwise question likely gets closed. – coffeemath Jul 10 '22 at 06:28
  • Mind that both factors in the denominator introduce poles. $z^2+a^2=0$ and $\cosh{(\pi z)}=0$. The latter has infinetly many zeros (that are poles of the function). – Luciano Jul 10 '22 at 14:57
  • The detailed solution of your problem via CI; just take the derivative with respect to $a$ - https://math.stackexchange.com/questions/4323748/trouble-understanding-blagouchines-extensions-to-the-malmsten-integral/4324160#4324160 – Svyatoslav Jul 11 '22 at 13:12

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