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For $a>0$, prove:

$$ \sum_{n=1}^{\infty}\frac{1}{n^2+a^2} = \left( \frac{\pi}{a}\cdot\frac{\exp(\pi a) + \exp(-\pi a)}{\exp(\pi a) - \exp(-\pi a)} - \frac{1}{a^2}\right) $$

One idea I have is to expand RHS and manually check that series expansions match, but that seems troublesome. There is a hint to use residue theorem, but I don't see how to translate this setting to integrals? Perhaps some Riemann sum on the left hand side?

jrg
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    See https://math.stackexchange.com/questions/3282692/computing-sum-frac1n2n2a2-using-the-residue-theorem. The original question is not quite the same as yours but the technique is the same (remove the $z^2$ from the denominator in the integrand and then follow the same ideas). Also some of the answers address exactly your problem. Sorry I don't have time to write a complete answer but maybe somebody else can. – David Feb 05 '25 at 23:53

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