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Does there exist an explicit formula for the sum of the series $$ \sum_{n=1}^\infty \frac{1}{n^2-z^2}? $$

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    Yes (the question is often asked here). See for example [this](http://math.stackexchange.com/questions/110494/possibility-to-simplify-sum-limits-k-infty-infty-frac-left/110495#110495).2012-12-19
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    That's very nice, many thanks! (The answer is accepted, so I hope it contains a correct proof.)2012-12-19
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    Thanks, of course Eric's answer is correct? Another [link](http://math.stackexchange.com/questions/3525/a-series-expansion-of-cot-pi-z) with links...2012-12-19

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Equation $(18)$ on this page states that $$\pi \cot(\pi z)=\frac{1}{z}+2z\sum_{n=1}^\infty \frac{1}{z^2-n^2}.$$

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This answer handles precisely this question by showing that the Cauchy Principal Value of $$ \sum_{k\in\mathbb{Z}}\frac1{k+z}=\pi\cot(\pi z) $$ and derives an explicit value for $\displaystyle\sum_{k=1}^\infty\frac{1}{k^2-z^2}$ in $(9)$.

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Yes, there does. Of course the sum is not defined if $z$ is a nonzero integer.