Formelsammlung Mathematik: Unendliche Reihen: Reihen aus Partialbruchentwicklungen

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$\sum _{k\in \mathbb {Z} }{\frac {\frac {\alpha ^{2}}{2}}{k^{4}+\left({\frac {\alpha ^{2}}{2}}\right)^{2}}}={\frac {\pi }{\alpha }}\,{\frac {\sinh \alpha \pi +\sin \alpha \pi }{\cosh \alpha \pi -\cos \alpha \pi }}$

$\sum _{k\in \mathbb {Z} }{\frac {k^{2}}{k^{4}+\left({\frac {\alpha ^{2}}{2}}\right)^{2}}}={\frac {\pi }{\alpha }}\,{\frac {\sinh \alpha \pi -\sin \alpha \pi }{\cosh \alpha \pi -\cos \alpha \pi }}$