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1

${\displaystyle \sum _{k=1}^{\infty }\arctan \left({\frac {2a^{2}}{k^{2}}}\right)=\lfloor a\rfloor \cdot \pi +{\frac {\pi }{4}}-\arctan \left({\frac {\tanh \pi a}{\tan \pi a}}\right)}$, wobei ${\displaystyle \arctan \left({\frac {\tanh \pi a}{\tan \pi a}}\right)}$ für ${\displaystyle a\in \mathbb {N} }$ als ${\displaystyle {\frac {\pi }{2}}}$ zu interpretieren ist.

2

${\displaystyle \sum _{k=1}^{\infty }\arctan \left({\frac {2xy}{k^{2}-x^{2}+y^{2}}}\right)=\arctan \left({\frac {y}{x}}\right)-\arctan \left({\frac {\tanh \pi y}{\tan \pi x}}\right)\qquad ({\text{mod}}\,\,2\pi )}$