ζ(2)のような総和

\begin{align*} \sum_{k=0}^{\infty}\frac{1}{k^{2}+1} & =\sum_{k=0}^{\infty}\frac{1}{\left(k-i\right)\left(k+i\right)}\\ & =\frac{1}{2i}\sum_{k=0}^{\infty}\left(\frac{1}{k-i}-\frac{1}{k+i}\right)\\ \\ & =\frac{1}{2i}\sum_{k=0}^{\infty}\left\{ \left(\frac{1}{k-i}-\frac{1}{k+1}\right)-\left(\frac{1}{k+i}-\frac{1}{k+1}\right)\right\} \\ & =\frac{1}{2i}\left\{ \left(-\gamma-\psi\left(-i\right)\right)-\left(-\gamma-\psi\left(i\right)\right)\right\} \cmt{\because\psi\left(z\right)=-\gamma-\sum_{k=0}^{\infty}\left(\frac{1}{z+k}-\frac{1}{k+1}\right)}\\ & =\frac{1}{2i}\left(\psi\left(i\right)-\psi\left(-i\right)\right)\\ & =\frac{1}{2i}\left(\psi\left(i\right)-\psi\left(1-i\right)-\frac{1}{i}\right)\cmt{\because\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}}\\ & =-\frac{1}{2i}\left(\psi\left(1-i\right)-\psi\left(i\right)-i\right)\\ & =-\frac{1}{2i}\left(\pi\tan^{-1}\left(\pi i\right)-i\right)\cmt{\because\psi\left(1-z\right)-\psi\left(z\right)=\pi\tan^{-1}\left(\pi z\right)}\\ & =-\frac{1}{2i}\left(\pi i^{-1}\tanh^{-1}\left(\pi\right)-i\right)\\ & =\frac{\pi}{2}\tanh^{-1}\left(\pi\right)+\frac{1}{2} \end{align*}