三角関数の部分分数展開
lim z → 0 d d z ( 1 z + ∑ n = 1 ∞ 2 z z 2 − n 2 ) = − 1 z 2 + ∑ n = 1 ∞ 2 z 2 − n 2 − ∑ n = 1 ∞ 4 z 2 ( z 2 − n 2 ) 2 = − 1 z 2 − ∑ n = 1 ∞ 2 n 2 + O ( z 2 ) {\displaystyle {\begin{aligned}\lim _{z\to 0}{\frac {d}{dz}}\left({\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {2z}{z^{2}-n^{2}}}\right)&=-{\frac {1}{z^{2}}}+\sum _{n=1}^{\infty }{\frac {2}{z^{2}-n^{2}}}-\sum _{n=1}^{\infty }{\frac {4z^{2}}{(z^{2}-n^{2})^{2}}}\\&=-{\frac {1}{z^{2}}}-\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}+O(z^{2})\\\end{aligned}}}