三角関数の部分分数展開
π sin  π z = lim N → ∞ ∑ n = − N N ( − 1 ) n z + n = 1 z + ∑ n = 1 ∞ ( − 1 ) n 2 z z 2 − n 2 {\displaystyle {\frac {\pi }{\sin {\pi }z}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}2z}{z^{2}-n^{2}}}}
π cos  π z = lim N → ∞ ∑ n = − N N ( − 1 ) n z + 1 2 + n = − ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) z 2 − ( n + 1 2 ) 2 {\displaystyle {\frac {\pi }{\cos {{\pi }z}}}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+{\frac {1}{2}}+n}}=-\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)}{z^{2}-\left(n+{\frac {1}{2}}\right)^{2}}}}