三角関数の部分分数展開
cot  θ + tan  θ = cos 2  θ + sin 2  θ sin  θ cos  θ = 2 sin  2 θ {\displaystyle \cot \theta +\tan \theta ={\frac {\cos ^{2}\theta +\sin ^{2}\theta }{\sin \theta \cos \theta }}={\frac {2}{\sin 2\theta }}}
π sin  π z = 1 2 cot  θ 2 + 1 2 tan  θ 2 = lim N → ∞ 1 2 ∑ n = − N N 2 z + 2 n − 1 2 ∑ n = − N N 2 z + 2 n + 1 = lim N → ∞ ∑ n = − N N ( − 1 ) n z + n = 1 z + ∑ n = 1 ∞ ( − 1 ) n 2 z z 2 − n 2 {\displaystyle {\begin{aligned}{\frac {\pi }{\sin {\pi }z}}&={\frac {1}{2}}\cot {\frac {\theta }{2}}+{\frac {1}{2}}\tan {\frac {\theta }{2}}\\&=\lim _{N\to \infty }{\frac {1}{2}}\sum _{n=-N}^{N}{\frac {2}{z+2n}}-{\frac {1}{2}}\sum _{n=-N}^{N}{\frac {2}{z+2n+1}}\\&=\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}}}\\\end{aligned}}}
数学において、三角関数は以下のように部分分数に展開される。
π cot  π z = lim N → ∞ ∑ n = − N N 1 z + n = 1 z + ∑ n = 1 ∞ 2 z z 2 − n 2 {\displaystyle \pi \cot {{\pi }z}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {2z}{z^{2}-n^{2}}}}
π tan  π z = lim N → ∞ ∑ n = − N N − 1 z + 1 2 + n = − ∑ n = 0 ∞ 2 z z 2 − ( n + 1 2 ) 2 {\displaystyle \pi \tan {{\pi }z}=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {-1}{z+\textstyle {\frac {1}{2}}+n}}=-\sum _{n=0}^{\infty }{\frac {2z}{z^{2}-\left(n+\textstyle {\frac {1}{2}}\right)^{2}}}}