逆三角関数
arccsc  z = arcsin  1 z = ∑ n = 0 ∞ ( 2 n n ) 4 n ( 2 n + 1 ) z − ( 2 n + 1 ) = z − 1 + ( 1 2 ) z − 3 3 + ( 1 ⋅ 3 2 ⋅ 4 ) z − 5 5 + ⋯ ; | z | ≥ 1 {\displaystyle {\begin{aligned}\operatorname {arccsc} z&=\arcsin {\frac {1}{z}}\\&=\textstyle \sum \limits _{n=0}^{\infty }{\dfrac {\binom {2n}{n}}{4^{n}(2n+1)}}z^{-(2n+1)}\\&=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\dotsb ;\quad |z|\geq 1\end{aligned}}}
レオンハルト・オイラー (Leonhard Euler) は逆正接関数のより効率的な級数を見つけた:
arctan  z = z 1 + z 2 ∑ n = 0 ∞ ∏ k = 1 n 2 k z 2 ( 2 k + 1 ) ( 1 + z 2 ) .