超幾何関数
超幾何関数は多くの初等関数や特殊関数を包含する。
対数関数、逆三角関数
log  ( 1 + z ) = z ∑ n = 0 ∞ ( − 1 ) n n + 1 z n = z ⋅ 2 F 1 [ 1 , 1 2 ; − z ] log  ( 1 + z 1 − z ) = 2 z ∑ n = 0 ∞ 1 2 n + 1 z 2 n = 2 z ⋅ 2 F 1 [ 1 2 , 1 3 2 ; z 2 ] sin − 1  z = z ∑ n = 0 ∞ ( 2 n − 1 ) ! ! ( 2 n ) ! ! ( 2 n + 1 ) z 2 n = z ⋅ 2 F 1 [ 1 2 , 1 2 3 2 ; z 2 ] tan − 1  z = z ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 z 2 n = z ⋅ 2 F 1 [ 1 2 , 1 3 2 ; − z 2 ] {\displaystyle {\begin{aligned}\log(1+z)&=z\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n+1}}z^{n}=z\cdot {_{2}F_{1}}\left[{\begin{matrix}1,1\\2\end{matrix}};-z\right]\\\log \left({\frac {1+z}{1-z}}\right)&=2z\sum _{n=0}^{\infty }{\frac {1}{2n+1}}z^{2n}=2z\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}},1\\{\frac {3}{2}}\end{matrix}};z^{2}\right]\\\sin ^{-1}z&=z\sum _{n=0}^{\infty }{\frac {(2n-1)!