テイラー展開
}}x^{2n-1}\quad {\mbox{ for }}|x|<{\frac {\pi }{2}}}
sinh − 1  x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 for | x | < 1 {\displaystyle \sinh ^{-1}x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}\quad {\mbox{ for }}|x|<1}
tanh − 1  x = ∑ n = 0 ∞ 1 2 n + 1 x 2 n + 1 for | x | < 1 {\displaystyle \tanh ^{-1}x=\sum _{n=0}^{\infty }{\frac {1}{2n+1}}x^{2n+1}\quad {\mbox{ for }}|x|<1}