逆双曲線関数
2 2 n ( n ! ) 2 ) x − 2 n ( 2 n ) , x > 1 {\displaystyle {\begin{aligned}\operatorname {arcosh} \,x&=\ln 2x-\left(\left({\frac {1}{2}}\right){\frac {x^{-2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-6}}{6}}+\cdots \right)\\&=\ln 2x-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{(2n)}},\qquad x>1\end{aligned}}}
artanh x = x + x 3 3 + x 5 5 + x 7 7 + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) , | x | < 1 {\displaystyle {\begin{aligned}\operatorname {artanh} \,x&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)}},\qquad \left|x\right|<1\end{aligned}}}