逆双曲線関数
}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{(2n+1)}},\qquad \left|x\right|>1\end{aligned}}}
arsech x = arcosh  1 x = ln  2 x − ( ( 1 2 ) x 2 2 + ( 1 ⋅ 3 2 ⋅ 4 ) x 4 4 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) x 6 6 + ⋯ ) = ln  2 x − ∑ n = 1 ∞ ( ( 2 n ) ! 2 2 n ( n ! ) 2 ) x 2 n 2 n , 0 < x ≤ 1 {\displaystyle {\begin{aligned}\operatorname {arsech} \,x=\operatorname {arcosh} {\frac {1}{x}}&=\ln {\frac {2}{x}}-\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 {\frac {2}{x}}-\sum _{n=1}^{\infty }\left({\frac {(2n)!
( − 1 ) n x 2 n + 1 ( 1 − x 2 ) n + 1 , | x | < 1 2 {\displaystyle \operatorname {artanh} \,x=\sum _{n=0}^{\infty }{\frac {2^{2n}(n!)^{2}}{(2n+1)!}}{\frac {(-1)^{n}x^{2n+1}}{(1-x^{2})^{n+1}}},\qquad \left|x\right|<{\frac {1}{\sqrt {2}}}}
( arsinh x ) 2 = ∑ n = 0 ∞ 2 2 n + 1 ( n ! ) 2 ( 2 n + 2 ) ! ( − 1 ) n x 2 n + 2 , | x | < 1 {\displaystyle (\operatorname {arsinh} \,x)^{2}=\sum _{n=0}^{\infty }{\frac {2^{2n+1}(n!)^{2}}{(2n+2)!}}(-1)^{n}x^{2n+2},\qquad \left|x\right|<1}
arsinh x に対する漸近展開は次の式で与えられる。
arsinh x = ln  2 x + ∑ n = 1 ∞ ( − 1 ) n − 1 ( 2 n − 1 ) ! ! 2 n ( 2 n ) ! ! 1 x 2 n {\displaystyle \operatorname {arsinh} \,x=\ln 2x+\sum \limits _{n=1}^{\infty }{\left({-1}\right)^{n-1}{\frac {\left({2n-1}\right)!!}{2n\left({2n}\right)!!}}}{\frac {1}{x^{2n}}}}