逆双曲線関数
sinh  ( arcosh  x ) = x 2 − 1 for | x | > 1 sinh  ( artanh  x ) = x 1 − x 2 for | x | < 1 cosh  ( arsinh  x ) = 1 + x 2 cosh  ( artanh  x ) = 1 1 − x 2 for | x | < 1 tanh  ( arsinh  x ) = x 1 + x 2 tanh  ( arcosh  x ) = x 2 − 1 x for | x | > 1 {\displaystyle {\begin{aligned}\sinh(\operatorname {arcosh} x)&={\sqrt {x^{2}-1}}&{\text{for }}|x|>1\\\sinh(\operatorname {artanh} x)&={\frac {x}{\sqrt {1-x^{2}}}}&{\text{for }}|x|<1\\\cosh(\operatorname {arsinh} x)&={\sqrt {1+x^{2}}}\\\cosh(\operatorname {artanh} x)&={\frac {1}{\sqrt {1-x^{2}}}}&{\text{for }}|x|<1\\\tanh(\operatorname {arsinh} x)&={\frac {x}{\sqrt {1+x^{2}}}}\\\tanh(\operatorname {arcosh} x)&={\frac {\sqrt {x^{2}-1}}{x}}&{\text{for }}|x|>1\end{aligned}}}