arcsinh z = ln ( 2 z ) + 1 2 1 2 z 2 - 1 ⋅ 3 2 ⋅ 4 1 4 z 4 + 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 1 6 z 6 - ⋯ \operatorname{arcsinh}z=\ln\left(2z\right)+\frac{1}{2}\frac{1}{2z^{2}}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^{6}}-\cdots