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