# Inverse Hyperbolic Sine Logarithmic Formulation

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## Theorem

For any complex number $z \in \C$:

- $\operatorname {arsinh} z = \map \ln {z + \sqrt {z^2 + 1} }$

where $\operatorname {arsinh} z$ is the inverse hyperbolic sine.

## Proof

\(\displaystyle z\) | \(=\) | \(\displaystyle \sinh \operatorname {arsinh} z\) | |||||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle z\) | \(=\) | \(\displaystyle \frac {e^{\operatorname{arsinh} z} - e^{-\operatorname{arsinh} z} } 2\) | Definition of Inverse Hyperbolic Sine | |||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle 2 z e^{\operatorname{arsinh} z}\) | \(=\) | \(\displaystyle e^{2 \operatorname{arsinh} z} - 1\) | Multiplication by $2 e^{\operatorname{arsinh} z}$ | |||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle 0\) | \(=\) | \(\displaystyle e^{2 \operatorname{arsinh} z} - 2 z e^{\operatorname{arsinh} z} - 1\) | ||||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle e^{\operatorname{arsinh} z}\) | \(=\) | \(\displaystyle z + \sqrt {z^2 + 1}\) | Quadratic Formula, $e^z > 0, \sqrt {z^2 + 1} > z$ | |||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle \operatorname{arsinh} z\) | \(=\) | \(\displaystyle \map \ln {z + \sqrt {z^2 + 1 } }\) |

$\blacksquare$