# Inverse Hyperbolic Sine Logarithmic Formulation

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## Theorem

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

- $\operatorname{arsinh} z = \ln \left({z + \sqrt{z^2 + 1}}\right)$

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

## Proof

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

\(\displaystyle \iff \ \ \) | \(\displaystyle z\) | \(=\) | \(\displaystyle \frac{e^{\operatorname{arsinh} z} - e^{-\operatorname{arsinh} z} } 2\) | Definition of inverse hyperbolic sine | |||||||||

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

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

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

\(\displaystyle \iff \ \ \) | \(\displaystyle \operatorname{arsinh} z\) | \(=\) | \(\displaystyle \ln\left({z + \sqrt{z^2+1} }\right)\) |

$\blacksquare$