# Inverse Hyperbolic Sine Logarithmic Formulation

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

$\blacksquare$