# Inverse Hyperbolic Sine Logarithmic Formulation

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

$\blacksquare$