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

$\blacksquare$