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

$\blacksquare$