\begin{definition}[Definition:Inverse Sine/Complex/Arcsine]
The principal branch of the complex inverse sine function is defined as:
:$\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$
where:
: $\Ln$ denotes the principal branch of the complex natural logarithm
: $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
\end{definition}