\begin{definition}[Definition:Inverse Hyperbolic Cosecant/Complex/Principal Branch]
The principal branch of the complex inverse hyperbolic cosecant function is defined as:
:$\forall z \in \C_{\ne 0}: \map \Arcsch z := \map \Ln {\dfrac {1 + \sqrt {z^2 + 1} } z}$
where:
:$\Ln$ denotes the principal branch of the complex natural logarithm
:$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.
\end{definition}