\begin{definition}[Definition:Inverse Hyperbolic Cosecant/Complex/Definition 2]
The '''inverse hyperbolic cosecant''' is a multifunction defined as:
:$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$
where:
:$\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$
:$\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$
:$\ln$ denotes the complex natural logarithm considered as a multifunction.
\end{definition}