\begin{definition}[Definition:Inverse Cosecant/Complex/Definition 2]
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
The '''inverse cosecant of $z$''' is the multifunction defined as:
:$\csc^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({\dfrac {i + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$
where:
: $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$
: $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$
: $\ln$ denotes the complex natural logarithm considered as a  multifunction.
\end{definition}