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