\begin{definition}[Definition:Inverse Secant/Complex/Definition 2]
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
The '''inverse secant of $z$''' is the multifunction defined as:
:$\sec^{-1} z := \set {\dfrac 1 i \map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi: k \in \Z}$
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.
\end{definition}