\begin{definition}[Definition:Inverse Sine/Complex/Definition 2]
Let $z \in \C$ be a complex number.
The '''inverse sine of $z$''' is the multifunction defined as:
:$\sin^{-1} \paren z := \set {\dfrac 1 i \ln \paren {i z + \sqrt {\cmod {1 - z^2} } \exp \paren {\dfrac i 2 \arg \paren {1 - z^2} } } + 2 k \pi: k \in \Z}$
where:
: $\sqrt {\cmod {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
: $\arg \paren {1 - z^2}$ denotes the argument of $1 - z^2$
: $\ln$ is the complex natural logarithm considered as a  multifunction.
\end{definition}