\begin{definition}[Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2]
Let $S$ be the subset of the complex plane:
:$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$
The '''inverse hyperbolic cotangent''' is a multifunction defined  on $S$ as:
:$\forall z \in S: \map {\coth^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {z + 1} {z - 1} } + k \pi i: k \in \Z}$
where $\ln$ denotes the complex natural logarithm considered as a  multifunction.
\end{definition}