Definition:Inverse Hyperbolic Tangent/Complex/Definition 2

Definition
Let $S$ be the subset of the complex plane:
 * $S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$

The inverse hyperbolic tangent is a multifunction defined on $S$ as:


 * $\forall z \in S: \map {\tanh^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

where $\ln$ denotes the complex natural logarithm considered as a multifunction.

Also see

 * Equivalence of Definitions of Complex Inverse Hyperbolic Tangent


 * Definition:Inverse Tangent/Complex/Definition 2