Definition:Inverse Hyperbolic Tangent/Complex/Definition 1

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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:

$\forall z \in S: \map {\tanh^{-1} } z := \set {w \in \C: z = \map \tanh w}$

where $\map \tanh w$ is the hyperbolic tangent function.

Also known as

The inverse hyperbolic tangent is also known as the area hyperbolic tangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arctangent, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic tangent.

Also see