Definition:Inverse Hyperbolic Tangent/Complex

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Definition

Let $S$ be the subset of the complex plane:

$S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$

Definition 1

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.


Definition 2

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.


Principal Branch

The principal branch of the complex inverse hyperbolic tangent function is defined as:

$\forall z \in \C: \map \Artanh z := \dfrac 1 2 \, \map \Ln {\dfrac {1 + z} {1 - z} }$

where $\Ln$ denotes the principal branch of the complex natural logarithm.


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.


In the complex plane, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term area hyperbolic tangent strictly for the principal branch.


Also see

  • Results about the inverse hyperbolic tangent can be found here.


Sources

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