Definition:Area Hyperbolic Tangent
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Definition
Complex
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.
Real
The inverse hyperbolic tangent $\artanh: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \map \artanh x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$
where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Symbol
The symbol used to denote the area hyperbolic tangent function is variously seen as follows:
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic tangent function is $\artanh$.
A variant symbol used to denote the area hyperbolic tangent function is $\operatorname {atanh}$.