# Definition:Area Hyperbolic Tangent

## 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.