# Definition:Area Hyperbolic Cotangent

## Definition

### Complex

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

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

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

### Real

The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.