# 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.

## Symbol

The symbol used to denote the area hyperbolic cotangent function is variously seen as follows:

arcoth

The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the area hyperbolic cotangent function is $\arcoth$.

acoth

A variant symbol used to denote the area hyperbolic cotangent function is $\operatorname {acoth}$.

actnh

A variant symbol used to denote the area hyperbolic cotangent function is $\operatorname {actnh}$.