# Definition:Inverse Hyperbolic Cotangent/Complex

## 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 cotangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\coth^{-1} } z := \set {w \in \C: z = \map \coth w}$

where $\map \coth w$ is the hyperbolic cotangent function.

### Definition 2

The inverse hyperbolic cotangent is a multifunction defined on $S$ as:

$\forall z \in S: \map {\coth^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {z + 1} {z - 1} } + k \pi i: k \in \Z}$

where $\ln$ denotes the complex natural logarithm considered as a multifunction.

## Principal Branch

The principal branch of the complex inverse hyperbolic 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.

## Also known as

The principal branch of the inverse hyperbolic cotangent is also known as the area hyperbolic cotangent, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arccotangent, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cotangent.

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

## Also see

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