# Definition:Inverse Hyperbolic Cotangent/Real

## Definition

Let $S$ denote the union of the unbounded open real intervals:

- $S := \openint \gets {-1} \cup \openint 1 \to$

### Definition 1

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

- $\forall x \in S: \arcoth x := y \in \R: x = \coth y$

where $\coth y$ denotes the hyperbolic cotangent function.

### Definition 2

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.

## Graph of Inverse Hyperbolic Cotangent

The graph of the real inverse hyperbolic cotangent function appears as:

## Also denoted as

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

- $\arcoth$
- $\operatorname {acoth}$
- $\operatorname {actnh}$

## Also known as

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

Note that as the real hyperbolic cotangent $\coth$ is injective, its inverse is properly a function on its domain.

Hence there is no need to make a separate distinction between branches in the same way as for real inverse hyperbolic cosine and real inverse hyperbolic secant.

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**.

## Also see

- Definition:Real Inverse Hyperbolic Sine
- Definition:Real Inverse Hyperbolic Cosine
- Definition:Real Inverse Hyperbolic Tangent
- Definition:Real Inverse Hyperbolic Secant
- Definition:Real Inverse Hyperbolic Cosecant

- Results about
**the inverse hyperbolic cotangent**can be found**here**.

## Sources

- Weisstein, Eric W. "Inverse Hyperbolic Cotangent." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicCotangent.html