# Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1

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## Definition

Let $S$ be the subset of the complex plane:

- $S = \C \setminus \set {-1 + 0 i, 1 + 0 i}$

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.

## 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**.

## Also see

## Sources

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