# Definition:Inverse Hyperbolic Tangent/Complex/Definition 1

Jump to navigation
Jump to search

## Definition

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

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

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

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

where $\map \tanh w$ is the hyperbolic tangent function.

## Also known as

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

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

## Also see

## Sources

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