# Definition:Inverse Hyperbolic Tangent/Real

## Definition

Let $S$ denote the open real interval:

- $S := \openint {-1} 1$

### Definition 1

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

- $\forall x \in S: \map \artanh x := y \in \R: x = \map \tanh y$

where $\map \tanh y$ denotes the hyperbolic tangent function.

### Definition 2

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

- $\forall x \in S: \map \artanh x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.

## Graph of Inverse Hyperbolic Tangent

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

## Also known as

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

Note that as the real hyperbolic tangent $\tanh$ 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 arctangent**, but this is strictly a misnomer, as there is nothing **arc** related about an **inverse hyperbolic tangent**.

## Also see

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

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

## Sources

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