# Definition:Inverse Hyperbolic Tangent/Real

## Definition

Let $S$ denote the open real interval:

$S := \left({-1 \,.\,.\, 1}\right)$

### Definition 1

The inverse hyperbolic tangent $\tanh^{-1}: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \tanh^{-1} \left({x}\right) := y \in \R: x = \tanh \left({y}\right)$

where $\tanh \left({y}\right)$ denotes the hyperbolic tangent function.

### Definition 2

The inverse hyperbolic tangent $\tanh^{-1}: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \tanh^{-1} \left({x}\right) := \dfrac 1 2 \operatorname{ln} \left({\dfrac {1 + x} {1 - x} }\right)$

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

## Also known as

The inverse hyperbolic tangent function is also known as the hyperbolic arctangent function.

## Also see

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