# 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

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