Definition:Inverse Hyperbolic Tangent

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Definition

Complex Plane

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

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

Definition 1

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.


Definition 2

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

$\forall z \in S: \map {\tanh^{-1} } z := \set {\dfrac 1 2 \map \ln {\dfrac {1 + z} {1 - z} } + k \pi i: k \in \Z}$

where $\ln$ denotes the complex natural logarithm considered as a multifunction.


Real Numbers

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.


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.


Examples

Inverse Hyperbolic Tangent of $i$

$\tanh^{-1} \paren i = \dfrac {\paren {4 k + 1} \pi i} 4$


Also see

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