Definition:Inverse Hyperbolic Tangent

From ProofWiki
Jump to: navigation, search

Definition

Complex Plane

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

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

Definition 1

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

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

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


Definition 2

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

$\forall z \in S: \tanh^{-1} \left({z}\right) := \left\{{\dfrac 1 2 \ln \left({\dfrac {1 + z} {1 - z} }\right) + k \pi i: k \in \Z}\right\}$

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


Hyperbolic Arctangent

The principal branch of the complex inverse hyperbolic tangent function is defined as:

$\forall z \in \C: \map {\Tanh^{-1} } z := \dfrac 1 2 \, \map \Ln {\dfrac {1 + z} {1 - z} }$

where $\Ln$ denotes the principal branch of the complex natural logarithm.


Real Numbers

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.


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.