Definition:Inverse Hyperbolic Cosine/Real

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Definition

Let $S$ denote the subset of the real numbers:

$S = \set {x \in \R: x \ge 1}$


Definition 1

The inverse hyperbolic cosine $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:

$\forall x \in S: \map {\cosh^{-1} } x := \set {y \in \R: x = \map \cosh y}$

where $\map \cosh y$ denotes the hyperbolic cosine function.


Definition 2

The inverse hyperbolic cosine $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:

$\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x \pm \sqrt {x^2 - 1} }$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number.
$\sqrt {x^2 - 1}$ denotes the square root of $x^2 - 1$


Hence for $x > 1$, $\map {\cosh^{-1} } x$ has $2$ values.

For $x < 1$, $\map {\cosh^{-1} } x$ is not defined.


Graph of Inverse Hyperbolic Cosine

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


Arcosh.png


Principal Branch

The principal branch of the real inverse hyperbolic cosine function is defined as:

$\forall x \in S: \map \arcosh x := \map \ln {x + \sqrt {x^2 - 1} }$

where:

$\ln$ denotes the natural logarithm of a (strictly positive) real number.
$\sqrt {x^2 - 1}$ specifically denotes the positive square root of $x^2 - 1$

That is, where $\map \arcosh x \ge 0$.


Also known as

The principal branch of the inverse hyperbolic cosine is known as the area hyperbolic cosine, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arccosine, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cosine.


In the real domain, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term area hyperbolic cosine strictly for the principal branch, that is, for $\map \arcosh x > 0$.


Also see

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


Sources