Definition:Inverse Hyperbolic Cosine/Real/Definition 2
Definition
Let $S$ denote the subset of the real numbers:
- $S = \set {x \in \R: x \ge 1}$
The inverse hyperbolic cosine $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:
- $\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x \pm \sqrt {x^2 - 1} }$
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.
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
Sources
- 1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Definitions of Inverse Hyperbolic Functions
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse hyperbolic function