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 function defined on $S$ as:


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

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

Also known as
The inverse hyperbolic cosine function is also known as the area hyperbolic cosine function, as they can be used for evaluating areas of regions bounded by hyperbolas.

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

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Cosine