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 see

 * Equivalence of Definitions of Real Area Hyperbolic Cosine