Definition:Inverse Hyperbolic Secant/Real/Definition 1
Definition
Let $S$ denote the half-open real interval:
- $S := \hointl 0 1$
The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:
- $\forall x \in S: \map {\sech^{-1} } x := y \in \R_{\ge 0}: x = \map \sech y$
where $\map \sech y$ denotes the hyperbolic secant function.
Hence for $0 < x < 1$, $\map {\sech^{-1} } x$ has $2$ values.
For $x > 0$ and $x > 1$, $\map {\sech^{-1} } x$ is not defined.
Principal Value
The principal branch of the real inverse hyperbolic secant function is defined as:
- $\forall x \in S: \map \arsech x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$
where:
- $\ln$ denotes the natural logarithm of a (strictly positive) real number.
- $\sqrt {1 - x^2}$ specifically denotes the positive square root of $x^2 - 1$
That is, where $\map \arsech x \ge 0$.
Also known as
The principal branch of the inverse hyperbolic secant is also known as the area hyperbolic secant, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Some sources refer to it as hyperbolic arcsecant, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic secant.
In the real domain, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term area hyperbolic secant strictly for the principal branch, that is, for $\map \arsech x > 0$.