# Definition:Inverse Hyperbolic Secant/Real/Definition 2

## 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:

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

where:

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

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 $\map \arsech x > 0$.

## 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