# Definition:Inverse Hyperbolic Secant/Real/Principal Branch

## Definition

Let $S$ denote the subset of the real numbers:

- $S := \hointl 0 1$

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 $\map \arsech x \ge 0$.

## Graph of Inverse Hyperbolic Secant

The graph of the real inverse hyperbolic secant function appears as:

## 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 $\map \arsech x > 0$.