# Definition:Inverse Hyperbolic Cosine/Real/Principal Branch

## Definition

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

- $S = \set {x \in \R: x \ge 1}$

The principal branch of the real inverse hyperbolic cosine function is defined as:

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

where:

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

That is, where $\map \arcosh x \ge 0$.

## Graph of Inverse Hyperbolic Cosine

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

## Also denoted as

The symbol used to denote the **area hyperbolic cosine function** is variously seen as:

- $\arcosh$
- $\operatorname {acosh}$

## Also known as

The principal branch of the **inverse hyperbolic cosine** is known as the **area hyperbolic cosine**, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

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

In the real domain, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term **area hyperbolic cosine** strictly for the principal branch, that is, for $\map \arcosh x > 0$.