# Definition:Inverse Hyperbolic Cosine/Principal Branch

## Definition

### Complex Plane

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

$\forall z \in \C: \map \Arcosh z := \map \Ln {z + \sqrt {z^2 - 1} }$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.

### Real Numbers

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 $\map \arcosh x \ge 0$.