Definition:Inverse Hyperbolic Cosine/Complex

Definition

Definition 1

The inverse hyperbolic cosine is a multifunction defined as:

$\forall z \in \C: \map {\cosh^{-1} } z := \set {w \in \C: z = \map \cosh w}$

where $\map \cosh w$ is the hyperbolic cosine function.

Definition 2

The inverse hyperbolic cosine is a multifunction defined as:

$\forall z \in \C: \map {\cosh^{-1} } z := \set {\map \ln {z + \sqrt {\size {z^2 - 1} } e^{\paren {i / 2} \map \arg {z^2 - 1} } } + 2 k \pi i: k \in \Z}$

where:

$\sqrt {\size {z^2 - 1} }$ denotes the positive square root of the complex modulus of $z^2 - 1$
$\map \arg {z^2 - 1}$ denotes the argument of $z^2 - 1$
$\ln$ denotes the complex natural logarithm considered as a multifunction.

Principal Branch

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 $z^2 - 1$.

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 complex plane, $\mathsf{Pr} \infty \mathsf{fWiki}$ reserves the term area hyperbolic cosine strictly for the principal branch.

Also see

• Results about the inverse hyperbolic cosine can be found here.