Definition:Inverse Hyperbolic Sine/Complex

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Definition

Definition 1

The inverse hyperbolic sine is a multifunction defined as:

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

where $\map \sinh w$ is the hyperbolic sine function.


Definition 2

The inverse hyperbolic sine is a multifunction defined as:

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

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 sine function is defined as:

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

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 sine is also known as the area hyperbolic sine, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

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


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


Notation

In general, the inverse hyperbolic functions are multifunctions.

When used in their multifunction form, the notation of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the inverse of a hyperbolic function $\operatorname h$ is $\operatorname h^{-1}$.


When the area hyperbolic function is specifically required, the following prefixes are used:

$\text {ar}$ for the real inverse hyperbolic functions
$\text {Ar}$ for the complex inverse hyperbolic functions

where $\text{ar}$ is an abbreviation for area.


The prefix $\text {arc}$, borrowing from the notation for the inverse trigonometric functions, is often seen to mean the same thing, but this is erroneous.

$\text{ar}$ is an abbreviation for area hyperbolic function, which is another name for an inverse hyperbolic function.


The forms $\sinh^{-1}$ and $\Sinh^{-1}$, and so on, are often seen in the literature for the area hyperbolic form, that is, as a single-value function.

This can cause confusion, for the following reasons:

$(1): \quad \sinh^{-1}$, for example, can be conflated with $\dfrac 1 {\sinh}$, as it conflicts with the similar notation $\sinh^2 x$ which means $\paren {\sinh x}^2$, and so on.
$(2): \quad h^{-1}$ is strictly interpreted as the inverse of a mapping, and for such hyperbolic functions that are not bijective, such inverses are not actually mappings.


Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ uses the notation $\text{ar-}$ or $\text{Ar-}$ for the area hyperbolic functions in preference to all others.


Also see

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


Sources