Definition:Complex Inverse Hyperbolic Function

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Definition

Let $h: \C \to \C$ be one of the hyperbolic functions on the set of complex numbers.

The inverse hyperbolic function $h^{-1} \subseteq \C \times \C$ is actually a multifunction, as in general for a given $y \in \C$ there is more than one $x \in \C$ such that $y = \map h x$.

As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $h^{-1}$ to be single-valued.


There are six basic hyperbolic functions, so each of these has its inverse functions:


Inverse Hyperbolic Sine

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.


Inverse Hyperbolic Cosine

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.


Inverse Hyperbolic Tangent

The inverse hyperbolic tangent is a multifunction defined on $S$ as:

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

where $\map \tanh w$ is the hyperbolic tangent function.


Inverse Hyperbolic Cotangent

The inverse hyperbolic cotangent is a multifunction defined on $S$ as:

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

where $\map \coth w$ is the hyperbolic cotangent function.


Inverse Hyperbolic Secant

The inverse hyperbolic secant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {w \in \C: z = \map \sech w}$

where $\map \sech w$ is the hyperbolic secant function.


Inverse Hyperbolic Cosecant

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$

where $\map \csch w$ is the hyperbolic cosecant function.


Also known as

The inverse hyperbolic functions are also known as the area hyperbolic functions, as they can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the term area hyperbolic function is specifically reserved for the principal branch of those inverse hyperbolic functions which are multifunctions.

Some sources refer to them as hyperbolic arc functions, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic function.


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 inverse hyperbolic function can be found here.


Sources