Definition:Real Inverse Hyperbolic Function

From ProofWiki
Jump to navigation Jump to search

Definition

Let $f: \R \to \R$ be one of the hyperbolic functions on the set of real numbers.

Certain of the inverse hyperbolic function $f^{-1} \subseteq \R \times \R$ are actually multifunctions, such that for a given $y \in \R$ there may be more than one $x \in \R$ such that $y = \map f x$.

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


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


Real Inverse Hyperbolic Sine

The inverse hyperbolic sine $\arsinh: \R \to \R$ is a real function defined on $\R$ as:

$\forall x \in \R: \map \arsinh x := \map \ln {x + \sqrt {x^2 + 1} }$

where:

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


Real Inverse Hyperbolic Cosine

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

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


The inverse hyperbolic cosine $\cosh^{-1}: S \to \R$ is a real multifunction defined on $S$ as:

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

where:

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


Real Inverse Hyperbolic Tangent

Let $S$ denote the open real interval:

$S := \openint {-1} 1$


The inverse hyperbolic tangent $\artanh: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \map \artanh x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.


Real Inverse Hyperbolic Cotangent

Let $S$ denote the union of the unbounded open real intervals:

$S := \openint \gets {-1} \cup \openint 1 \to$


The inverse hyperbolic cotangent $\arcoth: S \to \R$ is a real function defined on $S$ as:

$\forall x \in S: \arcoth x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$

where $\ln$ denotes the natural logarithm of a (strictly positive) real number.


Real Inverse Hyperbolic Secant

Let $S$ denote the half-open real interval:

$S := \hointl 0 1$


The inverse hyperbolic secant $\sech^{-1}: S \to \R$ is a real function defined on $S$ as:

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

where:

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


Real Inverse Hyperbolic Cosecant

The inverse hyperbolic cosecant $\arcsch: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:

$\forall x \in \R_{\ne 0}: \map \arcsch x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$

where:

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


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