area functions

The most usual area functions:

• The inverse function of the hyperbolic sine (in Latin sinus hyperbolicus) is $\mathrm{arsinh}$ (area sini hyperbolici):

  $$\mathrm{arsinh}x:=\ln (x+\sqrt{x^2+1})$$

• The inverse function of the hyperbolic cosine (in Latin cosinus hyperbolicus) is $\mathrm{arcosh}$ (area cosini hyperbolici):

  $$\mathrm{arcosh}x:=\ln (x+\sqrt{x^2-1})$$

  It is defined for  $x\geqq 1$.

• The inverse function of the hyperbolic tangent (in Latin tangens hyperbolica) is $\mathrm{artanh}$ (area tangentis hyperbolicae):

  $$\mathrm{artanh}x:=\frac{1}{2}\ln \frac{1+x}{1-x}$$

  It is defined for  .

• The inverse function of the hyperbolic cotangent (in Latin cotangens hyperbolica) is $\mathrm{arcoth}$ (area cotangentis hyperbolicae):

  $$\mathrm{arcoth}x:=\frac{1}{2}\ln \frac{x+1}{x-1}$$

  It is defined for  $|x|>1$.

These four functions are denoted also by ${\sinh }^{-1}x$, ${\cosh }^{-1}x$, ${\tanh }^{-1}x$ and ${\coth }^{-1}x$.

Derivatives:

$$\frac{d}{dx}\mathrm{arsinh}x=\frac{1}{\sqrt{x^2+1}}$$

$$\frac{d}{dx}\mathrm{arcosh}x=\frac{1}{\sqrt{x^2-1}}$$

$$\frac{d}{dx}\mathrm{artanh}x=\frac{1}{1-x^2}$$

$$\frac{d}{dx}\mathrm{arcoth}x=\frac{1}{1-x^2}$$

The functions  $\mathrm{arsinh}$  and  $\mathrm{artanh}$  have the Taylor series

$$\mathrm{arsinh}x=x-\frac{1}{2}\cdot \frac{x^3}{3}+\frac{1\cdot 3}{2\cdot 4}\cdot \frac{x^5}{5}-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\cdot \frac{x^7}{7}+-\mathrm{\cdots }\mathit{\hspace{1em}}(|x|\leqq 1),$$

Because the inverse tangent function (see the cyclometric functions) has the   $\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+-\mathrm{\cdots }(|x|\leqq 1)$, we see that

$$\mathrm{artanh}x=\frac{1}{i}\arctan ix;$$

similarly we get

$$\mathrm{arsinh}x=\frac{1}{i}\arcsin ix.$$

Some other formulae which may be obtained by means of the addition formulae of the hyperbolic functions:

$$\mathrm{arsinh}x\pm \mathrm{arsinh}y=\mathrm{arsinh}(x\sqrt{y^2+1}\pm y\sqrt{x^2+1})$$

$$\mathrm{arcosh}x\pm \mathrm{arcosh}y=\mathrm{arcosh}(xy\pm \sqrt{x^2-1}\sqrt{y^2-1})$$

$$\mathrm{artanh}x\pm \mathrm{artanh}y=\mathrm{artanh}\frac{x\pm y}{1\pm xy}$$

The classic abbreviations “$\mathrm{arsinh}$” and “$\mathrm{arcosh}$” are explained as follows:  The unit hyperbola  $x^2-y^2=1$ (its right half) has the parametric

$$\{\begin{array}{cc}x=\cosh A,\hfill & \\ y=\sinh A;\hfill & \end{array}$$

here $A$ means the area by the hyperbola and the straight line segments $OP$ and $OQ$, where $O$ is the origin, $P$ is the point  $(x,y)$  of the hyperbola and $Q$ is the point  $(x,-y)$  of the hyperbola.  Thus, conversely, $A$ is the area having hyperbolic cosine equal to $x$ (area cosini hyperbolici x), similarly $A$ is the area having hyperbolic sine equal to $y$ (area sini hyperbolici y).

Note.  In some countries the abbreviation “ar” in the symbols arsinh etc. is replaced by  “a”, “Ar”, “arc” or “arg”.

Title	area functions
Canonical name	AreaFunctions
Date of creation	2013-03-22 14:21:18
Last modified on	2013-03-22 14:21:18
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	38
Author	pahio (2872)
Entry type	Definition
Classification	msc 26A09
Synonym	inverse hyperbolic functions
Related topic	UnitHyperbola
Related topic	CyclometricFunctions
Related topic	HyperbolicAngle
Related topic	IntegralTables
Related topic	IntegrationOfSqrtx21
Related topic	IntegralRelatedToArcSine
Related topic	ArcLengthOfParabola
Related topic	ListOfImproperIntegrals
Related topic	InverseGudermannianFunction
Related topic	EulersSubstitutionsForIntegration
Related topic	ArcoshCurve
Related topic	EqualArcLength
Defines	arsinh
Defines	arcosh
Defines	artanh
Defines	arcoth