# area functions

The most usual area functions:

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

 $\operatorname{arsinh}{x}:=\ln{(x+\sqrt{x^{2}+1})}$
• The inverse function of the hyperbolic cosine (in Latin cosinus hyperbolicus) is $\operatorname{arcosh}$ (area cosini hyperbolici):

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

It is defined for  $x\geqq 1$.

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

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

It is defined for  $-1.

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

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

It is defined for  $|x|>1$.

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

 $\frac{d}{dx}\operatorname{arsinh}x=\frac{1}{\sqrt{x^{2}\!+\!1}}$
 $\frac{d}{dx}\operatorname{arcosh}x=\frac{1}{\sqrt{x^{2}\!-\!1}}$
 $\frac{d}{dx}\operatorname{artanh}x=\frac{1}{1\!-\!x^{2}}$
 $\frac{d}{dx}\operatorname{arcoth}x=\frac{1}{1\!-\!x^{2}}$

The functions  $\operatorname{arsinh}$  and  $\operatorname{artanh}$  have the Taylor series  $\operatorname{arsinh}{x}=x-\frac{1}{2}\!\cdot\!\frac{x^{3}}{3}+\frac{1\!\cdot% \!3}{2\!\cdot\!4}\!\cdot\!\frac{x^{5}}{5}-\frac{1\!\cdot\!3\!\cdot\!5}{2\!% \cdot\!4\cdot\!6}\!\cdot\!\frac{x^{7}}{7}+-\cdots\quad(|x|\leqq 1),$
 $\operatorname{artanh}x=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\frac{x^{7}}{7}+% \cdots\quad(|x|<1).$

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

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

similarly we get

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

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

 $\operatorname{arsinh}x\pm\operatorname{arsinh}y=\operatorname{arsinh}(x\sqrt{y% ^{2}\!+\!1}\pm y\sqrt{x^{2}\!+\!1})$
 $\operatorname{arcosh}x\pm\operatorname{arcosh}y=\operatorname{arcosh}(xy\pm% \sqrt{x^{2}\!-\!1}\sqrt{y^{2}\!-\!1})$
 $\operatorname{artanh}x\pm\operatorname{artanh}y=\operatorname{artanh}\frac{x% \pm y}{1\pm xy}$

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

 $\begin{cases}x=\cosh A,\\ y=\sinh A;\end{cases}$

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