PlanetMath: area functions

(more info)

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area functions

(Definition)

The most usual area functions:

The inverse function of the hyperbolic sine (in Latin sinus hyperbolicus) is ${\mathrm{arsinh}}$ (area sini hyperbolici):
$\displaystyle {\mathrm{arsinh}}{x} := \ln{(x+\sqrt{x^2+1})}$
The inverse function of the hyperbolic cosine (in Latin cosinus hyperbolicus) is ${\mathrm{arcosh}}$ (area cosini hyperbolici):
$\displaystyle {\mathrm{arcosh}}{x} := \ln(x+\sqrt{x^2-1})$
It is defined for $x \geqq 1$ .
The inverse function of the hyperbolic tangent (in Latin tangens hyperbolica) is ${\mathrm{artanh}}$ (area tangentis hyperbolicae):
$\displaystyle {\mathrm{artanh}}{x} := \frac{1}{2}\ln \frac{1+x}{1-x}$
It is defined for .
The inverse function of the hyperbolic cotangent (in Latin cotangens hyperbolica) is ${\mathrm{arcoth}}$ (area cotangentis hyperbolicae):
$\displaystyle {\mathrm{arcoth}}{x} := \frac{1}{2}\ln \frac{x+1}{x-1}$
It is defined for $\vert x\vert > 1$ .

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

Derivatives:

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

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

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

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

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

$\displaystyle {\mathrm{arsinh}}{x} = x-\frac{1}{2}\!\cdot\!\frac{x^3}{3} +\frac... ...2\!\cdot\!4\cdot\!6}\!\cdot\!\frac{x^7}{7} +-\cdots\quad (\vert x\vert\leqq 1),$

$\displaystyle {\mathrm{artanh}}x = x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots \quad (\vert x\vert < 1).$

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

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

similarly we get

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

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

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

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

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

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 representation