| Version 33 |
Version 32 |
| The most usual {\em area functions}: |
The most usual {\em area functions}: |
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| \begin{itemize} |
\begin{itemize} |
| \item The inverse function of the hyperbolic sine (in Latin {\em sinus hyperbolicus}) is $\arsinh$ ({\em area sini hyperbolici}): |
\item The inverse function of the hyperbolic sine (in Latin {\em sinus hyperbolicus}) is $\arsinh$ ({\em area sini hyperbolici}): |
| $$\arsinh{x} := \ln{(x+\sqrt{x^2+1})}$$ |
$$\arsinh{x} := \ln{(x+\sqrt{x^2+1})}$$ |
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| \item The inverse function of the hyperbolic cosine (in Latin {\em cosinus hyperbolicus}) is $\arcosh$ ({\em area cosini hyperbolici}): |
\item The inverse function of the hyperbolic cosine (in Latin {\em cosinus hyperbolicus}) is $\arcosh$ ({\em area cosini hyperbolici}): |
| $$\arcosh{x} := \ln(x+\sqrt{x^2-1})$$ |
$$\arcosh{x} := \ln(x+\sqrt{x^2-1})$$ |
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It is defined for\, $x \geqq 1$.
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It is defined for\, $|x| \geqq 1$.
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| \item The inverse function of the hyperbolic tangent (in Latin {\em tangens hyperbolica}) is $\artanh$ ({\em area tangentis hyperbolicae}): |
\item The inverse function of the hyperbolic tangent (in Latin {\em tangens hyperbolica}) is $\artanh$ ({\em area tangentis hyperbolicae}): |
| $$\artanh{x} := \frac{1}{2}\ln \frac{1+x}{1-x}$$ |
$$\artanh{x} := \frac{1}{2}\ln \frac{1+x}{1-x}$$ |
| It is defined for\, $-1 < x < 1$. |
It is defined for\, $-1 < x < 1$. |
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| \item The inverse function of the hyperbolic cotangent (in Latin {\em cotangens hyperbolica}) is $\arcoth$ ({\em area cotangentis hyperbolicae}): |
\item The inverse function of the hyperbolic cotangent (in Latin {\em cotangens hyperbolica}) is $\arcoth$ ({\em area cotangentis hyperbolicae}): |
| $$\arcoth{x} := \frac{1}{2}\ln \frac{x+1}{x-1}$$ |
$$\arcoth{x} := \frac{1}{2}\ln \frac{x+1}{x-1}$$ |
| It is defined for\, $|x| > 1$. |
It is defined for\, $|x| > 1$. |
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| \end{itemize} |
\end{itemize} |
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| These four functions are denoted also by $\sinh^{-1}x$, $\cosh^{-1}x$, $\tanh^{-1}x$ and $\coth^{-1}x$. |
These four functions are denoted also by $\sinh^{-1}x$, $\cosh^{-1}x$, $\tanh^{-1}x$ and $\coth^{-1}x$. |
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| Derivatives: |
Derivatives: |
| $$\frac{d}{dx} \arsinh x = \frac{1}{\sqrt{x^2\!+\!1}}$$ |
$$\frac{d}{dx} \arsinh x = \frac{1}{\sqrt{x^2\!+\!1}}$$ |
| $$\frac{d}{dx} \arcosh x = \frac{1}{\sqrt{x^2\!-\!1}}$$ |
$$\frac{d}{dx} \arcosh x = \frac{1}{\sqrt{x^2\!-\!1}}$$ |
| $$\frac{d}{dx} \artanh x = \frac{1}{1\!-\!x^2}$$ |
$$\frac{d}{dx} \artanh x = \frac{1}{1\!-\!x^2}$$ |
| $$\frac{d}{dx} \arcoth x = \frac{1}{1\!-\!x^2}$$ |
$$\frac{d}{dx} \arcoth x = \frac{1}{1\!-\!x^2}$$ |
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| The functions\, $\arsinh$\, and\, $\artanh$\, have the \PMlinkescapetext{simple} Taylor series |
The functions\, $\arsinh$\, and\, $\artanh$\, have the \PMlinkescapetext{simple} Taylor series |
| $$\arsinh{x} = x-\frac{1}{2}\!\cdot\!\frac{x^3}{3} |
$$\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}{2\!\cdot\!4}\!\cdot\!\frac{x^5}{5} |
| -\frac{1\!\cdot\!3\!\cdot\!5}{2\!\cdot\!4\cdot\!6}\!\cdot\!\frac{x^7}{7} |
-\frac{1\!\cdot\!3\!\cdot\!5}{2\!\cdot\!4\cdot\!6}\!\cdot\!\frac{x^7}{7} |
| +-\cdots\quad (|x|\leqq 1),$$ |
+-\cdots\quad (|x|\leqq 1),$$ |
| $$\artanh x = x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots |
$$\artanh x = x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots |
| \quad (|x| < 1).$$ |
\quad (|x| < 1).$$ |
| Because the inverse tangent function (see the cyclometric functions) has the \PMlinkescapetext{expansion} |
Because the inverse tangent function (see the cyclometric functions) has the \PMlinkescapetext{expansion} |
| \, $\arctan x = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+-\cdots\,\, |
\, $\arctan x = x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+-\cdots\,\, |
| (|x|\leqq 1)$, |
(|x|\leqq 1)$, |
| we see that |
we see that |
| $$\artanh x = \frac{1}{i}\arctan ix;$$ |
$$\artanh x = \frac{1}{i}\arctan ix;$$ |
| similarly we get |
similarly we get |
| $$\arsinh x = \frac{1}{i}\arcsin ix.$$ |
$$\arsinh x = \frac{1}{i}\arcsin ix.$$ |
| Some other formulae which may be obtained by means of the addition formulae of the hyperbolic functions: |
Some other formulae which may be obtained by means of the addition formulae of the hyperbolic functions: |
| $$\arsinh x\pm\arsinh y = \arsinh(x\sqrt{y^2\!+\!1}\pm y\sqrt{x^2\!+\!1})$$ |
$$\arsinh x\pm\arsinh y = \arsinh(x\sqrt{y^2\!+\!1}\pm y\sqrt{x^2\!+\!1})$$ |
| $$\arcosh x\pm\arcosh y = \arcosh(xy\pm\sqrt{x^2\!-\!1}\sqrt{y^2\!-\!1})$$ |
$$\arcosh x\pm\arcosh y = \arcosh(xy\pm\sqrt{x^2\!-\!1}\sqrt{y^2\!-\!1})$$ |
| $$\artanh x\pm\artanh y = \artanh\frac{x\pm y}{1\pm xy}$$ |
$$\artanh x\pm\artanh y = \artanh\frac{x\pm y}{1\pm xy}$$ |
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| The classic abbreviations ``$\arsinh$'' and ``$\arcosh$'' are explained as follows:\, The unit hyperbola\, $x^2\!-\!y^2 = 1$\,(its right half) has the parametric \PMlinkescapetext{representation} |
The classic abbreviations ``$\arsinh$'' and ``$\arcosh$'' are explained as follows:\, The unit hyperbola\, $x^2\!-\!y^2 = 1$\,(its right half) has the parametric \PMlinkescapetext{representation} |
| \[\begin{cases} |
\[\begin{cases} |
| x = \cosh A,\\ |
x = \cosh A,\\ |
| y = \sinh A; |
y = \sinh A; |
| \end{cases}\] |
\end{cases}\] |
| here $A$ means the area \PMlinkescapetext{bounded} 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 $x$ ({\em area cosini hyperbolici x}), similarly $A$ is the area having hyperbolic sine equal to $y$ ({\em area sini hyperbolici y}). |
here $A$ means the area \PMlinkescapetext{bounded} 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 $x$ ({\em area cosini hyperbolici x}), similarly $A$ is the area having hyperbolic sine equal to $y$ ({\em area sini hyperbolici y}). |
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| {\bf Note.}\, In some countries the abbreviation ``ar'' in the symbols arsinh etc. is replaced by\, ``a'', ``Ar'', ``arc'' or ``arg''. |
{\bf Note.}\, In some countries the abbreviation ``ar'' in the symbols arsinh etc. is replaced by\, ``a'', ``Ar'', ``arc'' or ``arg''. |
