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'area functions'
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| Title of object: |
area functions |
| Canonical Name: |
AreaFunctions |
| Type: |
Definition |
| Created on: |
2004-05-05 14:10:16 |
| Modified on: |
2004-06-18 16:31:23 |
| Classification: |
msc:26A09 |
| Synonyms: |
area functions=inverse hyperbolic functions |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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your knowledge
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% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
\DeclareMathOperator{\arsinh}{arsinh}
\DeclareMathOperator{\arcosh}{arcosh}
\DeclareMathOperator{\artanh}{artanh}
\DeclareMathOperator{\arcoth}{arcoth} |
Content:
The most usual {\em area functions}:
\begin{itemize}
\item The inverse function of the hyperbolic sine (in Latin {\em sinus hyperbolicus}) is $\arsinh$ ({\em area sini hyperbolici}):
$$\arsinh{x} := \ln{(x+\sqrt{x^2+1})}$$
\item The inverse function of the hyperbolic cosine (in Latin {\em cosinus hyperbolicus}) is $\arcosh$ ({\em area cosini hyperbolici}):
$$\arcosh{x} := \ln(x+\sqrt{x^2-1})$$
It is defined for $|x| \ge 1$.
\item The inverse function of the hyperbolic tangent (in Latin {\em tangens hyperbolica}) is $\artanh$ ({\em area tangentis hyperbolicae}):
$$\artanh{x} := \frac{1}{2}\ln \frac{1+x}{1-x}$$
It is defined for $-1 < x < 1$.
\item The inverse function os the hyperbolic cotangent (in Latin {\em cotangens hyperbolica}) is $\arcoth$ ({\em area cotangentis hyperbolicae}):
$$\arcoth{x} := \frac{1}{2}\ln \frac{x+1}{x-1}$$
It is defined for $|x| > 1$.
\end{itemize}
These four functions are denoted also by $\sinh^{-1}x$, $\cosh^{-1}x$, $\tanh^{-1}x$ and $\coth^{-1}x$.
Derivatives:
$$\frac{d}{dx} \arsinh x = \frac{1}{\sqrt{x^2+1}}$$
$$\frac{d}{dx} \arcosh x = \frac{1}{\sqrt{x^2-1}}$$
$$\frac{d}{dx} \artanh x = \frac{1}{1-x^2}$$
$$\frac{d}{dx} \arcoth x = \frac{1}{1-x^2}$$
The $\artanh$ has the \PMlinkescapetext{simple} Taylor series
$$\artanh x = x+\frac{x^3}{3}+\frac{x^5}{5}+\dots$$
The classic abbreviations ``$\arsinh$'' and ``$\arcosh$'' are explained as follows:
The {\em \PMlinkescapetext{unit} hyperbola} $x^2-y^2 = 1$ (its right half) has the parametric \PMlinkescapetext{representation}
$$x = \cosh A,$$
$$y = \sinh A;$$
here $A$ means the area \PMlinkescapetext{bounded} by the hyperbola and the straight lines $OP$ 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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