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Viewing Version
18
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'triangular-wave function'
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| Title of object: |
triangular-wave function |
| Canonical Name: |
SawBladeFunction |
| Type: |
Definition |
| Created on: |
2005-05-25 18:11:54 |
| Modified on: |
2007-06-04 05:05:37 |
| Classification: |
msc:26A06, msc:53A04 |
| Keywords: |
periodic function |
| Synonyms: |
triangular-wave function=saw blade function |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{pstricks}
\usepackage{pst-plot}
% 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
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem} |
Content:
The \PMlinkname{arcsin}{CyclometricFunctions} is the inverse function of the sine --- therefore the composition function
$$f:\,x\mapsto \arcsin(\sin{x})$$
is the identity map \, $x\mapsto x$\, on the interval \,
$[-\frac{\pi}{2},\,\frac{\pi}{2}]$.\, On this interval the \PMlinkescapetext{inner function $\sin$ increases monotonically and continuously from its least value $-1$ to its greatest value 1; then the outer} function $\arcsin$ (i.e. the angle corresponding the sine value) and the whole composition correspondingly grows from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.\, On the next equally long interval\, $[\frac{\pi}{2},\,\frac{3\pi}{2}]$,\, when the inner function decreases from 1 to $-1$, the composition thus decreases from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$,\, evidently again linearly.\, We now have run through a \PMlinkescapetext{period} interval\,
$[-\frac{\pi}{2},\,\frac{3\pi}{2}]$\, of the inner function and the composition $f$ and obtained a wedge-formed portion ($\wedge$) of the graph.\, Because of the periodicity the whole graph of $f$ consists of such successive wedges and thus is like a saw \PMlinkescapetext{blade}.\, The {\em triangular-wave function} is continuous.\, Its derivative (away from the singular points\,
$\frac{\pi}{2}+n\pi,\, n\in\mathbb{Z}$) is a
\PMlinkname{square-wave function}{CommonFourierSeries}.
\begin{center}
\begin{pspicture}(-6.6,-2.3)(6.7,2.3)
\psline[linecolor=blue](-6.6,-0.3168)(-4.7124,1.5708)
\psline[linecolor=blue](-4.7124,1.5708)(-1.5708,-1.5708)
\psline[linecolor=blue](-1.5708,-1.5708)(1.5708,1.5708)
\psline[linecolor=blue](1.5708,1.5708)(4.7124,-1.5708)
\psline[linecolor=blue](4.7124,-1.5708)(6.6,0.3168)
\rput[l](-6.6,0){.}
\psaxes[labels=none,Dx=1.5708,Dy=1.5708]{->}(0,0)(-6.6,-1.8)(7.0,2.3)
\rput[a](7.1,-0.25){$x$}
\rput[r](-0.22,2.35){$y$}
\rput[a](0,-2.3){\textbf{Figure:} Graph of $\arcsin(\sin x)$}
\rput[a](-6.2832,-0.4){$2\pi$}
\rput[a](-4.7124,-0.4){$-\frac{3\pi}{2}$}
\rput[a](-3.1416,-0.4){$\pi$}
\rput[a](-1.5708,-0.4){$-\frac{\pi}{2}$}
\rput[a](1.5708,-0.4){$\frac{\pi}{2}$}
\rput[a](3.1416,-0.4){$\pi$}
\rput[a](4.7124,-0.4){$\frac{3\pi}{2}$}
\rput[a](6.2832,-0.4){$2\pi$}
\rput[r](-0.3,-1.5708){$-\frac{\pi}{2}$}
\rput[r](-0.3,1.5708){$\frac{\pi}{2}$}
\end{pspicture}
\end{center}
Sometimes, such a function is called {\em saw-tooth function}, although this name usually refers to a discontinuous function with graph consisting of ascending ($/$) or descending ($\backslash$) line segments with jumps. |
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