PlanetMath: triangular-wave function

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triangular-wave function

(Definition)

The arcsine is the inverse function of the sine. Therefore the composition function

$\displaystyle f:\,x\mapsto \arcsin(\sin{x})$

is the identity map $x\mapsto x$ on the interval $[-\frac{\pi}{2},\,\frac{\pi}{2}]$ . On this interval, the inner function $\sin$ increases monotonically and continuously from its least value

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

, the composition thus decreases from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$ , evidently again linearly. We have now run through a period interval $[-\frac{\pi}{2},\,\frac{3\pi}{2}]$ of the inner function and the composition

and obtained a wedge-formed portion ( $\wedge$ ) of the graph. Because of the periodicity, the whole graph of

consists of such successive wedges and thus looks like a saw blade. The triangular-wave function is continuous. Its derivative (away from the singular points $\frac{\pi}{2}+n\pi,\, n\in\mathbb{Z}$ ) is a square-wave function.

$\begin{pspicture}(-6.6,-2.3)(6.7,2.3) \psline[linecolor=blue](-6.6,-0.3168)(-4.7... ...1.5708){$-\frac{\pi}{2}$} \rput[r](-0.3,1.5708){$\frac{\pi}{2}$} \end{pspicture}$

Sometimes, such a function is called a saw-tooth function, although this name usually refers to a discontinuous function with graph consisting of either ascending () or descending ( $\backslash$ ) line segments with jumps.

"triangular-wave function" is owned by pahio. [ full author list (3) ]

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Other names:

triangular wave function, saw-blade function, saw blade function

Keywords:

periodic function

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Cross-references: jumps, line segments, discontinuous, derivative, continuous, wedges, periodicity, graph, angle, interval, identity map, function, composition, sine, inverse function
There are 3 references to this entry.

This is version 23 of triangular-wave function, born on 2005-05-25, modified 2008-12-01.
Object id is 7115, canonical name is SawBladeFunction.
Accessed 8133 times total.

Classification:

AMS MSC:	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

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