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Sawtooth Wave

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SawtoothWave

The sawtooth wave, called the "castle rim function" by Trott (2004, p. 228), is the periodic function given by

S(x)=Afrac(x/T+phi),
(1)

where frac(x) is the fractional part frac(x)=x-|_x_|, A is the amplitude, T is the period of the wave, and phi is its phase. (Note that Trott 2004, p. 228 uses the term "sawtooth function" to describe a triangle wave.) It therefore consists of an infinite sequence of truncated ramp functions concatenated together.

The sawtooth wave is implemented in the Wolfram Language as SawtoothWave[x].

If phi=0, A=1, and T=2L, then the Fourier series is given by

f(x)=1/2-1/pisum_(n=1)^infty1/nsin((npix)/L)
(2)

and the function can be written

f(x)=1/2+i/(2pi)ln(e^(-ipix/L))
(3)
=1/2-1/(2pi)arg(-e^(-ipix/L))
(4)
=x/(2L)-|_x/(2L)_|
(5)
=1/(2L)mod(x,2L)
(6)
=1/2-tan^(-1)[cot((pix)/(2L))],
(7)

where |_x_| is the floor function.

See also

Fourier Series--Sawtooth Wave, Fractional Part, Ramp Function, Square Wave, Staircase Function, Triangle Wave

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References

Spanier, J. and Oldham, K. B. An Atlas of Functions. Washington, DC: Hemisphere, p. 74, 1987.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Cite this as:

Weisstein, Eric W. "Sawtooth Wave." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SawtoothWave.html

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