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# Sawtooth Wave

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

 (1)

where is the fractional part , is the amplitude, is the period of the wave, and 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 , , and , then the Fourier series is given by

 (2)

and the function can be written

 (3) (4) (5) (6) (7)

where is the floor function.

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