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.

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/.