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 floor function, 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 with , then the Fourier series is given by

(2)

At each jump point for integer , this series converges to , while the sawtooth wave defined above has value 0. Away from these points, the function can be written

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

where and take their principal values.