The sawtooth wave, called the "castle rim function" by Trott (2004, p. 228), is the periodic function given by
(1)
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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)
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and the function can be written
(3)
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(4)
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(5)
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(6)
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(7)
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where is the floor function.