Analytic representations the symmetric triangle wave with period 2 and varying between and 1 include
(1)
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(2)
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(3)
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where is the fractional part of .
The triangle wave is implemented in the Wolfram Language as TriangleWave[x].
The Fourier series for the triangle wave is given by
(4)
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which can be summed to yield the analytic expression
(5)
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where is a Lerch transcendent.
A form of triangle wave ranging between 0 and 1 with period 2 is given by
(6)
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(Trott 2004, p. 228), where is the nearest integer function.