Analytic representations the symmetric triangle wave with period 2 and varying between 
and 1 include
 |  | ![2/pisin^(-1)[sin(pix)]](/images/equations/TriangleWave/Inline4.svg) |
(1)
|
 |  | ![1-2|1-[2(1/2x+1/4 (mod 1))]|](/images/equations/TriangleWave/Inline7.svg) |
(2)
|
 |  |  |
(3)
|
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)
|
which can be summed to yield the analytic expression
![f(x)=i/(pi^2e^(piix))[Phi(x,2,1/2)-e^(2piix)Phi(-e^(2piix),2,1/2)],](/images/equations/TriangleWave/NumberedEquation2.svg) |
(5)
|
where 
is a Lerch transcendent.
A form of triangle wave ranging between 0 and 1 with period 2 is given by
 |
(6)
|
(Trott 2004, p. 228), where 
is the nearest
integer function.
