Analytic representations the symmetric triangle wave with period 2 and varying between 
and 1 include
 |  | ![2/pisin^(-1)[sin(pix)]](https://mathworld.wolfram.com/images/equations/TriangleWave/Inline4.svg) | (1) |
 |  | ![1-2|1-[2(1/2x+1/4 (mod 1))]|](https://mathworld.wolfram.com/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)],](https://mathworld.wolfram.com/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.
