Fourier Series/Sawtooth Wave/Special Cases/Half Interval Pi

Special Case of Fourier Series for Sawtooth Wave
Let $\map f x$ be the real function defined on the open interval $\openint {-\pi} \pi$ as:


 * $\map f x = x$

Then its Fourier series can be expressed as:

Proof
From Fourier Series for Sawtooth Wave, the real function $\map f x$ defined on the open interval $\openint {-l} l$ as:


 * $\map f x = \size x$

has a Fourier series which can be expressed as:

The result follows by setting $l = \pi$.