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

Special Case of Fourier Series for Sawtooth Wave
Let $\map S x$ be the sawtooth wave defined on the real numbers $\R$ as:


 * $\forall x \in \R: \map S x = \begin {cases}

x & : x \in \openint {-\pi} \pi \\ \map S {x + 2 \pi} & : x < -\pi \\ \map S {x - 2 \pi} & : x > +\pi \end {cases}$

Then its Fourier series can be expressed as:

Proof
From Fourier Series for Sawtooth Wave, the sawtooth wave defined on the real numbers $\R$ as:


 * $\forall x \in \R: \map S x = \begin {cases}

x & : x \in \openint {-l} l \\ \map S {x + 2 l} & : x < -1 \\ \map S {x - 2 l} & : x > +1 \end {cases}$

has a Fourier series which can be expressed as:

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