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

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

has a Fourier series which can be expressed as:

The result follows by setting $l = 1$.