Fourier Series/Sawtooth Wave

Theorem


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 {-l} l \\ \map S {x + 2 l} & : x < -l \\ \map S {x - 2 l} & : x > +l \end {cases}$ where $l$ is a given real constant.

Then its Fourier series can be expressed as:

Proof
Let $\map f x: \openint {-l} l \to \R$ denote the identity function on the open interval $\openint {-l} l$:
 * $\map f x = x$

From Fourier Series for Identity Function over Symmetric Range, $\map f x$ can immediately be expressed as:


 * $\ds \map f x \sim \frac {2 l} \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \dfrac {n \pi x} l$