Fourier Series/Sawtooth Wave

Theorem


Let $\map S x$ be the triangle 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
By definition, the identity function is an odd function:


 * $\map f {-x} = -x = -\map f x$

Thus by Fourier Series for Odd Function over Symmetric Range, $x$ can be expressed as:


 * $\displaystyle x \sim \sum_{n \mathop = 1}^\infty b_n \sin \dfrac {n \pi x} l$

Hence for all $n \in \Z_{\ge 1}$:

So for $x \in \openint {-l} l$:


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