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:

$\ds \map S x$

$\sim$

$\ds \frac 2 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n \pi x$

$\ds $

$=$

$\ds \frac 2 \pi \paren {\sin \pi x - \frac {\sin 2 \pi x} 2 + \frac {\sin 3 \pi x} 3 + \dotsb}$

$\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:

$\ds \map S x$

$\sim$

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

$\ds $

$=$

$\ds \frac {2 l} \pi \paren {\sin \dfrac {\pi x} l - \frac 1 2 \sin \dfrac {2 \pi x} l + \frac 1 3 \sin \dfrac {3 \pi x} l + \dotsb}$

The result follows by setting $l = 1$.

$\blacksquare$