# Fourier Series/Sawtooth Wave

## Theorem

Sawtooth Wave and $6$th Approximation

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:

 $\ds \map S x$ $\sim$ $\ds \frac {2 l} \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \dfrac {n \pi x} l$ $\ds$ $=$ $\ds \frac {2 l} \pi \paren {\sin \dfrac {\pi x} l - \frac 1 2 \sin \dfrac {2 \pi x} l + \frac 1 3 \sin \dfrac {3 \pi x} l + \dotsb}$

## 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:

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

## Special Cases

### Unit Half Interval

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:

 $\ds \map S x$ $\sim$ $\ds \frac 2 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n \pi x$ $\ds$ $=$ $\ds \frac 2 \pi \paren {\sin \pi x - \frac {\sin 2 \pi x} 2 + \frac {\sin 3 \pi x} 3 + \dotsb}$

### Half Interval $\pi$

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:

 $\ds \map S x$ $\sim$ $\ds 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin n x$ $\ds$ $=$ $\ds 2 \paren {\sin x - \frac {\sin 2 x} 2 + \frac {\sin 3 x} 3 + \dotsb}$