Fourier Series/Square Wave/Special Cases/Unit Half Interval
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Special Case of Fourier Series for Square Wave
Let $\map S x$ be the square wave defined on the real numbers $\R$ as:
- $\forall x \in \R: \map S x = \begin {cases}
1 & : x \in \openint 0 1 \\ -1 & : x \in \openint {-1} 0 \\ \map S {x + 2} & : x < -1 \\ \map S {x - 2} & : x > +1 \end {cases}$
Then its Fourier series can be expressed as:
\(\ds \map S x\) | \(\sim\) | \(\ds \frac 4 \pi \sum_{r \mathop = 0}^\infty \frac 1 {2 r + 1} {\sin \pi x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 4 \pi \paren {\sin \pi x + \dfrac {\sin 3 \pi x} 3 + \dfrac {\sin 5 \pi x} 5 + \dotsb}\) |
Proof
From Fourier Series for Square Wave, the real function $\map f x$ defined on the open interval $\openint {-l} l$ as:
- $\map f x = \begin {cases} -1 & : x \in \openint {-l} 0 \\ 1 & : x \in \openint 0 l \end {cases}$
has a Fourier series which can be expressed as:
\(\ds \map f x\) | \(\sim\) | \(\ds \frac 4 \pi \sum_{r \mathop = 0}^\infty \frac 1 {2 r + 1} {\sin \frac {\pi x} l}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 4 \pi \paren {\sin \frac {\pi x} l + \dfrac 1 3 \sin \frac {3 \pi x} l + \dfrac 1 5 \sin \frac {5 \pi x} l + \dotsb}\) |
The result follows by setting $l = 1$.
$\blacksquare$