Fourier Series/Square Wave/Special Cases

Special Cases of Fourier Series for Square Wave

Unit Half Interval

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:

\(\ds \map S x\)

\(\sim\)

\(\ds \frac 4 \pi \sum_{r \mathop = 0}^\infty \frac 1 {2 r + 1} {\sin \pi x}\)

\(\ds \)

\(=\)

\(\ds \frac 4 \pi \paren {\sin \pi x + \dfrac {\sin 3 \pi x} 3 + \dfrac {\sin 5 \pi x} 5 + \dotsb}\)

Half Interval $\pi$

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 \pi \\ -1 & : x \in \openint {-\pi} 0 \\ \map S {x + 2 \pi} & : x < -\pi \\ \map S {x - 2 \pi} & : x > +\pi \end {cases}$

Then its half-range Fourier sine series can be expressed as:

\(\ds \map S x\)

\(\sim\)

\(\ds \frac 4 \pi \sum_{r \mathop = 0}^\infty \frac 1 {2 r + 1} {\sin x}\)

\(\ds \)

\(=\)

\(\ds \frac 4 \pi \paren {\sin x + \dfrac {\sin 3 x} 3 + \dfrac {\sin 5 x} 5 + \dotsb}\)