Fourier Series/Triangle Wave/Special Cases

Special Cases of Fourier Series for Triangle Wave

Unit Half Interval

Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as:

$\forall x \in \R: \map T x = \begin {cases}

\size x & : x \in \closedint {-1} 1 \\ \map T {x + 2} & : x < -1 \\ \map T {x - 2} & : x > +1 \end {cases}$ where $\size x$ denotes the absolute value of $x$.

Then its Fourier series can be expressed as:

\(\ds \map T x\)

\(\sim\)

\(\ds \frac 1 2 - \frac 4 {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \paren {2 n + 1} \pi x\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 - \frac 4 {\pi^2} \paren {\cos \pi x + \frac 1 {3^2} \cos 3 \pi x + \frac 1 {5^2} 5 \pi x + \dotsb}\)

Half Interval $\pi$

Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as:

$\forall x \in \R: \map T x = \begin {cases}

\size x & : x \in \closedint {-\pi} \pi \\ \map T {x + 2 \pi} & : x < -\pi \\ \map T {x - 2 \pi} & : x > +\pi \end {cases}$ where $\size x$ denotes the absolute value of $x$.

Then its Fourier series can be expressed as:

\(\ds \map T x\)

\(\sim\)

\(\ds \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \paren {2 n + 1} x\)

\(\ds \)

\(=\)

\(\ds \frac \pi 2 - \frac 4 \pi \paren {\cos x + \frac 1 {3^2} \cos 3 x + \frac 1 {5^2} 5 x + \dotsb}\)