Fourier Series/Triangle Wave/Special Cases
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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:
\(\ds \map T x\) | \(\sim\) | \(\ds \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \paren {2 n + 1} x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 2 - \frac 4 \pi \paren {\cos x + \frac 1 {3^2} \cos 3 x + \frac 1 {5^2} 5 x + \dotsb}\) |