----- > [!proposition] Proposition. ([[Fourier series of a triangle wave]]) > The function obtained by [[periodic|periodically extending]] $f(t)=\begin{cases} \frac{1}{2}+t & -\frac{1}{2} \leq t \leq 0 {} \\ \frac{1}{2}-t & 0 \leq t \leq \frac{1}{2} \end{cases}$ has [[Fourier series|Fourier series]] $\frac{1}{4}+\sum_{k=0}^{\infty} \frac{2}{\pi^{2}(2k+1)^{2}}\cos\big(2\pi(2k+1)t\big).$ > [!proof]- Proof. ([[Fourier series of a triangle wave]]) > **[[Fourier series|Fourier coefficients.]]** Consider $n \in \zz$. We have $\begin{align} \hat{f}(n) = & \int _{-\frac{1}{2}} ^\frac{1}{2} f(t) e ^{-2 \pi i n t} \, dt \\ = & \int _{-\frac{1}{2}}^{0} \left( \frac{1}{2}+t \right)e ^{-2\pi i n t} \ dt + \int _{0} ^{\frac{1}{2}} \, \left( \frac{1}{2} - t \right) e ^{- 2 \pi i n t} \, dt \\ = & \begin{cases} - \frac{i \pi n - 1}{4 \pi^{2} n^{2}} + \frac{i \pi n + 1}{4 \pi^{2} n^{2}} - \frac{e^{i \pi n}}{4 \pi^{2} n^{2}} - \frac{e^{- i \pi n}}{4 \pi^{2} n^{2}} & n \neq 0 \\\frac{1}{4} & n=0 \end{cases} \ \ (\text{via } \texttt{sympy}) \\ = & \begin{cases} \frac{2 - (e ^{i\pi n} + e ^{- i \pi n})}{4 \pi^{2} n ^{2}} & n \neq 0 \\ \frac{1}{4} & n = 0 \\ \end{cases} \\ = & \begin{cases} \frac{2-2\cos n \pi}{4 \pi^{2} n^{2}} & n \neq 0 \\ \frac{1}{4} & n=0. \end{cases} \end{align}$ **[[Fourier series|Fourier Series.]]** Formally write $\begin{align} \sum_{n \in \mathbb{Z}}^{} \hat{ f}(n) e ^{2 \pi i n t}= & \frac{1}{4} + \sum_{ n \neq 0} \frac{2(\overbrace{1-\cos n \pi)}^{= \begin{cases} 2 & n \text{ even } \\ 0 & n \text{ odd} \end{cases}}}{4 \pi^{2} n ^{2}} e ^{ 2\pi i n t} \\ = & \frac{1}{4} + \sum_{n \text{ odd}}^{} \frac{1}{\pi^{2} n ^{2}} \textcolor{Skyblue}{e ^{2 \pi i n t}} \\ =& \frac{1}{4} + \sum_{n>0 \ \text{ odd}}^{} \frac{1}{\pi^{2} (-n) ^{2}} \textcolor{Skyblue}{e^{-2\pi i n t}} + \frac{1}{\pi^{2}n ^{2}} \textcolor{Skyblue}{e ^{2 \pi i n t}} \\ =& \frac{1}{4} + \sum_{n > 0 \ \text{ odd}} \frac{1}{\pi^{2} n ^{2}} \textcolor{Skyblue}{2 \cos 2\pi n t} \\ =& \frac{1}{4} + \sum_{k=0}^{\infty} \frac{2}{\pi^{2} (2k+1) ^{2}} \cos 2\pi (2k+1) t. \end{align}$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```