----- > [!proposition] Proposition. ([[Fourier series of a square wave]]) > The function obtained by [[periodic|periodically extending]] $f(t)=\begin{cases} +1 & 0 \leq t < \frac{1}{2} {} \\ -1 & \frac{1}{2} \leq t < 1 \end{cases}$ has [[Fourier series|Fourier series]] $\frac{4}{\pi} \sum_{k=0}^{\infty} \frac{1}{2k+1} \sin 2\pi (2k+1)t.$ > > [!proof]- Proof. ([[Fourier series of a square wave]]) > **[[Fourier series|Fourier coefficients.]]** Consider $n \in \zz$. When $n=0$, $\hat{f}(n)=-1+1=0$, so assume $n \neq0$ and write $\begin{align} \hat{f}(n) = & \int _{0}^{1} f(t)e ^{-2 \pi i n t} \, dt \\ = & \int _{0}^{\frac{1}{2}} e ^{- 2\pi i n t} \, dt - \int _{\frac{1}{2}}^{1} e ^{-2\pi i n t} \, dt \\ = & \left[ \frac{-e ^{-2\pi in t}}{2\pi in} \right]_{t=0} ^{1/2} - \left[ \frac{-e ^{-2\pi i nt}}{2\pi in} \right]_{t= 1 / 2}^{1} \\ = & \frac{-e ^{-i \pi n}+1}{2\pi i n}+ \frac{\cancel{e ^{-2 \pi i n}}^{1}-e ^{- \pi i n}}{2\pi i n} \\ = & \frac{1-e ^{-i\pi n}}{i \pi n}, \end{align}$ hence $\hat{f}(n)=\begin{cases} 0 & n=0 \\ \frac{1-e ^{-i \pi n}}{i \pi n} & n \neq 0. \end{cases}$ **[[Fourier Series|Fourier Series.]]** Formally write $\begin{align} \sum_{n \in \mathbb{Z}}^{} \hat{ f}(n) e ^{2 \pi i n t} = & \sum_{n \neq 0}^{} \frac{\textcolor{Thistle}{1-e ^{-i \pi n}}}{i \pi n} e ^{2 \pi i n t} \\ = & \sum_{n \neq 0}^{} \frac{e ^{2\pi in t}}{i \pi n} \begin{cases} \textcolor{Thistle}{2} & \textcolor{Thistle}{n \text{ even }} \\ \textcolor{Thistle}{0} & \textcolor{Thistle}{n \text{ odd}} \\ \end{cases} \\ = & \sum_{n \text{ odd}}^{} \frac{2 }{i \pi n}\textcolor{Skyblue}{e ^{2\pi i n t}} \\ = & \sum_{n > 0 \ \text{ odd}}-\frac{2}{i \pi n} \textcolor{Skyblue}{e ^{-2\pi i nt}} + \frac{2}{i \pi n} \textcolor{Skyblue}{e ^{2\pi i nt}} \\ = & \sum_{n > 0 \ \text{ odd} }^{} \frac{2}{i \pi n}(\textcolor{Skyblue}{ 2 i\sin 2\pi n t}) \\ = & \frac{4}{\pi}\sum_{k \in \mathbb{Z} _{\geq 0}}^{} \frac{1}{2k+1} \sin 2 \pi (2k+1) t. \end{align}$ ----- #### ---- #### 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 > ```