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UnkleRhaukus
 3 years ago
\(f(x)\) as a sine series
UnkleRhaukus
 3 years ago
\(f(x)\) as a sine series

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UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0\[ \qquad\text{\(f(x)\) as a sine series} \begin{equation*} % f(x) f(x)=1,\qquad0<x<\pi \end{equation*}\]

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0\[\begin{equation*} % g(x) g(x)= \begin{cases} 1,&0<x<\pi\\ 1,&\pi<x<0 \end{cases}\qquad\text{odd extension of \(f(x)\)} \end{equation*}\] \begin{equation*} % a_0, a_n a_0=a_n=0\qquad\text{odd function} \end{equation*} \begin{align*} % b_n b_n&=\frac1\pi\int\limits_{\pi}^{\pi} g(x)\sin(n x)\,\text dx\\ &=\frac2\pi\int\limits_0^\pi \sin(n x)\,\text dx\qquad\text{even integrand}\\ &=\frac2\pi\left(\frac{\cos(n x)}{n}\right)\Big_0^\pi\\ &=\frac2\pi\left(\frac{\cos(0)\cos(n\pi )}{n}\right)\\ &=\frac2{\pi}\left(\frac{1(1)^n}{n}\right)\\ \end{align*} \begin{align*} S(x)&=\frac2\pi\sum\limits_{n=1}^\infty\left(\frac{1(1)^n}{n}\right)\sin(nx)\\ &=\frac2\pi\sum\limits_{n=1,3,5,\dots}^\infty\left(\frac{2}{n}\right)\sin(nx)\\ &=\frac4\pi\sum\limits_{r=1}^\infty\frac{\sin\big((2r1)x\big)}{2r1} \end{align*}

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0just change L to pi http://mathworld.wolfram.com/FourierSeriesSquareWave.html

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0you got the opposite one .. shift the period by +pi

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0where did i go wrong?

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0don't know .. don't have much time right now!! but i often use that link as reference!!

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0sin(nx) is not even .. get rid of that minus

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0why does (4) on that link have a sin^2 term?

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0ah i found my mistake

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0\[\begin{align*} % b_n b_n&=\frac1\pi\int\limits_{\pi}^{\pi} g(x)\sin(n x)\,\text dx\\ &=\frac2\pi\int\limits_0^\pi \sin(n x)\,\text dx\qquad\text{even integrand}\\ &=\frac2\pi\left(\frac{\cos(n x)}{n}\right)\Big_0^\pi\\ &=\frac2\pi\left(\frac{\cos(0)\cos(n\pi )}{n}\right)\\ &=\frac2{\pi}\left(\frac{1(1)^n}{n}\right)\\ \end{align*}\]

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0\begin{align*} S(x)&=\frac2\pi\sum\limits_{n=1}^\infty\left(\frac{1(1)^n}{n}\right)\sin(nx)\\ &=\frac4\pi\sum\limits_{n=1,3,5,\dots}^\infty\frac{\sin(nx)}{n}\\ &=\frac4\pi\sum\limits_{r=1}^\infty\frac{\sin\big((2r1)x\big)}{2r1} \end{align*}

Echdip
 3 years ago
Best ResponseYou've already chosen the best response.0guy please help me with this. This is impossible tough for me

UnkleRhaukus
 3 years ago
Best ResponseYou've already chosen the best response.0i havent studies solving PDEs yet sorry @Echdip
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