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

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

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

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

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

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

UnkleRhaukus
 one year 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
 one year 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*}

UnkleRhaukus
 one year ago
Best ResponseYou've already chosen the best response.0just a minus sign

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

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