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UnkleRhaukusBest 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*}\]
 one year ago

UnkleRhaukusBest 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*}
 one year ago

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

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

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

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

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

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

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

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

UnkleRhaukusBest 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*}
 one year ago

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

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

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