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ece205: fix bug

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eggy 2023-11-21 22:04:05 -05:00
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docs/2a/ece205.md
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@ -395,7 +395,7 @@ Thus if a Fourier series on $(0,L)$ exists, it can be expressed as either a **Fo
We have $L=\pi,a=\sqrt 2$. We have $L=\pi,a=\sqrt 2$.
\begin{align*} \begin{align*}
u(x,t)&=\sum^\infty_{n=1}\alpha_ne^{-left(\frac{n\pi\sqrt 2}{\pi}\right)^2t}\sin(\frac{n\pi x}{\pi}) u(x,t)&=\sum^\infty_{n=1}\alpha_ne^{left(\frac{n\pi\sqrt 2}{\pi}\right)^2t}\sin(\frac{n\pi x}{\pi})
&=\sum^\infty_{n=1}\alpha_ne^{-2n^2t}\sin(nx) \\ &=\sum^\infty_{n=1}\alpha_ne^{-2n^2t}\sin(nx) \\
\alpha_n&=\frac 2 L\int^L_0f(x)\sin(\frac{n\pi x}{L})dx \\ \alpha_n&=\frac 2 L\int^L_0f(x)\sin(\frac{n\pi x}{L})dx \\
&=\frac2\pi\int^{\pi/2}_0\frac\pi 2\sin(nx)dx+\frac2\pi\int^\pi_{\pi/2}(x-\frac\pi2\sin(nx)dx \\ &=\frac2\pi\int^{\pi/2}_0\frac\pi 2\sin(nx)dx+\frac2\pi\int^\pi_{\pi/2}(x-\frac\pi2\sin(nx)dx \\