Answer to Question #282523 in Real Analysis for Khaja

Question #282523
  1. Find the fourier cosine Series for the function f(x)=x^2 for 0<x<π
  2. Obtain a fourier expression for f(x)=x^2 for -π<x<π where f(x) is a even function
1
Expert's answer
2021-12-27T08:16:28-0500

1.


a0=1π0πx2dx=1π[x33]π0=π23a_0=\dfrac{1}{\pi}\displaystyle\int_{0}^{\pi}x^2dx=\dfrac{1}{\pi}[\dfrac{x^3}{3}]\begin{matrix} \pi \\ 0 \end{matrix}=\dfrac{\pi^2}{3}

an=2π0πx2cos(nx)dxa_n=\dfrac{2}{\pi}\displaystyle\int_{0}^{\pi}x^2\cos(nx)dx

x2cos(nx)dx=1nx2sin(nx)2nxsin(nx)dx\int x^2\cos(nx)dx=\dfrac{1}{n}x^2\sin(nx)-\dfrac{2}{n}\int x\sin(nx)dx

=1nx2sin(nx)+2n2xcos(nx)2n2cos(nx)dx=\dfrac{1}{n}x^2\sin(nx)+\dfrac{2}{n^2}x\cos(nx)-\dfrac{2}{n^2}\int \cos(nx)dx

=1nx2sin(nx)+2n2xcos(nx)2n3sin(nx)+C=\dfrac{1}{n}x^2\sin(nx)+\dfrac{2}{n^2}x\cos(nx)-\dfrac{2}{n^3}\sin(nx)+C

an=2π[x2nsin(nx)+2xn2cos(nx)2n3sin(nx)]π0a_n=\dfrac{2}{\pi}[\dfrac{x^2}{n}\sin(nx)+\dfrac{2x}{n^2}\cos(nx)-\dfrac{2}{n^3}\sin(nx)]\begin{matrix} \pi \\ 0 \end{matrix}

=4n2cos(nπ)=4(1)nn2=\dfrac{4}{n^2}\cos(n\pi)=\dfrac{4(-1)^n}{n^2}

Therefore, we have


f(x)π234cos(x)+cos(2x)49cos(3x)+...f(x)\sim \dfrac{\pi^2}{3}-4\cos(x)+\cos(2x)-\dfrac{4}{9}\cos(3x)+...

2.


a0=12π02πx2dx=12π[x33]2π0=4π23a_0=\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi}x^2dx=\dfrac{1}{2\pi}[\dfrac{x^3}{3}]\begin{matrix} 2\pi \\ 0 \end{matrix}=\dfrac{4\pi^2}{3}

an=1π02πx2cos(nx)dxa_n=\dfrac{1}{\pi}\displaystyle\int_{0}^{2\pi}x^2\cos(nx)dx

x2cos(nx)dx=1nx2sin(nx)2nxsin(nx)dx\int x^2\cos(nx)dx=\dfrac{1}{n}x^2\sin(nx)-\dfrac{2}{n}\int x\sin(nx)dx

=1nx2sin(nx)+2n2xcos(nx)2n2cos(nx)dx=\dfrac{1}{n}x^2\sin(nx)+\dfrac{2}{n^2}x\cos(nx)-\dfrac{2}{n^2}\int \cos(nx)dx

=1nx2sin(nx)+2n2xcos(nx)2n3sin(nx)+C=\dfrac{1}{n}x^2\sin(nx)+\dfrac{2}{n^2}x\cos(nx)-\dfrac{2}{n^3}\sin(nx)+C

an=1π[x2nsin(nx)+2xn2cos(nx)2n3sin(nx)]2π0a_n=\dfrac{1}{\pi}[\dfrac{x^2}{n}\sin(nx)+\dfrac{2x}{n^2}\cos(nx)-\dfrac{2}{n^3}\sin(nx)]\begin{matrix} 2\pi \\ 0 \end{matrix}

=4n2=\dfrac{4}{n^2}

Therefore, we have


f(x)4π23+4cos(x)+cos(2x)+49cos(3x)+...f(x)\sim \dfrac{4\pi^2}{3}+4\cos(x)+\cos(2x)+\dfrac{4}{9}\cos(3x)+...


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