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Math Help - find Fourier coefficients

  1. #1
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    find Fourier coefficients

    Fourier coefficients
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  2. #2
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    Let's see. It's been a while since I worked a Fourier.

    \begin{Bmatrix}f(x)=\frac{x}{2}, \;\ {-\pi}<x<0\\f(x)=0, \;\ 0<x<{\pi}\end{Bmatrix}

    a_{0}=\frac{1}{\pi}\int_{-\pi}^{0}\frac{x}{2}dx=\frac{-\pi}{4}

    a_{n}=\frac{1}{\pi}\int_{-\pi}^{0}\frac{x}{2}cos(nx)dx

    = \frac{-cos(n\pi)}{2n^{2}\pi}-\frac{sin(n\pi)}{2n}+\frac{1}{2n^{2}\pi}

    But cos(n\pi)=(-1)^{n} \;\ and \;\ sin(n\pi)=0

    So, we have a_{n}=\frac{1-(-1)^{n}}{2n^{2}\pi}

    b_{n}=\frac{1}{\pi}\int_{-\pi}^{0}\frac{x}{2}sin(nx)dx

    = \frac{sin(n\pi)}{2n^{2}\pi}-\frac{cos(n\pi)}{2n}=\frac{-(-1)^{n}}{2n}

    Check and make sure I didn't err somewhere.
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