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Computing Fourier series

  1. Mar 14, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the Fourier series for [tex]y(x)=\begin{cases}
    A\sin(\frac{2\pi x}{L}) & 0\leq x\leq\frac{L}{2}\\
    0 & \frac{L}{2}\leq x\leq L\end{cases}[/tex]

    2. Relevant equations
    [tex]B_{n}=\frac{2}{L}\int_{0}^{L}y(x)\sin(\frac{n\pi x}{L})dx[/tex]

    3. The attempt at a solution

    [tex]B_{n}=\frac{2}{L}\int_{0}^{L/2}A\sin(\frac{2\pi x}{L})\sin(\frac{n\pi x}{L})dx[/tex]

    [tex]=\frac{2}{L}\int_{0}^{\frac{L}{2}}A\sin(\frac{\pi x}{L/2})\sin(\frac{(n/2)\pi x}{L/2})dx[/tex]

    [tex]=\frac{1}{p}\int_{0}^{p}A\sin(\frac{\pi x}{p})\sin(\frac{\frac{n}{2}\pi x}{p})dx[/tex]

    0 & \frac{n}{2}=1\\
    \frac{A}{2} & \frac{n}{2}\in\mathbb{Z}\text{ and }\frac{n}{2}\neq1\end{cases}[/tex]

    So this takes care of the even values of n, but I'm not sure what to do when n is odd.

    [tex]=\frac{1}{p}\int_{0}^{p}A\sin(\frac{\pi x}{p})\sin(\frac{(\frac{2k+1}{2})\pi x}{p})dx[/tex]

    [tex]=\frac{1}{p}\int_{0}^{p}A\sin(\frac{\pi x}{p})\sin(\frac{(k\pi x+\frac{1}{2}\pi x}{p})dx[/tex]

    [tex]=\frac{1}{p}\int_{0}^{p}A\sin(\frac{\pi x}{p})[\sin(\frac{k\pi x}{p})\cos(\frac{\frac{1}{2}\pi x}{p})+\cos(\frac{k\pi x}{p})\sin(\frac{\frac{1}{2}\pi x}{p})]dx[/tex]

    I'm not really sure if this is going anywhere. The final answer should be [tex]B_{n}=\frac{4A(-1)^{\frac{n+1}{2}}}{\pi(n^{2}-4)}[/tex], where n is odd.

    Any ideas? Thank you.
    Last edited: Mar 14, 2010
  2. jcsd
  3. Mar 14, 2010 #2


    User Avatar
    Science Advisor
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    Gold Member

    Try using a trig identity at this point.

    [tex]\sin(a) \sin (b) = ?[/tex]
  4. Mar 14, 2010 #3
    Thanks, that worked perfectly!
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