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Math Help - Integrating a fourier series

  1. #1
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    May 2010
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    Integrating a fourier series

    Hi there. I wanted to intagrate the fourier series for g(t)=t^2 to get the fourier series for f(x)=t^3
    So I thought making something like:

    f(t)=3\int_0^t x^2 dx

    I know that g(t)=t^2\sim \frac{p^2}{3}+\sum_{n=1}^{\infty}\frac{4p^2(-1)^n}{n^2\pi^2}\cos\left (\frac{n\pi t}{p}\right)

    p is for the period.

    Then
    f(t)\sim 3\left [\int_0^tp^2dx+\sum_{n=1}^{\infty}\frac{4p^2(-1)^n}{n^2\pi^2}\int_0^t\cos\left (\frac{n\pi t}{p}\right)dx \right ]
    f(t)\sim 3\left [p^2t+\sum_{n=1}^{\infty}\frac{4p^3(-1)^n}{n^3\pi^3}\sin\left (\frac{n\pi t}{p}\right)dx \right ]

    Now this is wrong, but I don't know why. What I get with this looks like x, I think that's because of the term p^2t. But I don't know what I'm doing wrong. Perhaps I have to do something else, but I don't know what, it actually doesn't look pretty much like a Fourier series.
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  2. #2
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    May 2011
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    Well, you DID lose the factor of a 1/3 in your frequency-0 term. You multiplied p^2/3 by 3 and integrated it w.r.t. t and somehow ended up with 3p^2*t, in other words you multiplied it by 9 when you should have only multiplied it by 3. It should only be p^2*t.
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  3. #3
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    You're right, I've missed that, but anyway that's not the problem. I just can't use the series expansion that I was trying to use, I know that now :P
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