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Complex Fourier series has a singular term

  1. Apr 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the complex Fourier series for f(t)=t(1-t), 0<t<1

    2. Relevant equations

    [tex]\sum_{n=-\infty}^{\infty}c_{n}e^{2in\pi t}[/tex]

    where [tex]c_{n}=\int_{0}^{1}f(t)e^{-2in\pi t}dt[/tex]

    3. The attempt at a solution

    I've worked out that c[tex]_{n}=-1/(2n^2 \pi^2)[/tex]. The problem is that for n=0, it is singular. Is there some way around this or does it mean that the complex Fourier series doesn't exist?
    I tried using maple to graph the series with the n=0 term omitted and it comes out to the right shape, but is shifted vertically down some, leading me to believe that the singular term should be replaced by a constant or something.
  2. jcsd
  3. Apr 6, 2009 #2
    [tex] c_0=\int_0^1 f(t)\,dt [/tex]
  4. Apr 6, 2009 #3
    Well that was easy, just like a real Fourier series. Thanks.
    I'm interested in knowing why that's the case though, I haven't seen anything about doing anything special for [tex] c_{0} [/tex] in anything I've seen about complex Fourier series.
  5. Apr 7, 2009 #4
    Just plug in n=0 before integrating instead of after!
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