Sine series for cos(x) (FOURIER SERIES)

  1. I was finally able to figure out how to find the sine series for cos(x), but only for [0,2pi]. A question i have though is what is the interval of validity? is it only [0,pi]?
    Ie if I actually had to sketch the graph of the sum of the series, on all of R, would I have cosine or just a periodic extension of cosine from [0,2pi]?
     
  2. jcsd
  3. CompuChip

    CompuChip 4,299
    Science Advisor
    Homework Helper

    Hey Konrad, welcome to PF.
    I am afraid I don't entirely understand your question. You say that you have managed to write cos(x) as a(n infinite) sum of sines on the interval [0, 2pi].
    But both cos(x) and the sines you used are periodic with period 2pi, aren't they? So if the infinite sum converges to cos(x) on an interval with a length of at least one period, then it converges to cos(x) everywhere, doesn't it?
     
  4. LCKurtz

    LCKurtz 8,369
    Homework Helper
    Gold Member

    If you have expanded cos(x) in a sine series using [itex]p = 2\pi[/itex] in the formula
    [tex] b_n = \frac 2 p \int_0^p \cos(x) \sin{\frac{n\pi x}{p}}\,dx[/tex]
    what you are representing is the [itex]4\pi[/itex] periodic odd extension of cos(x).

    [edit - corrected typo: bn not an]
     
    Last edited: Nov 30, 2009
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