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Math Help - Fourier sine series...

  1. #1
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    Fourier sine series...

    Let f(x) = cos x on 0<= x <= pi. Calculate the Fourier sine series representation of f(x).

    Can anybody possibly start me off on this question or give me any help at all as I don't understand my lecture notes on this topic at all

    Thanks in advance!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by hunkydory19 View Post
    Let f(x) = cos x on 0<= x <= pi. Calculate the Fourier sine series representation of f(x).

    Can anybody possibly start me off on this question or give me any help at all as I don't understand my lecture notes on this topic at all

    Thanks in advance!
    What do you mean by fourier sine? is it the sine part of the Fourier Series?
    if does, this should be a good hint:

    If f(x) is an even function, then b_n = 0, i.e f(x) \approx \frac{a_0}{2} + \sum a_n \, \cos nx

    If f(x) is an odd function, then a_n = 0, i.e f(x) \approx \frac{a_0}{2} + \sum b_n \, \sin nx
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  3. #3
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    Thanks kalagota,

    So a_n = 0

    and b_n = \frac{2}{\pi}\int^\pi_0 cos x sin nx \, \mathrm{d}x

    But how do integrate this, I tried doing it by parts but realised that it will just go on forever!
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  4. #4
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    Krizalid's Avatar
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    You can turn a product of sine and cosine into a sum, do you know that formula? Applyin' that the rest follows.
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  5. #5
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    You need to know the orthogonal properties of sine and cosine.
    (If L>0 we have):

    \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m.

    \int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}.

    \int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0.

    Here L=\pi.
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You need to know the orthogonal properties of sine and cosine.
    (If L>0 we have):

    \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m.

    \int_{-L}^L \cos \frac{\pi n x}{L} \cos \frac{\pi m x}{L} ~ dx= \mbox{ same as above}.

    \int_{-L}^L \sin \frac{\pi n x}{L}\cos \frac{\pi m x}{L} ~ dx = 0.

    Here L=\pi.
    indeed! and the first two are 0 if m \ne n

    you know i didn't know about these rules until after my differential equations class was over! i always had to do it the hard way (the product to sum formulas)
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