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Math Help - Sine and Cosine Fourier series

  1. #1
    Klo
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    Sine and Cosine Fourier series

    Hi!
    Im asked to write sine AND cosine series of sin(x) ; x \in <0,\pi)

    *what does that mean? i thought the fourier series can exist for the sine part being 0 OR the cosine being 0 OR both sine and cosine being included in the formula.. how do i do then one SINE and another COSINE serie of the same function?

    *Im confused with the condition: first of all it doesn't look like a periodic function to me, but im supposed to do a fourier transform on it...

    if i imagine it's a periodic function(somehow) then:
    *how does the function look like? is it the positive parts of sine one next to the other, thus with period = \pi ... or is it the complete sine with a constant 0 put in between the positive parts thus the period would be 2\pi?

    *Also, how do i know what the function looks like on x < 0 ? (considering it is periodic, is the function even or odd or none of these?

    Thanks!
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  2. #2
    MHF Contributor chisigma's Avatar
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    Probably is requested the Fourier expansion of the output of a 'full wave rectifier' ...

    f(x)= |\sin x| ,  - \pi < x < \pi (1)

    The function is 'even' so that there are only the terms in 'cosine' and is...

    \displaystyle f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} a_{n}\ \cos nx (2)

    ... where...

    \displaystyle a_{0} = \frac{2}{\pi}\ \int_{0}^{\pi} \sin x \ dx = \frac{4}{\pi}

    \displaystyle a_{n} = \frac{2}{\pi}\ \int_{0}^{\pi} \sin x \ \cos nx \ dx = \left\{\begin{array}{ll} - \frac{4}{\pi\ (n^{2}-1)}  ,\,\, n \ even\\{}\\0 ,\,\, n \ odd \end{array}\right. (3)

    Kind regards

    \chi \sigma
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  3. #3
    Klo
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    It definitely makes much more sense now... so i guess the question to find the sine fourier form would lead to b_k = 0 which would be the result - silly.
    Thank you chisigma
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  4. #4
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    No, the sine Fourier series for sin(x) would have all b_k= 0 except for k= 1: b_1= 1 because the sine Fourier sine series for sin(x) is precisely "sin(x)".
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