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Math Help - Fourier Series

  1. #1
    Super Member Aryth's Avatar
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    Fourier Series

    I just need to find a function representation for the following graph. I'm not sure why I can't figure it out, but nothing I've tried yields a correct series. :\

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    Quote Originally Posted by Aryth View Post
    I just need to find a function representation for the following graph. I'm not sure why I can't figure it out, but nothing I've tried yields a correct series. :\

    Could we see your working please?

    The first thing you should do is create an analytical equation for the function in the period  0 \leq x \leq \pi . It's a straight line graph.
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    Super Member Aryth's Avatar
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    Specifically I need the range -\pi \leq x \leq \pi

    I'm not terribly great with these types of functions... But I got:

    10\left(\frac{x}{\pi} + 1\right) \ for \ -\pi \leq x < 0

    and

    \frac{10x}{\pi} \ for \ 0 < x \leq \pi

    I think I recognize the graph as a sawtooth wave... But I don't know how to formulate one...
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    Super Member Aryth's Avatar
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    Anyone?
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    Quote Originally Posted by Aryth View Post
    Specifically I need the range -\pi \leq x \leq \pi

    I'm not terribly great with these types of functions... But I got:

    10\left(\frac{x}{\pi} + 1\right) \ for \ -\pi \leq x < 0

    and

    \frac{10x}{\pi} \ for \ 0 < x \leq \pi

    I think I recognize the graph as a sawtooth wave... But I don't know how to formulate one...
    What you have done is correct. All you must do now is integrate them to find the Fourier coefficients, and since your function is piecewise, you must split the integral into 2 integrals for each interval.

    For example:

     A_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx = \frac{1}{\pi} \bigg( \int_{-\pi}^0 \big(10(\frac{x}{\pi} + 1\big)) \cos(nx) \, dx + \int_{0}^{\pi} 10\big(\frac{x}{\pi}\big) \cos(nx) \, dx \bigg)
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