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Math Help - Fourier Series

  1. #1
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    Fourier Series

    I am having some trouble with the following problem:

    Show that the Fourier series:

    \frac{1}{2} a_{0} +\sum^{\infty}_{n=1} (a_{n} \cos nx +b_{n} \sin nx)

    can be written in the form

    \frac{1}{2} \rho_{0} +\sum^{\infty}_{n=1} \rho_{n} \cos (nx + \theta_{n})

    where \rho_{n} = \sqrt{a^{2}_{n} +b^{2}_{n} }. Express \theta_{n} in terms of a_{n} and b_{n}.

    Ok so I got down to the fourier series being equal to this:

    = \frac{1}{2\pi} \int^{\pi}_{-\pi} f(\theta) d\theta + \frac{1}{\pi} \int^{\pi}_{-\pi} f(\theta) \sum^{\infty}_{n=1}  \cos n(x - \theta) d\theta

    which is not exactly what we want especially the \cos n(x - \theta).
    Any suggestion??
    Also any idea how to do "Express \theta_{n} in terms of a_{n} and b_{n}."???
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  2. #2
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    For each n, write (a_n,-b_n) in terms of polar coordinates \rho_n,\,\theta_n. So a_n = \rho_n\cos\theta_n, b_n = -\rho_n\sin\theta_n. (For n=0, take \rho_0=a_0,\ \theta_0=0.) Then a_n\cos nx + b_n\sin nx = \rho_n(\cos nx\cos\theta_n -\sin nx\sin\theta_n) = \rho_n\cos(nx+\theta_n).
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  3. #3
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    I thought that in a Fourier Series the coefficients had to be:

    a_{n} = \frac{1}{\pi}\int^{\pi}_{- \pi}f(\theta)\cos n \theta d\theta
    <br />
b_{n} = \frac{1}{\pi}\int^{\pi}_{- \pi}f(\theta)\sin n \theta d\theta<br />

    So can I still just rewrite the way you show me??? Sorry I am just trying to make sure I have a good understanding of it all.

    Also to express \theta_{n} in terms of a_{n} and b_{n}, I am guessing it would be something like:

    \theta_{n}= \tan^{-1} \frac{b_{n}}{a_{n}}
    Last edited by ynn6871; March 30th 2009 at 11:05 AM.
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  4. #4
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    Quote Originally Posted by ynn6871 View Post
    I thought that in a Fourier Series the coefficients had to be:

    a_{n} = \frac{1}{\pi}\int^{\pi}_{- \pi}f(\theta)\cos n \theta d\theta
    <br />
b_{n} = \frac{1}{\pi}\int^{\pi}_{- \pi}f(\theta)\sin n \theta d\theta<br />

    So can I still just rewrite the way you show me??? Sorry I am just trying to make sure I have a good understanding of it all.
    Yes, but you are given a_n and b_n in the statement of the question (and you are not given the function f, except in terms of its Fourier coefficients). So those formulas for a_n and b_n are not relevant to this question.

    Quote Originally Posted by ynn6871 View Post
    Also to express \theta_{n} in terms of a_{n} and b_{n}, I am guessing it would be something like:

    \theta_{n}= \tan^{-1} \frac{b_{n}}{a_{n}}
    \theta_n will be either \tan^{-1} \bigl(\tfrac{b_{n}}{a_{n}}\bigr) or \tan^{-1} \bigl(\tfrac{b_{n}}{a_{n}}\bigr)+\pi, depending on the signs of a_{n} and b_{n}.
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