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Calculate the Fourier series of the function

  1. Sep 10, 2012 #1
    Calculate the Fourier series of the function $f$ defined on the interval [itex][\pi, -\pi][/itex] by
    $$
    f(\theta) =
    \begin{cases} 1 & \text{if} \ 0\leq\theta\leq\pi\\
    -1 & \text{if} \ -\pi < \theta < 0
    \end{cases}.
    $$
    [itex]f[/itex] is periodic with period [itex]2\pi[/itex] and odd since [itex]f[/itex] is symmetric about the origin.
    So [itex]f(-\theta) = -f(\theta)[/itex].
    Let [itex]f(\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{in\theta}[/itex].
    Then [itex]f(-\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{-in\theta} = \cdots + a_{-2}e^{2i\theta} + a_{-1}e^{i\theta} + a_0 + a_{1}e^{-i\theta} + a_{2}e^{-2i\theta}+\cdots[/itex]
    [itex]-f(\theta) = \sum\limits_{n = -\infty}^{\infty}a_ne^{-in\theta} = \cdots - a_{-2}e^{-2i\theta} - a_{-1}e^{-i\theta} - a_0 - a_{1}e^{i\theta} - a_{2}e^{2i\theta}-\cdots[/itex]

    [itex]a_0 = -a_0 = 0[/itex]

    I have solved many Fourier coefficients but I can't think today.

    What do I need to do next?
     
  2. jcsd
  3. Sep 10, 2012 #2

    micromass

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    Why don't you just find the Fourier coefficients by using the usual formula?? That is

    [tex]a_n=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}dx[/tex]
     
  4. Sep 10, 2012 #3

    LCKurtz

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    Or use the half-range sine expansion since it is an odd function.
     
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