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Fourier Series for a piecewise function help

  1. Apr 24, 2013 #1
    1. The problem statement, all variables and given/known data

    I'm trying to find a Fourier series for the piecewise function where f(x)=

    [itex] 0 \in -\pi \leq x \leq 0 [/itex]
    [itex] -1 \in 0 \leq x \leq \frac{\pi}{2} [/itex]
    [itex] 1 \in \frac{\pi}{2} \leq x \leq \pi [/itex]



    2. Relevant equations

    [tex]a_{n} = \frac{1}{\pi} \int_{0}^{2\pi}\cos(nx)y(x)\,dx[/tex]
    [tex]b_{n} = \frac{1}{\pi} \int_{0}^{2\pi}\sin(nx)y(x)\,dx[/tex]


    3. The attempt at a solution

    I found the pattern that every even a is 0, so that becomes
    [tex]a_{m} = \sum_{m=1}^{\infty}\frac{(-1)^{m}2}{(2m-1)\pi}[/tex]

    and the b coefficients are 0 when n is odd and 0 for every other even n, so that becomes
    [tex]b_{m} = \sum_{m=1}^{\infty}\frac{-2}{\frac{4m-2}{2}\pi}[/tex]

    however when I plot this, the plot between -pi and -pi/2 is switched with the plot between pi/2 and pi

    I attached a picture for reference along with a plot of f(x)

    Any ideas of where i went wrong?
     
    Last edited: Apr 24, 2013
  2. jcsd
  3. Apr 24, 2013 #2

    vela

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    Not really since you didn't show your work. Just posting your final, probably wrong answer isn't very helpful.
     
  4. Apr 24, 2013 #3
    Whoops sorry, I forgot I didn't do this all in mathematica:

    [itex]a_{0} = 0 [/itex]

    [itex]a_{1} = \frac{-2}{\pi}[/itex]

    [itex]a_{2} = 0[/itex]

    [itex]a_{3} = \frac{2}{3\pi}[/itex]

    [itex]a_{4} = 0[/itex]

    [itex]a_{5} = \frac{-2}{5\pi}[/itex]

    [itex]a_{6} = 0 [/itex]


    [itex]b_{1} = 0 [/itex]

    [itex]b_{2} = \frac{-2}{\pi}[/itex]

    [itex]b_{3} = 0 [/itex]

    [itex]b_{4} = 0 [/itex]

    [itex]b_{5} = 0 [/itex]

    [itex]b_{6} =\frac{-2}{3\pi}[/itex]

    [itex]b_{7} = 0 [/itex]

    [itex]b_{8} = 0 [/itex]

    [itex]b_{9} = 0 [/itex]

    [itex]b_{10} =\frac{-2}{5\pi}[/itex]

    [itex]b_{11} = 0 [/itex]

    [itex]b_{12} = 0 [/itex]

    [itex]b_{13} = 0 [/itex]

    [itex]b_{14} = \frac{-2}{7\pi}[/itex]
     
    Last edited: Apr 24, 2013
  5. Apr 24, 2013 #4
    And I figured out the summations from just looking at these in terms of n:

    [itex]a_{n} = \sum\frac{(-1)^{n}2}{n\pi}[/itex]

    [itex]b_{n} = \sum\frac{-2}{\frac{n}{2}\pi}[/itex]

    and then to get rid of the 0s I made it in terms of m:

    [itex]a_{m} = \sum\frac{(-1)^{m}2}{(2m-1)\pi}[/itex]

    [itex]b_{m} = \sum\frac{-2}{\frac{4m-2}{2}\pi}[/itex]

    The function is then:

    [tex]g(x) = \sum_{m=1}^{\infty} a_m Cos((2m-1)x) + \sum_{m=1}^{\infty} b_m Sin((2m-1)x)[/tex]
     
    Last edited: Apr 24, 2013
  6. Apr 24, 2013 #5
    I figured out what I did wrong, but thanks!
     
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