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Fourier Series of an Odd Piecewise function

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Fourier Series of the following function f(x).

    f(x) is -1 for -.5<x<0
    f(x) is 1 for 0<x<.5


    2. Relevant equations
    b[itex]_{n}[/itex] = [itex]\frac{1}{L}[/itex][itex]\int[/itex][itex]^{L}_{-L}[/itex]f(x)sin(nπx/L)dx
    Where L is half the period.


    3. The attempt at a solution
    Graphing the solution, I know that it is odd, which is why I didn't include the given function for the "even" cos(nπ) portion of the Fourier series.
    Since the period is 1, L would be .5. So the function would look like this:

    2[itex]\int[/itex][itex]^{0}_{-.5}[/itex](–sin(2nπx)dx) + 2[itex]\int[/itex][itex]^{.5}_{0}[/itex](sin(2nπx)dx)

    To cut some work short, I get to:
    [itex]\frac{1}{πn}[/itex] - [itex]\frac{cos(nπ}{πn}[/itex] - [itex]\frac{cos(nπ}{πn}[/itex] + [itex]\frac{1}{πn}[/itex]

    Which becomes:
    [itex]\frac{4}{πn}[/itex]sin[itex]^{2}[/itex]([itex]\frac{πn}{2}[/itex])

    I check my solution on Mathematica using these commands:
    a = If[-.5 < x < 0, -1, If[0 < x < .5, 1]]
    FourierTrigSeries[a, x, 5]

    And see that there are terms for values where my b[itex]_{n}[/itex] should be zero.

    EDIT: I was looking at another attempt I had made to this problem, so I corrected for that.
     
    Last edited: Apr 28, 2012
  2. jcsd
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