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Fourier coefficient calculations

  1. Feb 2, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    I've been following closely a book on PDE's working through lots of steps but here I'm stuck.
    Basically I have the relation [itex]f(x)= \sum _{n=1}^\infty F_n \sin \left ( \frac{n\pi x }{L} \right )[/itex]. I want to calculate the Fourier coefficients [itex]F_n[/itex].
    I look at the definition in the same book and I see that if [itex]f(x)= \sum _{n=1}^\infty c_n g_n(x)[/itex] then [itex]c_n=\frac{1}{||g_n||^2} \int _a ^b f(x)g_n (x)dx[/itex].
    Since I'm solving the 1 dimensional wave equation between [itex]x=0[/itex] and [itex]x=L[/itex], the limits of the integral are in my case [itex]0[/itex] and [itex]L[/itex].
    I get that [itex]F_n =\frac{1}{|| \sin \left ( \frac{n\pi x }{L} \right )||^2} \int _0^L f(x) \sin \left ( \frac{n\pi x }{L} \right ) dx[/itex]. I don't really know how to calculate the modulus squared of the denominator. When I look in the book it says that [itex]F_n=\frac{2}{L} \int _0^L f(x) \sin \left ( \frac{n\pi x }{L} \right ) dx[/itex]. I've absolutely no idea how he did this calculation.


    2. Relevant equations
    I don't know if there are any other than the one I posted.


    3. The attempt at a solutionOnly thoughts so far...
    Any clarification is welcome.
     
  2. jcsd
  3. Feb 2, 2012 #2

    jbunniii

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    What is the definition of the modulus squared of a function?
     
  4. Feb 2, 2012 #3

    fluidistic

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    Ok thank you, I just found out the result... I didn't know to tell the truth, now I know. Problem solved.
     
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