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Complex fourier series

  1. Feb 26, 2012 #1
    The fourier series can also be written as:

    f(x) = Ʃcr*exp(r*2π*i*x/L) where sum if from -∞ to ∞

    My book says this at least, but I can't really determine the realitionship between the coefficients of an ordinary fourier and the complex one. How do you get rid of the i that would appear in front of every sin factor, and how do you overall translate the coefficients cr to ar and br of an ordinary fourier series?
  2. jcsd
  3. Feb 26, 2012 #2


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    Use Cauchy's formula:
    [itex]e^{2ir\pi x/L}= cos(2r\pi x/L)+ i sin(2r\pi x/L)[/itex]

    Cosine is an even function and sine is an odd function so
    [itex]e^{-\pi x/L}= cos(2r\pi x/L)- i sin(2r\pi x/L)[/itex]
    which is why you do not need negative values of r in the sine, cosine series.

    [tex]\sum_{r=-\infty}^\infty a_re^{2ir\pi x/L}= (a_r+ a_{-r}) cos(2r\pi x/L)+ (a_r- a_{-r})i sin(2r\pi x/L)[/tex]
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