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Complex Fourier Series

  1. Apr 30, 2012 #1
    Hi, I don't understand why does n goes from -∞ to +∞ in the complex Fourier series, but it goes from n=1 to n=+∞ in the real Fourier series?
  2. jcsd
  3. Apr 30, 2012 #2


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  4. Apr 30, 2012 #3


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    "Real Fourier Series" are in the form [itex]\sum a_ncos(nx)+ b_nsin(nx)[tex]
    cosine is an even function and sine is an odd function so that if we did use negative values for n, it wouldn't give us anything new: [itex]a_{-n}cos(-nx)+ b_n sin(-nx)= a_{-n}cos(nx)- b_{-n} sin(nx)[/itex] and would can then combine that with the corresponding "n" term: [itex]( a_n+ a_{-n})cos(nx)+ (b_n- b_{-n})sin(nx)[/itex]

    Another, but equivalent, way of looking at it is that [itex]cos(nx)= (e^{inx}+ e^{-inx})/2[/itex] and [itex]sin(nx)= (e^{inx}+ e^{-inx})/2i[/itex] so that sin(nx) and cosine(nx) with only positive n includes exponentials with both positive and negative n.
  5. May 1, 2012 #4
    Thanks :smile:
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