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Complex Fourier Series & Full Fourier Series

  1. Dec 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Claim: If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series.

    This is a claim stated in my textbook, but without any proof. I also searched some other textbooks, but still I have no luck of finding the proof.
    I've already spent an hour thinking about how to show that this is true, but still I am not having much progress. Here is what I've got so far:

    Full Fourier series is:
    fourier_series.gif
    where
    coefficient.gif

    Complex Fourier series is:
    complex.gif
    where
    complex_coefficient.gif
    And now I am having trouble with this...how can I use the last part to show that if f(x) is REAL-valued, the complex Fourier series can be reduced to the full Fourier series. Can someone please show me how to continue from here? I also don't see how a sum from negative infinity to infinity (for complex Fourier series) can possibly be reduced to a sum from 0 to infinity (for full Fourier series). It seems like I have no hope...

    2. Relevant equations
    As shown above

    3. The attempt at a solution
    As shown above

    I am really frustrated now and any help is very much appreciated! :)
     

    Attached Files:

    Last edited: Dec 9, 2009
  2. jcsd
  3. Dec 9, 2009 #2

    HallsofIvy

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    Science Advisor

    An obvious first step is to use the fact that [itex]e^{in\pi x/L}= cos(n\pi x/L)+ i sin(n\pi x/L)[/itex]. Multiply it out and use the fact that cos(-x)= cos(x), sin(-x)= sin(x).
     
  4. Dec 9, 2009 #3
    OK, the following is what I got.
    (for simplicity I am taking the interval to be from -pi to pi)

    pde3.JPG

    Is this a correct proof??

    Thanks!
     
  5. Dec 9, 2009 #4
    The claim is
    "If f(x) is a REAL-valued function on x E [-L,L], then the full Fourier series is exactly equivalent to the complex Fourier series."

    But nowhere in the proof have I assumed f(x) is real-valued. Is it absolutely necessary for f(x) to be REAL-valued in order to prove that the full Fourier series is exactly equivalent to the complex Fourier series??
     
  6. Dec 10, 2009 #5

    HallsofIvy

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    Staff Emeritus
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    That's because the equality of those sums does not depend upon real or complex numbers. We require that F be real valued in order to have the coefficients real numbers.
     
  7. Dec 10, 2009 #6
    But looking at my proof above, I believe that the full Fourier series and the complex Fourier series are equivalent in general, even when f(x) is complex-valued. Right??
     
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