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Math Help - Complex Fourier Series & Full Fourier Series

  1. #1
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    Complex Fourier Series & Full Fourier Series

    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:

    where


    Complex Fourier series is:

    where

    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...

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

    [note: also under discussion in s.o.s. math cyberboard]
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  2. #2
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    Quote Originally Posted by kingwinner View Post
    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:

    where


    Complex Fourier series is:

    where

    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...

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

    [note: also under discussion in s.o.s. math cyberboard]
    Use the fact that e^{\frac{i n\pi x}{L}}= cos\left(\frac{n\pi x}{L}\right)+ i sin\left(\frac{n\pi x}{L}\right)

    The sum "collapses" from "-infinity to infinity" to "0 to infinity" because cos(-x)= cos(x) and sin(-x)= -sin(x).
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