Complex Fourier Series & Full Fourier Series

Quote:

Originally Posted by kingwinner

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:
http://www.efunda.com/math/fourier_s...ier_series.gif
where
http://www.efunda.com/math/fourier_s...oefficient.gif

Complex Fourier series is:
http://www.efunda.com/math/fourier_s...es/complex.gif
where
http://www.efunda.com/math/fourier_s...oefficient.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...

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).