How is the constant pi/L deduced in Fourier series?

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How is pi/L part deduced in (n*pi*x)/L?
 

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mathman
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Essentially it has to do with the interval that the series is defined for. The length of the interval is 2L while [itex]\pi[/itex] is the normalization (based on the fact that sines and cosines have period [itex]2\pi[/itex]).
 
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fourierSeries.png

How is pi/L part deduced in (n*pi*x)/L?
We're requiring that the fundamental frequency is periodic in space, with spatial period ##2L##. So we have for the fundamental frequency: $$\cos \omega x = \cos(\omega(x+2L)) = \cos(\omega x + 2L \omega) \Rightarrow 2L \omega = 2\pi \Rightarrow \omega = \frac{\pi}{L}$$
 
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Thank you guys, I understand now.
 

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