Author Topic: Fourier expansion not centering at origin  (Read 790 times)

Jingxuan Zhang

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Fourier expansion not centering at origin
« on: February 24, 2018, 05:44:53 PM »
In Promblems for 4.3 questions 4,5 ask for expansion on $[0,\pi]$. Are the same formulae valid there? or do we have to first make the change of variable or other manipulation? In particular for function such as $x(\pi-x)$, is there any trick that can shorten the direct computation?

Thanks!
 

Victor Ivrii

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Re: Fourier expansion not centering at origin
« Reply #1 on: February 24, 2018, 07:23:08 PM »
Complete Fourier series are used on any interval of the length $2l$ and formulae are the same (with integral over interval), but for incomplete ones interval must be either $[0,l]$ or $[-l,0]$, If it is not so––shift the origin.

Jingxuan Zhang

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Re: Fourier expansion not centering at origin
« Reply #2 on: February 24, 2018, 09:47:22 PM »
But then how would you justify the "double the integral" in computing the incomplete series coefficients?

Victor Ivrii

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Re: Fourier expansion not centering at origin
« Reply #3 on: February 25, 2018, 05:32:41 AM »
But then how would you justify the "double the integral" in computing the incomplete series coefficients?
To deal with incomplete series (sin or cosine) we use complete series on $[-l,l]$.

Jingxuan Zhang

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Re: Fourier expansion not centering at origin
« Reply #4 on: February 26, 2018, 06:51:03 AM »
To deal with incomplete series (sin or cosine) we use complete series on $[-l,l]$.

And imagine them to be even on $[-l,l]$, to deal with either case?