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Fourier series on a general interval [a, a + T]

  1. Apr 9, 2012 #1
    Are there general formulas for Fourier coefficients on an integral [a, a + T], where T is the period. There is a general formula for the coefficients of exponential Fourier series. Are there general formulas for the coefficients of the trigonometric Fourier series that would work on any interval [a, a + T]?

    http://en.wikipedia.org/wiki/Fourie...general_interval_.5Ba.2C.C2.A0a_.2B_.CF.84.5D
     
  2. jcsd
  3. Apr 9, 2012 #2

    Ray Vickson

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    Of course there are; the article http://en.wikipedia.org/wiki/Fourier_series contains all the formulas you need. Just make sure you do not give up too soon; you need to go almost half way through the article.

    RGV
     
  4. Apr 9, 2012 #3
    I cannot find it. I could not find it anywhere. I am looking for the most general formulas. Like the one for the exponential Fourier series (formulas which can be applied to any interval of the form [a, a + T]). Usually, there are different formulas for different intervals. I am looking the the most general one.
     
  5. Apr 9, 2012 #4

    Ray Vickson

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    Look again. The link I gave you has a section entitled "Fourier Series on an Interval [a,a+T]". I just now looked at it again! Anyway, if you cannot find it on the web you can find it in many books.

    RGV
     
  6. Apr 10, 2012 #5
    Yes, I know. I actually have the link to that section in my first post. I am looking for similar formulas for the coefficients of the trigonometric Fourier series.
     
  7. Apr 10, 2012 #6

    Ray Vickson

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    It is simple.
    (1) Either express the exponentials in terms of sin and cos and use the formulas given in the link (exactly the same way as one does for the interval [-π,π]); or (2) change variables to x = α + βt, so that a <= t <= a+T becomes -π <= x <= π, then just change variables in the integrations, etc.

    RGV
     
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