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Fourier series convergence 2

  1. Jun 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Screenshot2012-06-17at15610AM.png

    Screenshot2012-06-17at15615AM.png



    3. The attempt at a solution

    Obviously brackets mean something other than parentheses because .5[0 + 0] ≠ .5
     
  2. jcsd
  3. Jun 17, 2012 #2

    Mark44

    Staff: Mentor

    The brackets mean the same thing as parentheses...

    I was unable to "Dirichlet's theorem" on the web, except as it applies to number theory and prime numbers. In Kreyszig's Advanced Engineering Mathematics, 3rd Ed., there is a theorem (identified only as Theorem 1) in the section titled "Fourier Series. Euler Formulas," and I believe that's the one you're mentioning.

    The numbers in parentheses are, respectively, the usual definition of a Fourier series, and the formulas for the coefficients of the series.

    The theorem is talking more about the function represented by the Fourier series, and not so much about the function the series represents. At most points, the two are identical, but at discontinuities, they are different.

    The theorem doesn't say that f(##-\pi/2##) = 1/2. It says that the value of the series at x = ##-\pi/2## is 1/2.
     
  4. Jun 17, 2012 #3

    LCKurtz

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    That's just a typo. It should read$$\frac 1 2[f(0^+)+f(0^-) =\frac 1 2 [0+1]=\frac 1 2$$and it isn't f(0); it is the value of the FS at 0.
     
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