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Sum of a serie involving Fourier coefficients

  1. Aug 23, 2013 #1
    1. The problem statement, all variables and given/known data

    Let [tex]\hat{u}_k[/tex] the Fourier coefficients of 2-periodic function [tex]u(t)=t[/tex] with [tex]t\in [0,2)[/tex]. Evaluate the sum of the serie:

    [tex]\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t}[/tex] for [tex]t= 2[/tex]


    Ok, I think there is a trick that I don't know...

    [tex]\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t}[/tex] for [tex]t= 2[/tex]

    becomes

    [tex]\sum_{k=-\infty}^{\infty}\hat{u}_k [/tex]

    What can i do now? Any help will be appreciated :)
     
  2. jcsd
  3. Aug 23, 2013 #2

    LCKurtz

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    Draw the graph of the periodic extension of ##f(t) = t## on [0,2). What happens at ##t = 2##? What do you know about the sum of the series there?
     
  4. Aug 25, 2013 #3
    Well.. u(t) is discontinue for t=2 so the Fourier serie converges to [u(2+)+ u(2-)]/2 where

    [tex]u(2+)= \lim_{t\to 2^+}u(t)= 0[/tex]
    [tex]u(2-)= \lim_{t\to 2^-}u(t)= 2[/tex]

    so

    [tex]\sum_{k=-\infty}^{\infty}\hat{u}_k= 1[/tex]

    Right?
     
    Last edited: Aug 25, 2013
  5. Aug 25, 2013 #4

    LCKurtz

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    Yes. That's the idea.
     
  6. Aug 25, 2013 #5
    Thank you so much LCKurtz :)
     
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