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Derivate of generalized function

  1. Apr 25, 2008 #1
    Hi,

    I have to show that if the derivate f'(x) of a generalized function f(x) is defined by the sequence f'_n(x) where f(x) is defined

    [tex]f_n(x)[\tex]

    then

    [tex]\int_{-\infty}^{\infty}f'(x)F(x) dx = - \int_{-\infty}^{\infty}f(x)F'(x) dx [/tex]

    I use the limits for generalized functions and get

    [tex]lim_{n \to \infty} \int_{-\infty}^{\infty}f'_n(x)F(x) dx = - \int_{-\infty}^{\infty}f_n(x)F'(x) dx [/tex]

    which should show the above - I am a liltte confused where the minus sign comes from?
    [tex]- \int_{-\infty}^{\infty}f(x)F'(x) dx [/tex]

    Any help appreciated - thanks in advance
     
  2. jcsd
  3. Apr 25, 2008 #2

    lurflurf

    User Avatar
    Homework Helper

    it is from integration by parts
    let f,F be functions
    d(fF)=(fF)'dx=f'Fdx+fF'dx
    f'Fdx=d(fF)-fF'dx
    integrating gives (assume f,F->0)
    ʃf'Fdx=ʃd(fF)-ʃfF'dx
    ʃd(fF)=0 so
    ʃf'Fdx=-ʃfF'dx
    now extent this to generalized functions by limits
     
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