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Homework Help: Proof for this

  1. Sep 6, 2005 #1
    Show that [tex]x \frac{d(\delta (x))}{dx} = -\delta (x)[/tex]
    where [itex]\delta (x)[/itex] is a Dirac delta function.

    My work:

    Let f(x) be a arbitrary function. Using integration by parts:
    [tex]\int_{-\infty}^{+\infty}f(x)\left (x \frac{d(\delta (x))}{dx}\right)dx = xf(x)\delta (x)\vert _{-\infty}^{+\infty} - \int_{-\infty}^{+\infty}d\left (\frac{xf(x) \delta (x)}{dx}\right)dx[/tex]

    The first term is zero, since [itex]\delta (x) = 0 [/itex]
    at [itex]-\infty, +\infty[/itex].
    How is the second term evaluated?
     
  2. jcsd
  3. Sep 6, 2005 #2

    arildno

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    Well, if we are to do this symbol manipulation properly, you should have:
    [tex]\int_{-\infty}^{\infty}xf(x)\delta{'}dx=xf(x)\delta\mid_{-\infty}^{\infty}-\int_{-\infty}^{\infty}(f(x)+xf'(x))\delta(x)dx=-(f(0)+0*f'(0))=-f(0)[/tex]
     
  4. Sep 7, 2005 #3
    Wow, thanks Arildno! I got it!
     
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