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Dividing both sides by a Dirac delta function - ok?

  1. Sep 8, 2010 #1
    Suppose I wind up with the relation

    [tex]f(x)\delta (x-x')=g(x)\delta (x-x')[/tex]

    true for all x'.

    Can I safely conclude that f(x) = g(x) (for all x)? Or am I overlooking something? this is a little too close to dividing both sides by zero for comfort.
     
    Last edited: Sep 8, 2010
  2. jcsd
  3. Sep 8, 2010 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    You could integrate your relation over any interval containing x, and use the definition of the delta function (or rather, distribution):

    If
    [tex]
    f(x)\delta (x-x')=g(x)\delta (x-x')
    [/tex]
    then
    [tex]\int_{x - \epsilon}^{x + \epsilon} f(x)\delta (x-x') \, \mathrm{d}x' = \int_{x - \epsilon}^{x + \epsilon} g(x)\delta (x-x') \, \mathrm{d}x'
    [/tex]
    which evaluates (by definition of the delta) to
    [tex]f(x) = g(x)[/tex]
     
  4. Sep 8, 2010 #3
    Thank you!
     
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