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A question about Dirac Delta Function

  1. Sep 9, 2014 #1
    For proving this equation:

    [tex]
    \delta (g(x)) = \sum _{ a,\\ g(a)=0,\\ { g }^{ ' }(a)\neq 0 }^{ }{ \frac { \delta (x-a) }{ \left| { g }^{ ' }(a) \right| } }
    [/tex]

    We suppose that
    [tex] g(x)\approx g(a) + (x-a)g^{'}(a) [/tex]

    Why for Taylor Expansion we just keep two first case and neglect others? Are those expressions so small? if yes how we can explain it?
     
  2. jcsd
  3. Sep 9, 2014 #2
    OK. Let me explain it to you. We start from decomposing the integral

    ##\int_{+\infty}^{-\infty} f(x)\ \delta(g(x))\,dx = \sum_{a} \int_{a + ε}^{a - ε} f(x)\ \delta((x - a) g^{'}(a)) ##

    into a sum of integrals over small intervals containing the zeros of g(x). In these intervals, since x is supposed to be very near to the a we can approximate g(x) as g(a) + (x - a)##g^{'}(a)## (note also that it is just the definition of derivative of g(x) when x goes toward a). Now we have proved it if we employ the equation ##\delta(αx) = 1/α\ \delta(x)## on the right-hand side of the integral.
     
    Last edited: Sep 9, 2014
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