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Integrating with a dirac delta function

  1. Dec 23, 2012 #1
    1. The problem statement, all variables and given/known data

    I have to integrate:

    [itex]\int_0^x \delta(x-y)f(y)dy[/itex]

    2. Relevant equations



    3. The attempt at a solution

    I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity:
    [itex]\delta(0)=\infty[/itex]
    I have to express the integral in terms of function f only and i am unsure how to do this.
    Do I have to use integration by parts?

    Thank you very much in advance
     
  2. jcsd
  3. Dec 23, 2012 #2
    Dirac's Delta is...weird. You have to integrate [itex]\int_0^x \delta(x-y)f(y)dy[/itex]?

    Integration by parts sounds...okay? However, one of the properties of the Dirac Delta "function" is that [itex]\displaystyle \forall ε > 0, \int_{a-ε}^{a+ε} f(y)\delta(y-a) \ dy = f(a)[/itex]. Can you manipulate your integral into that form? :wink:

    By my calculations, you should end up with [itex]f(x)(2H(x)-1)[/itex], where H is the antiderivative of δ.
     
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