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Homework Help: Please help with finding the limit of the integral

  1. Dec 26, 2011 #1
    1. The problem statement, all variables and given/known data

    I want to show that

    [tex]\lim_{\epsilon\to 0}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\epsilon\over 2\pi [(x-x')^2+(y-y')^2+\epsilon^2]^{3\over2}}f(x,y)\;dx\,dy\,\,=f(x',y')[/tex]? I am not sure what conditions there is on [itex]f(x,y)[/itex], though I do know that [tex]\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{z\over 2\pi [(x-x')^2+(y-y')^2+z^2]^{3\over2}}f(x,y)\;dx\,dy[/tex] is well-defined for all [itex]x,y\in R[/itex] and [itex]z>0[/itex].

    2. Relevant equations

    Please see above section.

    3. The attempt at a solution

    It is possible that we couldchange variables or sth? Or maybe show that in the limit, the green's function is the delta function? Please help!
     
    Last edited: Dec 26, 2011
  2. jcsd
  3. Dec 26, 2011 #2

    Dick

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    Yes, you need to show that in the limit it is a representation of a delta function. You need to i) as [itex]\epsilon[/itex]->0 that that [tex]{\epsilon\over [(x-x')^2+(y-y')^2+\epsilon^2]^{3\over2}}[/tex]
    approaches zero unless x=x' and y=y'. And ii) that for fixed [itex]\epsilon[/itex] that the integral of that is 1. Use polar coordinates around the point (x',y'). I think you actually need a bit more than that but that's a good start. To go beyond that I think you do need some conditions on f(x,y).
     
    Last edited: Dec 26, 2011
  4. Dec 27, 2011 #3
    Thanks, @Dick . So we are showing that that bit is acting like a distribution, right? What conditions are required for f? Is it enough to have f tending to 0?
     
  5. Dec 27, 2011 #4

    Dick

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    The main condition you need is that f is continuous at x',y'. And sure, it shouldn't blow up at infinity so fast the integral doesn't exist.
     
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