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Delta function

  1. Mar 5, 2009 #1
    1. The problem statement, all variables and given/known data

    How does one prove that [tex]\int^\infty_{-\infty}\lim_{\epsilon \rightarrow 0}(1/\pi)\frac{\epsilon g(x)}{(x-a)^{2}+\epsilon^{2}}dx=g(a)[/tex]?
     
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  3. Mar 5, 2009 #2

    George Jones

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    One doesn't, since the result isn't true. The limit has to be outside the integral sign.

    Mathematical Physics by Butkov has a nice proof on pages 238-239. The idea is, for positive [itex]\epsilon[/itex] to write

    [tex]\lim_{\epsilon \rightarrow 0} \int_{-\infty}^\infty = \lim_{\epsilon \rightarrow 0} \int_{-\infty}^{a-\epsilon} + lim_{\epsilon \rightarrow 0} \int_{a-\epsilon}^{a+\epsilon} + lim_{\epsilon \rightarrow 0} \int_{a+\epsilon}^{\infty},[/tex]

    and then to assume [itex]g[/itex] is bounded to show that the first and last terms go to zero.

    For [itex]\epsilon[/itex] small and [itex]g[/itex] continuous, [itex]g(x)[/itex] is approximately equal to the constant value [itex]g(a)[/itex] over the middle interval, so pull this outside of the middle integral, or, more rigorously, use the mean value theorem for integrals.
     
  4. Mar 6, 2009 #3
    Thanks, got it right now. The limit was indeed before the integral sign.
     
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