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Dirac delta

  1. Sep 25, 2007 #1
    1. The problem statement, all variables and given/known data
    How would one show that dirac delta is the limit of the normal distribution?
    http://en.wikipedia.org/wiki/Dirac_delta
    using the definition [tex] \delta(k) = 1/(2\pi)\int_{-\infty}^{\infty}e^{ikx}dx [/tex]

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Sep 26, 2007
  2. jcsd
  3. Sep 25, 2007 #2

    Hurkyl

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    Equality for distributions is defined pointwise. You just have to prove you get the same value if you convolve either one with a test function. I.E. for any test function f, you have to prove

    [tex]
    \int_{-\infty}^{+\infty} \delta(k) f(k) \, dk
    =
    \frac{1}{2\pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}
    e^{ikx} f(k) \, dk \, dx
    [/tex]
     
  4. Sep 26, 2007 #3

    Avodyne

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    One way (not so rigorous mathematically) to define the delta function is that it is a function that satisfies [tex]\delta(x)=0[/tex] if [tex]x\ne 0[/tex], and [tex]\textstyle \int_{-\infty}^{+\infty}dx\;\delta(x)=1.[/tex] So you need to show (1) that the limit of the normal distribution has these properties, and (2) that [tex]\textstyle{1\over2\pi}\int_{-\infty}^{+\infty}dk\;e^{ikx}[/tex] has these properties. Part (1) is easy. Amusingly, the easiest way to do part (2) is to define it by inserting a convergence factor of [tex]\exp(-\epsilon^2 k^2/2)[/tex] into the integrand, which turns it into a normal distribution that becomes a delta function in the limit [tex]\epsilon\to 0.[/tex]
     
  5. Sep 26, 2007 #4

    Hurkyl

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    Oh, hah, I misread the problem. I thought the equation the OP posted was the equation he wanted to prove.

    The idea is the same, though. For a distribution F(_) and a family of distributions G(_, y), to prove

    [tex]
    F(x) = \lim_{y \rightarrow 0} G(x, y)
    [/tex]

    you have to show

    [tex]
    \int F(x) f(x) \, dx = \lim_{y \rightarrow 0} \int G(x, y) f(x) \, dx
    [/tex]
     
  6. Sep 26, 2007 #5
    Can I choose any f(x)?
     
  7. Sep 26, 2007 #6

    Avodyne

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    It has to be infinitely differentiable, or something like that (not up on my rigorous defs, sorry), but otherwise yes.
     
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