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The Alternate form of the Dirac Delta Function

  1. Sep 29, 2011 #1
    Hello,

    I am trying to show that:
    [itex]\delta(x) = \lim_{\epsilon \to 0} \frac{\sin(\frac{x}{\epsilon})}{\pi x}[/itex]
    Is a viable representation of the dirac delta function. More specifically, it has to satisfy:
    [itex]
    \int_{-\infty}^{\infty} \delta(x) f(x) dx = f(0)
    [/itex]

    I know that the integral of sin(x)/x over the reals is [itex]\pi[/itex], and so far as I can tell, it doesn't depend on epsilon. What I've tried so far is integration by parts, which leads me to:
    [itex]
    f(x) - \int_{-\infty}^{\infty} f'(x) dx
    [/itex]

    Which isn't really getting me somewhere, and the limit drops off due to the integral not caring what epsilon is. Is there another way of approaching this? Or am I on the right track, I just can't pull out an f(0) from this.

    Any help is appreciated,

    Thanks.
     
  2. jcsd
  3. Sep 29, 2011 #2
    Try substituting z = x/epsilon.
     
  4. Sep 29, 2011 #3

    vela

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    Are you familiar with complex analysis and the residue theorem?
     
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