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Strange representation of Heaviside and Delta function

  1. Jun 4, 2008 #1
    Last edited by a moderator: Apr 23, 2017
  2. jcsd
  3. Jun 4, 2008 #2


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    Probably the idea is, that we make it a contour integration. Then the only pole is at z = 0 and the result of an integration would be e.g.
    [tex]\int \delta(x) f(x) \, dx = \int \frac{-1}{2i\pi} \frac{f(z)}{z} \, dz = (2i\pi) \operatorname{Res}_{z = 0} \frac{f(z)}{2 \pi i} = f(0)[/tex]
    if f(x) does not have any poles in the upper half plane.
    but I actually doubt how valid this is (even if it works, one would need requirements on f(x) for [itex]x \to \pm i\infty[/itex] to close the countour; and I wonder what happens if f(x) itself has poles).

    So I think that is the idea, but I wonder if it works and under what conditions.
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