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Problem on integrating dirac delta function

  1. May 6, 2012 #1
    Hi there,
    I am trying to integrate this: http://imm.io/oqKi
    I should get the second line from the integral, but I can't show it.
    This should somehow relate to the Heaviside step function, or I am completely wrong.
    Any ideas?

    Sorry about the url, I fixed it.
    Last edited by a moderator: May 6, 2012
  2. jcsd
  3. May 6, 2012 #2


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    What is "imm.io/oqKi "??????
  4. May 6, 2012 #3


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    Anyhow, whatever "imm.io/oqKi " is:

    The Dirac delta "function" isn't a function in the usual sense, so can't integrate it in the usual sense, either.
  5. May 6, 2012 #4


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    The "Dirac delta funtion" is not really a function, as arildno says. It is a "generalized function" or "distribution"- a linear operator on functions.

    By definition
    [tex]\int_a^b f(x)\delta(x)dx= f(0)[/tex]
    if a< 0< b, 0 otherwise.

    That means that
    [tex]\int_a^b \delta(g(x))f(x) dx= f(c)[/tex]
    if g(c)= 0 for some c between a and b.
  6. May 6, 2012 #5
    This makes sense for why they put [tex](x-x_0)\cos(\theta)+(y-y_0)\sin(\theta)=0[/tex] after the second line.

    Actually I am in the middle of proving the simple backprojection of the Radon transform of a dot can be viewed as a two dimensional convolution of [tex]\frac{1}{\sqrt{x^2+y^2}}[/tex] and the original function. I used the Dirac Delta to formulate the dot, so this is just for the convenience of prove.

    The Dirac Delta also has this property:[tex]\int\delta(\alpha x)dx = \frac{1}{|\alpha|}[/tex], I think this might help.
    Last edited: May 6, 2012
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