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Dirac Distribution with 2 Diracs

  1. Nov 30, 2012 #1

    It is well known that

    \int f(x) \delta(x-a) dx = f(a) \quad\mathrm{and}\\
    \int f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a)

    Similarly, Is there a way to express a distribution integral with multiplicative Diracs in a compact form (e.g., a sum)?

    \int f(x) \delta(t-x-l_1) \delta(t-x-l_2) dx

  2. jcsd
  3. Dec 1, 2012 #2


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    The way you have written this, with a single x, is a bit misleading. What you show would be simply f(t- l) if [itex]l_1= l_2= l[/itex] and 0 if [itex]l_1\ne l_2[/itex].

    But I suspect you mean a multi-dimensional version where x, t, and l are all vectors or shorthand for multiple variables. In that case,
    [tex]\int\int f(x,y)\delta(t_x- x- l_1)\delta(t_y- y- l_2)dxdy= f(t_x-l_1, t_y- l_2)[/tex]
  4. Dec 3, 2012 #3
    Thank you, your answer was exactly what I was looking for!
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