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Integration of Step Function

  1. Sep 15, 2010 #1
    I have a question about dirac-delta function. Integral of step function from -infinity to +infinity equals to infinity from the graph of step function.

    [tex]\int_{-\infty}^{\infty}\theta (x) dx=\infty[/tex]

    But calculating mathematically

    [tex]\lim_{a\rightarrow\infty}\int_{-a}^{a}\theta(x) dx=\lim_{a\rightarrow\infty}(\delta \left(a \right)-\delta \left(-a\right))=0[/tex]

    Because delta function is 0 except x=0. Please explain to me. Thanks
  2. jcsd
  3. Sep 15, 2010 #2

    George Jones

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    It seems that you're using

    [tex]\theta \left( x \right) = \frac{d \delta}{dx} \left( x \right),[/tex]

    when actually (in a distributional sense)

    [tex]\delta \left( x \right) = \frac{d \theta}{dx} \left( x \right).[/tex]

    Also, for [itex]a > 0[/itex],

    [tex]\int^a_{-a} \theta \left( x \right) dx = a.[/tex]
  4. Sep 15, 2010 #3
    Oh, yes, thanks for your help.
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