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Delta function

  1. Jun 21, 2013 #1
    In Dirac definition ##\delta(x)## is ##\infty## when ##x=0##, and ##0## when ##x\neq 0##. My question is when I have some ##\alpha \delta(x)## could I interpretate this like function which have value ##\alpha## in point ##x=0##?
     
  2. jcsd
  3. Jun 21, 2013 #2

    Fredrik

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    That interpretation would suggest that
    $$\int \alpha\delta(x)=0$$ which is wrong. The right-hand side should be 1. Makes more sense to think of that value as ##\alpha\cdot(+\infty)=+\infty##, if ##\alpha>0##. Note that since
    $$\int\delta(x)f(x)dx=f(0)$$ for all f, we should have
    $$\int \alpha\delta(x)f(x)dx=\int\delta(x)(\alpha f)(x)dx =(\alpha f)(0)=\alpha f(0).$$
     
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