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Problem in distribution theory

  1. Jul 8, 2009 #1
    I have a little problem with the following exercise:
    "Consider the temperate distribution

    [tex] f\left(x\right)=\frac{1}{\left(x-i0\right)^2} [/tex]

    Write f(x) like function of elementary temperate distributions and calculate its Fourier-transform."
    I am almost sure I have to use the identity

    [tex] \frac{1}{x-i0}=PP\frac{1}{x}+i\pi\delta\left(x\right) [/tex]

    But the square makes appear terms like [tex] \delta^2\left(x\right) [/tex], that is not a distribution.

    Any idea?
     
  2. jcsd
  3. Jul 9, 2009 #2

    turin

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    Homework Helper

    What is "the temperate distribution", and what is an "elementary temperate distribution"?

    What is "i0"? E.g. is 0 simply an arbitrarily small positive number?

    The original equation defines a function with one double pole (assuming my example interpretation of 0 above). I don't see any need to use distribution theory. You can evaluate the Fourier transform using the Cauchy Integral Theorem (assuming my example interpretation of 0 above).
     
  4. Jul 9, 2009 #3

    George Jones

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    I think this means "tempered distribution" and "regular distribution that is also a tempered distribution."
    I think so. This is usually denoted [itex]x- i \epsilon[/itex].
    But I think the idea behind the question is to gain familiarity with distribution theory.
    [tex]\frac{1}{\left( x - i \epsilon \right)^2} = - \frac{d}{dx} \left[ \frac{1}{ x - i \epsilon} \right][/tex]
     
  5. Jul 10, 2009 #4
    Thanks, that was exactly the answer I got myself after a while. Thanks again.
     
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