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Integral transform

  1. Apr 24, 2007 #1

    tpm

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    let be the next linear integral transform:

    [tex] g_{k} (x)= \int_{-\infty}^{\infty}dt \frac{f(t)}{(t-x)^{k}} [/tex]

    no matter what f(t) is there is a singularity at the points where t=x how could you define it so it's finite avoiding the poles at t=x where k is a positive integer.
     
  2. jcsd
  3. Apr 24, 2007 #2
    False. When f(t) has a zero of order [itex]\ge k[/itex] in x, i.e. [itex]f(t)=(t-x)^k f_0(t)[/itex], and [itex]f_0(t)\in L^1(0,\infty)[/itex], such integral is well defined.

    Another way the integral can be well defined is verified with complex variable and residue theory.
     
    Last edited: Apr 24, 2007
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