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Inverse fourier transform

  1. Jun 3, 2008 #1
    1. The problem statement, all variables and given/known data
    calculate the inverse fourier transform of [tex]\left( a^2 + \left( bk \right)^2 \right)^{-1}[/tex]

    3. The attempt at a solution

    I know that [tex]FT[e^{-|x|)}](k) = ( \pi (k^2 + 1 ) )^{-1}[/tex]. I've tried to to concatenate the shift FT or the strech FT, but the "+1" in the known FT is in the way.

    Sorry for my bad English, it's not my native language.

    thanks.
     
  2. jcsd
  3. Jun 3, 2008 #2

    Defennder

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    Homework Helper

    I can't quite read what you're writing here. The FT you've quoted is supposed to be [tex]F[e^{-\alpha|x|}] = \frac{2\alpha}{\alpha^2+ \omega^2}[/tex], correct?

    Are a and b supposed to be arbitrary constants? If so, then I take [itex]k=\omega[/itex]. You must then express the question in terms of the result above.

    Note that [tex]\frac{1}{a^2 + b^2 \omega^2} = \frac{1}{b^2} \ \frac{1}{\frac{a^2}{b^2} + \omega^2}[/tex].

    A factor of 2 is still required. Can you see it now?
     
  4. Jun 4, 2008 #3
    thanks for answering. You helped me "getting it" :)

    I think our normalization factors are different. I know that [tex]F[e^{-|x|}]=\frac{1}{\pi (\omega^2 +1)}[/tex], so if a,b>0 (i assume it, since it's the only way to get to the following result) i can write [tex]F[e^{-\frac{a}{b} |x|}]=\frac{b}{a} \hat{f}(\frac{\omega b}{a}) = \frac{a b e^{i \omega \frac{b}{a}}{\pi (\omega^2 b^2 + a^2)}[/tex]

    i'm using [tex]F[f(x)]=\frac{1}{2\pi}\int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx[/tex]

    thanks.
     
    Last edited: Jun 4, 2008
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