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Inverse Fourier transform

  1. Aug 13, 2009 #1
    Hi all,

    I'm having a bit trouble computing the Inverse Fourier Transform of the following:

    [tex]\frac{\alpha}{2\pi}\exp\left(\frac{1}{2} \alpha^2 C^2(K) \tau \omega^2\right)[/tex]

    Here, [tex]C^2(K)[/tex], [tex]\alpha[/tex] and [tex]\tau[/tex] can be assumed to be constant. Hence, we have an integral with respect to [tex]\omega[/tex].

    Who can help me out?
     
  2. jcsd
  3. Aug 13, 2009 #2
    So you want to find the inverse Fourier transform of
    [tex]\frac{\alpha}{2\pi}\exp(A \omega^2)[/tex]?

    It should be:

    [tex]\frac{\alpha}{2\pi}\frac{1}{\sqrt{-2 A}}\exp\left(\frac{t^2}{4A}\right)[/tex]
     
  4. Aug 13, 2009 #3

    mathman

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    A < 0 is necessary.
     
  5. Aug 13, 2009 #4

    Redbelly98

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    One can do a suitable variable transform to get the integral in the form


    e-x2 dx
    with limits
    -∞ to +∞

    which can be looked up in a standard table of integrals. I suspect the answer is what jpreed gave in post #2.
     
  6. Aug 14, 2009 #5
    Whoops.. I just figured that there are two small mistakes in my first post, I would like to have the Inverse Fourier Transform of:
    [tex]\frac{\alpha^2}{2\pi}\exp\left(-\frac{1}{2} \alpha^2 C^2(K) \tau \omega^2\right)[/tex]

    Here, note that [tex]\alpha[/tex] is squared, and a minus sign is added in the argument of exp.

    Don't know if that makes a lot of difference?
     
  7. Aug 14, 2009 #6

    Redbelly98

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    Not really. Just replace A with -A in all the responses.

    That would become

    -A < 0​
    or in other words
    A > 0​
     
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