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Cauchy theorem and fourier transform

  1. Nov 12, 2007 #1
    1. The problem statement, all variables and given/known data
    I have this problem and I dont know how to finish it:

    Using the Cauchy Theorem, prove that the fourier tranform of [tex]\frac{1}{(1+t^2)}[/tex] is
    [tex]\pi.e^{-2.\pi.|f|}[/tex] .( you must show the intergration contour) Stetch the power spectrum.

    I applied the fourier transform formula but then tried to break down the
    1/(1+t^2) but I get stuck to apply the Cauchy theorem.
    Please can I have some help?

    Thank you
  2. jcsd
  3. Nov 12, 2007 #2
    You want to find, [tex]\int_{-\infty}^{\infty} e^{-2\pi i ft}\frac{1}{1+t^2}dt[/tex]. Use the popular semi-circular contour and proceede by Cauchy-Goursat theorem. I can post all the detail but it is too long. It is better if you start doing the problem and we help when you need it.
  4. Nov 12, 2007 #3
    Thank you Kummer

    this iswhere my problem is. I dont get the part of the contour.
    But first, just say I have this function [tex]f(z)=e^{-\omega*i.z}\frac{1}{1+z^2}[/tex]

    I tried to break it down [tex]f(z)=
    \frac{e^{-i\omega z}}{z^2+1} = \frac{\frac{1}{2i}e^{-i\omega z}}{z-i} - \frac{\frac{1}{2i}e^{-i\omega z}}{z+i}
    Please, what Iam am doing from here?
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