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Fourier Transform

  1. Sep 18, 2010 #1

    kreil

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    Gold Member

    1. The problem statement, all variables and given/known data
    I have to find the fourier transform of

    [tex]f(x)=\frac{\beta^2}{\beta^2+x^2}[/tex]


    2. Relevant equations
    Fourier Transform is given by

    [tex]F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}f(x) dx[/tex]

    3. The attempt at a solution
    I'm having trouble with the integration after I separate into two integrals using partial fractions:

    [tex]F(k)=\frac{\beta^2}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \frac{e^{-ikx}}{\beta^2+x^2}dx[/tex]

    Note
    [tex]\frac{1}{\beta^2+x^2}=\frac{1}{2i\beta} \left( \frac{1}{x-i\beta} - \frac{1}{x+i\beta} \right) [/tex]

    [tex]F(k)=\frac{1}{\sqrt{2\pi}} \frac{\beta}{2i} \left[ \int_{-\infty}^{\infty} \frac{e^{-ikx}}{(x-i\beta)} dx - \int_{-\infty}^{\infty} \frac{e^{-ikx}}{(x+i\beta)} dx \right][/tex]


    Are there any suggestions on how to proceed?
     
  2. jcsd
  3. Sep 19, 2010 #2
    Use the residue theorem.
     
  4. Sep 19, 2010 #3

    kreil

    User Avatar
    Gold Member

    excellent advice, thank you
     
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