Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Fourier Transform

  1. Sep 18, 2010 #1


    User Avatar
    Gold Member

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


    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]

    [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


    User Avatar
    Gold Member

    excellent advice, thank you
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook