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

How to calculate the fourier transform of a gaussion?

  1. Aug 28, 2013 #1
    Hi all,

    I want to calculate [itex]\int_0^{\infty}e^{-a t^2}\cos(2xt)dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{-x^2}{a}}[/itex]. The answer is known from the literature, but I don't know how to do it step by step. Any one has a clue? Thanks.

    Jo
     
  2. jcsd
  3. Aug 28, 2013 #2

    rubi

    User Avatar
    Science Advisor

    Use [itex]\cos(x) = \frac{1}{2}(e^{\mathrm i x} + e^{-\mathrm i x})[/itex], then complete the squares and perform a countour integral.
     
  4. Aug 28, 2013 #3
    Hi rubi,

    Thank you. Yes, the problem is actually starting in the form of [itex]\int_{-\infty}^{\infty}e^{-a t^2}e^{-i2\pi x t}dt=?[/itex], which is the fourier transform of a gaussian. I actually tried using complex contour integral, but I was stuck there. Could you be more specific? I guess I don't some tricks...

    Thank you

    Jo
     
  5. Aug 28, 2013 #4

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Use the completion of the square and do a direct integration.
    at2+i2πxt =a(t +iπx/a)2+(πx)2/a. Calculate the t integral.
     
  6. Aug 28, 2013 #5
    O, I see. Yes, I was not careful. Thank you both.

    Jo
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook