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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.

  2. jcsd
  3. Aug 28, 2013 #2


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    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

  5. Aug 28, 2013 #4


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    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.

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