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Help please!: Fourier Transform of a Gaussian function showing integral equals 1

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi, the question is from a piece of coursework and before hand we were asked to find the Fourier transform G(K) of the function g(x)= e^(-∏(x^2)) (where g(x)= ∫ G(K)e^2∏ikx dx (integral from -∞ to ∞)). We were told to find G(K) by forming a differential equation in H(K), where H(K) is the Fourier transform for h(x)=g'(x), and using the fact that H(K)=2∏iK*G(K).
    We had to show that G(K) = e^(-∏(k^2)) ∫ e^(-∏(x^2)) dx (integral from -∞ to ∞).

    The question I am stuck on is the following:

    Use the above results to show that ∫ e^(-∏(x^2) dx = 1 (integral from -∞ to ∞)

    ( i.e ∫ g(x) dx =1 (integral from -∞ to ∞) )

    2. Relevant equations

    Another previous part involved showing that the Fourier transform for l(x)=xg(x) was given by L(K) = (i/2∏) * d/dk ( G(K) ) , so this may be relevant too.


    3. The attempt at a solution

    I don't know how to proceed at all and have tried manipulating all the different expressions but to no avail. Any help would be much appreciated! Thanks
     
  2. jcsd
  3. Feb 28, 2012 #2

    jbunniii

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    Try integrating your expression for G(k) from k = -infinity to infinity. What does this give you?
     
    Last edited: Feb 28, 2012
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