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

  • Thread starter tjphop
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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
 

jbunniii

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Try integrating your expression for G(k) from k = -infinity to infinity. What does this give you?
 
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