# How to calculate the fourier transform of a gaussion?

1. Aug 28, 2013

### jollage

Hi all,

I want to calculate $\int_0^{\infty}e^{-a t^2}\cos(2xt)dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{-x^2}{a}}$. 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. Aug 28, 2013

### rubi

Use $\cos(x) = \frac{1}{2}(e^{\mathrm i x} + e^{-\mathrm i x})$, then complete the squares and perform a countour integral.

3. Aug 28, 2013

### jollage

Hi rubi,

Thank you. Yes, the problem is actually starting in the form of $\int_{-\infty}^{\infty}e^{-a t^2}e^{-i2\pi x t}dt=?$, 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

4. Aug 28, 2013

### mathman

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.

5. Aug 28, 2013

### jollage

O, I see. Yes, I was not careful. Thank you both.

Jo