How to calculate the fourier transform of a gaussion?

[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

 

[rubi]

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

 

[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

 

[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.

 

[jollage]

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.

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

Jo