# complex integral

• Nov 21st 2008, 04:31 AM
enricokr
complex integral
Hi. I wish help to solve this complex integral:
$\displaystyle \int \limits_{-\infty}^{+\infty} {e^{-\pi x^2} \cdot e^{-2 \pi \cdot i \cdot x y} \ dx}$
Thanks everyone!
• Nov 21st 2008, 05:31 AM
ThePerfectHacker
Quote:

Originally Posted by enricokr
Hi. I wish help to solve this complex integral:
$\displaystyle \int \limits_{-\infty}^{+\infty} {e^{-\pi x^2} \cdot e^{-2 \pi \cdot i \cdot x y} \ dx}$
Thanks everyone!

It is a known result from complex analysis (I can derive it if you wish) that:
$\displaystyle \int \limits_{-\infty}^{+\infty} {e^{-t^2} \cdot \cos( \alpha t) \ dt}=\sqrt{\pi} \cdot e^{-\alpha^2/4}$
If you write your integral out it becomes,
$\displaystyle \int \limits_{-\infty}^{\infty} e^{-\pi x^2} \cos (2\pi yx) dx + i\int \limits_{-\infty}^{\infty} e^{-\pi x^2} \sin (2\pi yx) dx$
However, the second integral vanishes! (Happy)

Now use substitution $\displaystyle x\to x\sqrt{\pi}$ to bring that integral into the form above.