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I have got a homework for Advanced Quantum Mechanics, actually I've tried to solve it in many ways my own, but I'm always forced to use computer at the end (For infinite series or improper integrals), I want to solve it my self, so I can do it also in the class!

The problem is:

The problem is:

Find the fourier transform of exp((a x^2)/2)

Actually the first way I tried is by splitting the fourier factor exp(ikx) to sine and cosine, by Euler form, but no use! because I reached "Abramowitz and Stegun integral", which have a result without details.

The second way is in the PDF, where I tried to release the exp((a x^2)/2) term to an infinite series, But also no use, I reached the Erf error function due to an infinite series!!!!

Thanks for your help :)