For what reason is exp(-1/2 x^2) a rapidly decreasing function? I cannot see it. I know that |exp(-1/2 x^2)| <= 1, but I don't think this helps me.
Also, we know that it satisfies the differential equation y' + xy = 0. I have to show that it's Fourier transform also satisfies it (this is in a proof of showing that it equals its own FT, so we cannot use this fact).
I have tried to use the definition and differentiation under the integral sign, but I can't get it to 0. I end up with an 'x' term inside one integral and an 'x' term on the outside of another.