The other day I was playing around with gaussian functions and I noticed that the fourier transform of a gaussian function looked an awful lot like another gaussian function. I managed to find a single blurb about this fact in the Wikipedia article, and indeed, my hunch was correct. However, the article gave no clue as to how to demonstrate this is true. I've been playing around with the integral for a day and I haven't gotten anywhere with it! Can someone help me along with a proof that the FT of a gaussian function is another with the parameters slightly tweaked? Additionally, the article said this was only true for when the function is symmetric about the y-axis. However, it seems natural enough to assume that shifted gaussian functions would have a similarly regular FT.