I am using Fourier Analysis tool in Excel to transform a Gaussian $\exp(-x^2)$ on a uniform and symmetric grid of $x$ values. I am expecting the result to contain only real numbers (which should furthermore look like a Gaussian), but what I get is a collection of complex coefficients. I am at a loss. A Fourier transform of a Gaussian should be another Gaussian, why complex coefficients? How to get from them expected Gaussian?
A delayed or offset Gaussian would have complex coefficients. I suspect that the $x=0$ point you are assuming is not where Excel assumes. The magnitude would have a Gaussian shape and the phase would be Linear, which you can use as a diagnostic. Gaussian is also infinite in extent so, there is going to be some effects due to truncation.
The issue that confuses most people is that the DFT has the character of "assuming" that it is operating on one period of a periodic sequence. In your case the Gaussian is symmetric around $x=0$ but the first point in the sequence corresponds to $x=0$ so where do the values for $x<0$ go? They go with the next cycle of the sequence so if the period has $N$ data points, depending if $N$ is odd or even the periodic portion corresponding to $x<0$ would be in the samples from $N/2$ to $N-1$.