# Gaussian integral w/ imaginary coeff. in the exponential

## Main Question or Discussion Point

So I've seen this type of integral solved. Specifically, if we have

∫e-i(Ax2 + Bx)dx then apparently you can perform this integral in the same way you would a gaussian integral, completing the square etc. I noticed on wikipedia it says doing this is valid when "A" has a positive imaginary part. I can see why that might be important, we would then have an overall decaying magnitude of the integrand, as opposed to purely oscillating.

What if A has no imaginary part, how would one go about doing such an integral?

Ben Niehoff
If A is real, then the integral does not strictly converge. However, sometimes you can get a finite answer by replacing $A \to A + i \varepsilon$ and then take $\varepsilon \to 0$ at the end of the calculation.