I'm trying to find the Fourier Transform of a Gaussian, and I end up having to complete the square in the argument of an exponential so I can use the standard Gaussian integral. I basically have:
$$\bar f(k) = \int_{-\infty}^{\infty}e^{\frac{(x-c)^2}{a^2}-ikx}dx$$
And I have trouble finding the polynomial I need. I end up doing something like:
We require a function of the form $(\frac{x-c}{a} + A)^2$ such that
$$-2c+2Aa = ikx$$
as I believe $(\frac{x-c}{a} + A)^2 = (x + (-c + Aa))^2$ should be equivalent?
$$\implies A = \frac{ikx + 2c}{2a}$$
However, this doesn't solve the problem. I need $A$ to not be a function of $x$. How should I instead approach this?
