series summation of quadratic exponentials

Question

I need to prove the equation $\sum_{n=-\infty}^{\infty} e^{-\frac{1}{2} An^2+iBn} = \sqrt{\frac{2\pi}{A}} \sum_{l=-\infty}^{\infty} e^{-\frac{1}{2A}(B-2\pi l)^2}$, where A and B are constants(possibly can be complex number). First I tried to use the direct expansion $e^x=\sum_{n=-\infty}^{\infty} \frac{x^n}{n!}$, but it is quite hard to using binomial expansion and summarizing them. The second one I tried is the discrete fourier transform, but in wikipedia, there are only finite sum case, where I want to do is infinite sum. In fact, I can't understand why the coefficient $\sqrt{\frac{2\pi}{A}}$ appears. How can I prove this?