Showing $\sum_{n=-\infty}^{\infty}\exp\left(-\pi an^2+2\pi ibn\right)=a^{-\frac{1}{2}}\sum_{m=-\infty}^{\infty}\exp\left(-\frac{\pi(m-b)^2}{a}\right)$

Question

How do I show that

\begin{align} \sum_{n=-\infty}^{\infty} \exp\left(-\pi a n^2 + 2 \pi i bn\right) = a^{-\frac{1}{2}} \sum_{m=-\infty}^{\infty} \exp\left(-\frac{\pi(m-b)^2}{a}\right) \end{align}

is a valid identity?

Answer 1

It is a consequence of Poisson summation formula. You just have to prove that, if $$ f(x) = \exp\left(-\pi a x^2+2\pi i b x\right), $$ then its Fourier transform is: $$ \widehat{f}(s) = \frac{1}{\sqrt{2\pi a}}\exp\left(-\frac{(2b\pi+s)^2}{4a\pi}\right).$$