Prove that $\sum _{n=-\infty }^{\infty } e^{-n^2 \pi x}=\frac{1}{\sqrt{x}}\sum _{n=-\infty }^{\infty } e^{-\frac{n^2 \pi }{x}}$
Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula