functional equation for the theta function

The lemma used in the derivation of the functional equation for the Riemann Xi function. This functional equation is not as remarkable as the one for the Xi function, because it does not actually extend the domain of the function.

$$\theta \left(\frac{1}{x}\right)=\sqrt{x}\theta (x)$$

The proof relies on the Cauchy integral formula and the Poisson summation formula.