Riemann $\varpi$ function

The Riemann $\varpi$ function is used in the proof of the analytic continuation for the Riemann Xi function to the whole complex plane. It is defined as:

$$\varpi (x)=\sum _{n=1}^{\mathrm{\infty }}{e}^{-n^2\pi x}$$

This function is a special case of a Jacobi $\vartheta$ function (http://planetmath.org/JacobiVarthetaFunctions):

$$\varpi (x)={\vartheta }_3(0|ix)$$

As such the $\varpi$ function satisfies a functional equation, which a special case of Jacobi’s Identity for the $\vartheta$ function (http://planetmath.org/JacobisIdentityForVarthetaFunctions).