I am working with periodic Green's functions for a scattering problem and the form of the Green's function given is as follows -
$$G(x, y) = -\sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha} \int_E^{\infty} \frac{1}{4\pi t} e^{\omega^2 t - \frac{|x - n - y|^2}{4t}} dt$$
This is supposed to be in a computationally friendly form in that it will rapidly converge (Ewald's method - http://www.math.ntua.gr/~papanico/publications/paper19.pdf)
This is related to my previous question - Can this integral be evaluated numerically? - and the integral in that post is in fact possible to integrate as shown in the answer. However with the integral in this post we have an integrand with $e^t$ and a limit of infinity so I can't see how this can be valid?
(As an aside, this Green's function and the one in my previous post are added together to obtain the true Green's function.)