I am working with periodic Green's functions for a scattering problem and the form of the Green's function given is as follows -
$$G(x, y) = -\sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha} \int_0^E \frac{1}{4\pi t} e^{\omega^2 t - \frac{|x - n - y|^2}{4t}} dt$$
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)
However this integral doesn't seem valid to me considering the domain is from $0$ to some arbitrary constant $E$, yet we have $\frac{1}{t}$ in the integrand. So is it possible to integrate this? Am I missing something?
It's true that the integrand is not defined at $t=0$, but because $\exp(-c/t)$ goes very rapidly to $0$ as $t \to 0+$ for any $c > 0$, the integral will converge as long as $x - n - y \ne 0$, and the sum will converge as long as $x - y \notin \mathbb Z^2$. If you're going to use a numerical integration method that will evaluate the integrand at the endpoints, it's important to tell it that the integrand is $0$ at $t=0$.