The idea is that for each Fourier component of our source, we want to evaluate the following integral:
A = \int[+-inf] dx (\sin x) / \sqrt(x^2 + y^2)
= Im \int[+-inf] dx (\exp ix) / \sqrt{(x+iy)(x-iy)}
So this is an integral over a particular contour (the real axis) in the complex plane, with poles at +-iy. We can play the usual contour games and say it equals a different contour integral
A = \int(something far away that vanishes)
+ \int(once around one of the poles)
The second integral is an exponential decay because you get two factors of $i$.
For discrete point-lattices you have an periodic array of delta functions rather than a single sinusoid. Summing the Fourier components thus gives a sum of exponentials, each one dopping off faster than the last. So I guess you get an overall function like 1 /(1 - e^x).
A = \int[+-inf] dx (\sin x) / \sqrt(x^2 + y^2) = Im \int[+-inf] dx (\exp ix) / \sqrt{(x+iy)(x-iy)}
So this is an integral over a particular contour (the real axis) in the complex plane, with poles at +-iy. We can play the usual contour games and say it equals a different contour integral
The second integral is an exponential decay because you get two factors of $i$.For discrete point-lattices you have an periodic array of delta functions rather than a single sinusoid. Summing the Fourier components thus gives a sum of exponentials, each one dopping off faster than the last. So I guess you get an overall function like 1 /(1 - e^x).