Some aspects of electromagnetism are better discussed in terms of complex analysis. I believe the first sentence is referring to this: https://en.wikipedia.org/wiki/Euler%27s_formula The function e^(i x) is a (complex) sinusoidal oscillation when x is a real number, but is a (complex) exponential funciton when x is an imaginary number. The phrasing "in the direction at right angles in the complex plane" refers to the fact that the real and imaginary axis are perpendicular to each other.

I'm not sure about the second phrase. A contour integral is an integral over a closed path on the complex plane, and there's a theorem that says that if the function and the path have certain properties, the result of this path integral is just some coefficients (called residues). But I'm not sure how that's connected to the rest of the conversation. https://en.wikipedia.org/wiki/Residue_theorem

The residue theorem implies that you can compute some integrals on the real line by closing up the contour in the complex plane and accounting for any poles that you've enclosed. The trick is to choose your contour so that the contribution to the integral of the new piece goes to zero as the contour gets larger and larger (think of a real piece which goes from -R to R, and a semicircle in the upper half-plane connecting those two points; as R->infty, the real part of the integral goes from -infty to infty). One reason the new piece may go to zero is that oscillating functions on the real line turn in to exponential decay as you go up or down along the imaginary axis (as you point out), so as the new piece of the contour moves up or down, its contribution to the integral gets exponentially smaller.

(This is hard to explain without pictures and formulas, but you can find some examples here: http://web.williams.edu/Mathematics/sjmiller/public_html/302...).

I know what it is, how to use it and all that. I just don't know how to make sense of the specific phrase that the parent asked about.

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).