In complex analysis and the calculus of "residues," Cauchy's integral theorem gives a "shortcut": The integral is $2\pi i$ times the sum of the "residues." This works because there are "singularities," in the area in question.
Normally, Green's theorem (real case) is a fairly cut and dried matter. But if there is a "singularity" at say, $(0,0)$ then you need to multiply by $2 \pi$ to get the value of the integral.
These two phenomena look suspiciously similar, except for the fact that in the real case, you multiply by $2\pi$, and in the complex case, by $2 \pi i$. Are they, in fact, somehow connected? Or is this a "false" analogy that happens to be coincidental?