In this very informative and interesting answer about the relation between residues and representation of complex functions as vector fields the author states that the function $$f(z) = \frac{1}{z}$$ is in a special role since the corresponding Polya field $$ V(x,y) = \frac{x \hat{x} + y \hat{y}}{x^2 + y^2} = \frac{\hat{r}}{|r|}$$ is the only vector field defined everywhere except the origin with vanishing divergence but a finite $$ \oint_C V \cdot N$$ where $C$ is a simple curve which contains the origin, and $N$ is a normal vector the curve.
Can somebody comment/prove what the author stated? The magic of the residue theorem seems to hint that the function defined is indeed in a very special role, but could somebody supply an intuitive reasoning or proof that this is the ONLY function which satisfies this condition?