One has function of two complex variables $f(x, y) = \frac{x^y}{y^2}$. From some simple calculations from wikipedia: https://en.wikipedia.org/wiki/Essential_singularity we get $\lim_{y\rightarrow0}\frac{x^y}{y^2}= \tilde{\infty}$ or alternatively if we do not want to use complex limit complexity: $\lim_{y\rightarrow0}|\frac{x^y}{y^2}|= \infty$ (as I understand only in the first case we have complex limit, i.e. in all complex directions). The inverse complex limit is simpler: $\lim_{y\rightarrow0}\frac{y^2}{x^y}= 0$.
So that means for any $x$ we have a pole at $y=0$. Now the problem is I want to find the residue but all naive formulas require one to have just 1 variable, though I can do contour integration around $y = 0$ (that gives residue 0 and multiplicity 2 for all values), or I can do Laurent expansion, that gives residue $ 2 iπ \ln(x)$. Looks like last result is correct, can someone confirm?