antiholomorphic derivative on z^a

Question

It is a somewhat well-known statement that $\partial_{\bar z} \frac{ 1 }{ z } \propto \delta^2(z, \bar z)$. I wonder if it there is generalization to $z^\alpha$ with $\alpha\in \mathbb{R}$? For $\alpha \in \mathbb{Z} \backslash \{-1\}$ the answer should be zero, but I'm not sure about more generic $\alpha$.