In 2 Dimensional Conformal field theory, We have two coordinates $ z=x+i y$ and $\bar{z}=x-i y$. Here $x$ and $y$ are both real. So, one can work directly with $z$ and $\bar{z}$.
I want to define $\partial_z$ and $\partial_{\bar{z}}$. But there is a little subtlety that $\partial_z \frac{1}{\bar{z}}=2 \pi \delta^2(z,\bar{z})$. One can prove this identity in many ways. The action of the derivative operator on a more general function is like $\partial_z \frac{1}{\bar{z}}f[\bar{z}]=2 \pi f[0]$. Here $f[\bar{z}]$ is well-behaved near 0.
Is there a way to define such an operator in Mathematica? Ordinary derivatives $"D"$ won't really work because of subtlety regarding Delta function.