In the theoretical physics literature on conformal field theory, one encounters distributional formulas like $$ \frac{1}{\pi}\partial_{\bar z}\frac{1}{z} = \delta(z), $$ where $\partial_{\bar z}$ is the usual Wirtinger derivative operator defined by $$ \partial_{\bar z} = \frac{1}{2}(\partial_x + i\partial_y). $$ The usual justification for the formula above is that given an analytic function $\varphi$, and given a disk $D$ containing the origin, one has \begin{align} \int_{D}\varphi(z)\frac{1}{\pi}\partial_{\bar z}\frac{1}{z} \, dx \, dy &= \frac{1}{\pi}\int_D \partial_{\bar z}\left(\frac{\varphi(z)}{z}\right)\, dx\, dy = \frac{1}{2\pi i}\int_{\partial D}\frac{\varphi(z)}{z}\,dz = \frac{1}{2\pi i}(2\pi i \varphi(0)) \\ &= \varphi(0) \end{align} where the second equality follows from Green's Theorem, and the third equality follows from Cauchy's Integral Formula.
It seems that when one performs such computations, one is using some notion of an analytic test function $\varphi$.
Is there a standard theory of distributions on the complex plane which addresses what sorts of test functions $\varphi$ one ought to consider? If so, what are they? (references more than welcome).