TITLE: Representation of Dirac Delta Function in complex plane
QUESTION [7 upvotes]: I am stuck on a representation of the Dirac Delta function that is used
in several books I am reading. They state:
$\begin{equation}
\delta^{(2)} = \frac{1}{2\pi} \partial_{\bar{z}} \frac{1}{z} = \frac{1}{2\pi}
\partial_{z} \frac{1}{\bar{z}}.
\end{equation}
$
They all either do not give a proof or I am not understanding the short motivations they give for this. Does anyone know how to understand this? Do
you have some references where I can read on this?
Help is greatly appreciated.

REPLY [8 votes]: According to the definition of distributional derivative, the statement $\delta =  \frac{1}{2\pi}
\partial_{z} (1/ \bar{z})$ means: for every compactly supported $C^\infty$ function $\phi$, 
$$\phi(0)= - \frac{1}{2\pi} \int_{\mathbb C} \frac{1}{\bar z} \frac{\partial \phi}{\partial z}\tag{1}$$ 
The formula (1)  is known as the Cauchy-Pompeiu formula, the generalization of the Cauchy integral formula to smooth (not necessarily holomorphic) functions. There is no   boundary term in (1) since $\phi$ is compactly supported.
One can express  (1) by saying that convolution with $-\frac{1}{2\pi}\frac{1}{\bar z}$ (the Cauchy kernel) is an operator inverting $z$-derivative. This operator is known as the Cauchy transform. You can read about it in The Cauchy Transform, Potential Theory and Conformal Mapping by Bell.