# A complex integral

1. Mar 4, 2010

### ismaili

1. The problem statement, all variables and given/known data

Prove the following identity,
$$\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}\ln|z|^2 = 2\pi \delta^2(z,\bar{z})$$
where the delta function is defined such that
$$\int dz d\bar{z} \detla^2(z,\bar{z}) = 1$$

2. Relevant equations

3. The attempt at a solution

While $$z$$ is not zero, the identity is easily seen to be hold, because,
$$\ln|z|^2 = \ln z + \ln\bar{z}$$
So both sides are zero.
To include the point $$z=0$$, I tried to integrate both sides,
$$\int dzd\bar{z}...$$
The right hand side is obviously $$2\pi$$.
But I don't know how to deal with the left hand side?

Anyone got any ideas?
Thanks!