(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the following identity,

[tex]

\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}\ln|z|^2

= 2\pi \delta^2(z,\bar{z})

[/tex]

where the delta function is defined such that

[tex]

\int dz d\bar{z} \detla^2(z,\bar{z}) = 1

[/tex]

2. Relevant equations

3. The attempt at a solution

While [tex]z[/tex] is not zero, the identity is easily seen to be hold, because,

[tex] \ln|z|^2 = \ln z + \ln\bar{z} [/tex]

So both sides are zero.

To include the point [tex]z=0[/tex], I tried to integrate both sides,

[tex] \int dzd\bar{z}... [/tex]

The right hand side is obviously [tex]2\pi[/tex].

But I don't know how to deal with the left hand side?

Anyone got any ideas?

Thanks!

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# Homework Help: A complex integral

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