- #1
ismaili
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Homework Statement
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]
Homework Equations
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!