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A complex integral

  1. Mar 4, 2010 #1
    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!
     
  2. jcsd
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