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[Complex Functions]

  1. Aug 24, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]f(z)[/tex] is a complex function that belongs to [tex]C^1[/tex]. Prove that:

    [tex]\lim_{r\to{0}}\frac{1}{r^2}\oint_{\tiny{|z-z_0|=r}}{f(z)dz}=2\pi{i}\frac{\partial f}{\partial \overline z}(z_0) [/tex]

    3. The attempt at a solution
    Using Green's Theorem:
    [tex]\oint_{{C}}{f(z,\overline z)dz}=2{i}\iint\limits_D {\frac{\partial f}{\partial \overline z}} \, dA[/tex]

    I got:
    [tex]\lim_{r\to{0}}\frac{1}{r^2}\oint_{\tiny{|z-z_0|=r}}{f(z)dz}= \lim_{r\to{0}}\frac{1}{r^2}2{i}\iint\limits_{|z-z_0| \le r} {\frac{\partial f}{\partial \overline z}} \, dA[/tex]

    It would be perfect if [tex]\iint\limits_{|z-z_0| \le r} {\frac{\partial f}{\partial \overline z}} \, dA = {\pi}{r^2}\frac{\partial f}{\partial \overline z}(z_0)[/tex] but I don't know how to prove it (acctualy I can't even believe that it is true).

    Any directions will be appreciated.
  2. jcsd
  3. Aug 24, 2010 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    This is analysis. You don't need perfect, you just need close enough. :smile:
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