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Grothendieck decompostion simple example

  1. Jul 30, 2011 #1
    Say you have [tex]exp{(\omega)}[/tex] and [tex]\omega[/tex] has a simple pole show that
    [tex]exp{(\omega)}=exp{(F+)}z^{-1}exp{(F-)}[/tex] where F+ is holomorphic and F- is antiholomorphic. My basic thought is if [tex]\omega[/tex] has a simple pole then [tex]\omega z[/tex] is holomorphic and on the punctured disk it can be represented by a Laurant series. The problem is that I'm missing a factor of log on z.
     
  2. jcsd
  3. Aug 10, 2011 #2
    That can't be right - consider w=1/z, so exp(w) has an essential singularity at z=0 and thus hits almost every value in a neighbourhood of 0. But if F+ and F- are holomorphic and antiholomorphic then exp(F+) and exp(F-) would be close to 1 near z=0, which means that |exp(F+)*exp(F-)/z| is bounded from below near z=0.
     
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