Grothendieck decompostion simple example

[Jim Kata]

Say you have $$exp{(\omega)}$$ and $$\omega$$ has a simple pole show that
$$exp{(\omega)}=exp{(F+)}z^{-1}exp{(F-)}$$ where F+ is holomorphic and F- is antiholomorphic. My basic thought is if $$\omega$$ has a simple pole then $$\omega z$$ 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.

 

[bpet]

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.