```Date: Oct 13, 2012 2:48 AM
Author: Stuart M Newberger
Subject: Re: existence of holomorphic log in simply-connected region not<br> containing {0}

On Thursday, October 11, 2012 4:37:37 PM UTC-7, (unknown) wrote:> Hi, All:> > > >  I'm looking for a proof of the existence of a holomorphic log in> > > >  a region R that are simply-connected but do not wind around the origin.> > > >  My idea is:> > > >   logz is defined as the integral Int_Gamma dz/z , for Gamma a simple-closed> > > >   curve. The log is then well-defined , since, in simply-connected regions,> > > >   the integral is independent of path. In addition,  1/z is holomorphic> > > >   since z=/0 in R . Then the integral is well-defined and holomorphic,> > > >   (integral of holomorphic function is holomorphic ) , so the log exists.> > > >   Is this O.K?> >  > > > >   thatAs George said ,in your region where where the Cauchy theorem holds the function 1/z z|=0 has a primitive f(z) ,that means a function with f'(z)=1/z in the region. If a is in the region, then by adding a constant to f, arrange that exp(f(a)=a  Now show that d/dz( (1/z)exp(f(z))) =0 in the region so the indicated function is a constant which is 1 since that is its value at a.Regards,smn
```