On Thu, 11 Oct 2012 16:37:37 -0700 (PDT), firstname.lastname@example.org 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.
A simply connected region that does not _include_ the origin cannot "wind around" the origin.
> > My idea is: > > logz is defined as the integral Int_Gamma dz/z , for Gamma a simple-closed > > curve.
No, not a _closed_ curve. In a simply connected region not containing the origin, the integral of 1/z over a closed curve is 0.
You meant to choose a base point p, choose a particular number L with e^L = p, and then say that log(z) is defined as the integral of 1/z over any curve from p to z. Definitely not a 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?