
Re: existence of holomorphic log in simplyconnected region not containing {0}
Posted:
Oct 13, 2012 12:20 PM


On Thu, 11 Oct 2012 16:37:37 0700 (PDT), baclesback@gmail.com wrote:
>Hi, All: > > I'm looking for a proof of the existence of a holomorphic log in > > a region R that are simplyconnected 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 simpleclosed > > 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 welldefined , since, in simplyconnected regions, > > the integral is independent of path. In addition, 1/z is holomorphic > > since z=/0 in R . Then the integral is welldefined and holomorphic, > > (integral of holomorphic function is holomorphic ) , so the log exists. > > Is this O.K?
With some corrections as above, yes.
> > that

