email@example.com 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.
Not the most efficiently chosen conditions for such a result: you could make a stronger version by dropping the requirement for simple connectedness or by changing the second requirement to being that the origin not be in R.
> 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?
Well, I would suggest gamma being an arc, not a closed curve, or your integral will always be zero.
Then, for a fixed starting point z_0 and an appropriate additive constant C (existence of which may require proof?) I'd say you would be home free.