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Topic: existence of holomorphic log in simply-connected region not
containing {0}

Replies: 4   Last Post: Oct 13, 2012 12:20 PM

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George Cornelius

Posts: 65
Registered: 12/12/04
Re: existence of holomorphic log in simply-connected region not containing {0}
Posted: Oct 12, 2012 2:12 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply 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.

> that


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