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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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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: existence of holomorphic log in simply-connected region not containing {0}
Posted: Oct 13, 2012 12:20 PM
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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 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?


With some corrections as above, yes.

>
> that





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