The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
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
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Thu, 11 Oct 2012 16:37:37 -0700 (PDT), 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

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.