Re: Conformal mapping
From: Robert Israel (israel_at_math.ubc.ca)
Date: 08/11/04
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Date: 11 Aug 2004 05:26:59 GMT
In article <waderameyxiii-D55416.21153210082004@news.supernews.com>,
The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote:
>In article <cf6tak$nkp$1@nntp.itservices.ubc.ca>,
> israel@math.ubc.ca (Robert Israel) wrote:
>
>> In article <waderameyxiii-771C3F.20364008082004@news.supernews.com>,
>> The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote:
>> >In article <080820040728092400%edgar@math.ohio-state.edu.invalid>,
>> > "G. A. Edgar" <edgar@math.ohio-state.edu.invalid> wrote:
>>
>> >> > >Does there exist a holomorphic mapping of the disc onto C?
>>
>> >> > Yes. For example, F(z) = ((z-1)/(z+1)+1)^2.
>>
>> >> > But not if you want it to be one-to-one.
>>
>> >> ...because if there were, the inverse would be a bounded nonconstant entire
>> >> function, contradicting Liouville's Theorem.
>>
>> >Right, although the map can be conformal at each point, for example f(z) =
>> >exp[i(1+z)/(1-z)].
>>
>> That's not onto: it misses 0. However, I think an antiderivative of that
>> would work.
>
>Egads, of course you're right. Does its antiderivative actually work?
I think so, but I have no proof. A plot of F(r e^{it}) for r near 1
appears to show winding numbers >= 1 in a large disk around the origin.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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