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Topic: conformal mapping problem (Read 12333 times) 

immanuel78
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conformal mapping problem
« on: Feb 11^{th}, 2009, 6:21am » 
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Show that there is an analytic function defined on G=ann(0;0;1) such that f'(prime) never vanishes and f(G)=B(0;1). This problem comes from the textbook by John. ConwayIn particular, Riemann mapping theorem section.


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Eigenray
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Re: conformal mapping problem
« Reply #1 on: Feb 11^{th}, 2009, 5:17pm » 
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That's a good question. If we didn't require f to be locally injective we could simply take an automorphism of the unit disk that moves 0, and then compose with squaring. By the Riemann mapping theorem, it suffices to show that there is a surjective conformal map from X \ {p} to Y, where X,Y are proper simply connected domains, and p X. Then you can compose with the biholomorphic maps between X,Y, and , and an automorphism taking 0 to the appropriate point. In fact we can replace \ {p} with \ {p_{1},...,p_{n}}, where n 1. That is, for any p_{1},...,p_{n} , we can find q_{1},...,q_{n1} and a surjective conformal map from \ {q_{1},...,q_{n1}} to \ {p_{1},...,p_{n}}. To see this, apply an automorphism to take p_{n} = 0, and pick J an interval of length > 2 which contains exactly one argument of each of p_{1},...,p_{n1}. Then exp takes U = { x + i y  x < 0, y J } surjectively onto \ {0}, and we just remove the preimages of p_{1},...,p_{n1}. The simplest example I can think of is to take X a large disk around the origin, and f(z) = z^{2}(z1). We need to remove {0, 2/3} since these are the points where f'(z)=0. But f(1) = f(0) and f(1/3) = f(2/3), so as long as X is big enough, f(X \ {0,2/3}) = f(X). It remains to show that for X large enough, f(X) is simply connected. Actually I'm not sure how to do this, but it should be true for any polynomial.


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Eigenray
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Re: conformal mapping problem
« Reply #2 on: Feb 12^{th}, 2009, 3:20am » 
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Maybe this is easier: Let (z) = z+1/z, and for r > 1, let A_{r} = { z : 1/r < z < r }. Then (A_{r}) is the interior of the ellipse (x/(r+1/r))^{2} + (y/(r1/r))^{2} = 1. In particular, it is simply connected. Now, let F(z) = z^{3}  3z. Since F((z)) = (z^{3}), F((A_{r})) = (A_{r^3}) is also simply connected. F is conformal on (A_{r}), except at z = 1. But we can remove these points without changing the image because F(2) = F(1). To sum up: (disk minus a point) (halfstrip minus a point) (disk minus 2 points) ((A_{r}) \ {1}) (A_{r^3}) disk, where the maps are exp and F, respectively.

« Last Edit: Feb 12^{th}, 2009, 3:43am by Eigenray » 
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Michael Dagg
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Re: conformal mapping problem
« Reply #3 on: Feb 15^{th}, 2009, 10:09am » 
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Clever example! How did you come with it? I have done a similar thing to produce a locally univalent but not univalent holomorphic map of the unit disk onto a convex image. You should be able to see how to do that from what you did here.

« Last Edit: Feb 15^{th}, 2009, 10:18am by Michael Dagg » 
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Eigenray
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Re: conformal mapping problem
« Reply #4 on: Feb 15^{th}, 2009, 10:49am » 
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I had the same idea: try to find a noninjective conformal map between two simply connected regions. The exponential won't work (since the image of a path between z and z + 2i k has winding number k), so I thought about taking a polynomial and removing the roots of the derivative, since we can add as many holes as we want. A quadratic won't work, so I tried a cubic. I wasn't sure how to show the image was simply connected though so I checked the section of Ahlfors on conformal mappings, where he uses the trick of pulling back via z = w+1/w to reduce a general cubic to w w^{3}.

« Last Edit: Feb 15^{th}, 2009, 10:55am by Eigenray » 
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Michael Dagg
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Re: conformal mapping problem
« Reply #5 on: Feb 18^{th}, 2009, 8:31am » 
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Thanks. For some a, 0 < a < 1 , then z(z^2  a^2)/(a^2 z^2  1) maps D onto D and branches at exactly two points z1,z2 . (D \ {z1,z2}) maps to the unit disk. We can get a conformal map from D onto D by considering a composed universal covering map of the domain. Now just restrict the domain to the annulus.

« Last Edit: Feb 18^{th}, 2009, 8:48am by Michael Dagg » 
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Michael Dagg
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Re: conformal mapping problem
« Reply #6 on: Mar 9^{th}, 2009, 6:46pm » 
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Here are some similar constructions to yours: 1. D.S.Greenstein, Monthly 65 ( 1958 ) , p.214. 2. Due to John V.Ryff (in a totally inaccessible 1969 journal (Math Reviews, vol. 42, review #1981.


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