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        general >> complex analysis >> conformal mapping problem
        
(Message started by: immanuel78 on Feb 11^th, 2009, 6:21am)

Title: conformal mapping problem
Post by immanuel78 on Feb 11^th, 2009, 6:21am

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. Conway-In particular, Riemann mapping theorem
section.

Title: Re: conformal mapping problem
Post by Eigenray on Feb 11^th, 2009, 5:17pm

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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif 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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gif X. Then you can compose with the biholomorphic maps between X,Y, and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif, and an automorphism taking 0 to the appropriate point.

In fact we can replace http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif \ {p} with http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif \ {p₁,...,p_n}, where n http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ge.gif 1. That is, for any p₁,...,p_n http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif, we can find q₁,...,q_n-1 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif and a surjective conformal map from http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif \ {q₁,...,q_n-1} to http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif\ {p₁,...,p_n}. To see this, apply an automorphism to take p_n = 0, and pick J an interval of length > 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif which contains exactly one argument of each of p₁,...,p_n-1. Then exp takes U = { x + i y | x < 0, y http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gif J } surjectively onto http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdelta.gif\ {0}, and we just remove the preimages of p₁,...,p_n-1.

The simplest example I can think of is to take X a large disk around the origin, and f(z) = z²(z-1). 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.

Title: Re: conformal mapping problem
Post by Eigenray on Feb 12^th, 2009, 3:20am

Maybe this is easier:

Let http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(z) = z+1/z, and for r > 1, let A_r = { z : 1/r < |z| < r }. Then http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(A_r) is the interior of the ellipse (x/(r+1/r))² + (y/(r-1/r))² = 1. In particular, it is simply connected.

Now, let F(z) = z³ - 3z. Since F(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(z)) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(z³), F(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(A_r)) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(A_r^3) is also simply connected.

F is conformal on http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(A_r), except at z = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pm.gif 1. But we can remove these points without changing the image because F(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mp.gif2) = F(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pm.gif1).

To sum up:
(disk minus a point) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cong.gif (half-strip minus a point) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/twoheadrightarrow.gif (disk minus 2 points) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cong.gif (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(A_r) \ {http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pm.gif1}) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/twoheadrightarrow.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varphi.gif(A_r^3) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cong.gif disk,
where the http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/twoheadrightarrow.gif maps are exp and F, respectively.

Title: Re: conformal mapping problem
Post by Michael Dagg on Feb 15^th, 2009, 10:09am

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.

Title: Re: conformal mapping problem
Post by Eigenray on Feb 15^th, 2009, 10:49am

I had the same idea: try to find a non-injective conformal map between two simply connected regions. The exponential won't work (since the image of a path between z and z + 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifi 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 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/mapsto.gifw³.

Title: Re: conformal mapping problem
Post by Michael Dagg on Feb 18^th, 2009, 8:31am

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.

Title: Re: conformal mapping problem
Post by Michael Dagg on Mar 9^th, 2009, 6:46pm

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.