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Typical proofs of the Riemann mapping theorem are not terribly explicit (one maximizes a functional, or something equivalent, such as using Dirichlet's principle).

The theorem states that if $U$ is a simply connected open subset of the plane, then there is a biholomorphism between $U$ and the unit disk. I imagine, due to the wild generality of the result, no explicit construction can be expected in general. However, in many concrete cases, I would think a construction "by hand" should be possible; and in applications (to problems in engineering, for example), this would almost be a requirement.

Do you know of such a construction, or of a reference where these constructions are discussed? (The answer may of course only apply to certain families of open sets.)

I know of a very nice reference: "Schwarz-Christoffel Mapping", by Tobin A. Driscoll and Lloyd N. Trefethen, Cambridge Monographs on Applied and Computational Mathematics (No. 8). The Schwarz-Christoffel Mappings explicitly give us biholomorphisms between the upper half plane and the interior of simple polygons. I am hoping for additional examples.

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Google for the Schwarz-Christoffel formula and keep your favorite manual on elliptic integrals handy. –  t.b. Jun 7 '11 at 1:47
    
Hi Theo, thanks! I actually have been working through "Schwarz-Christoffel Mapping" by Driscoll and Lloyd (cambridge.org/cz/knowledge/isbn/item1169156/?site_locale=cs_CZ) It is what inspired the question. (I'll edit the body of the question to reflect this.) –  Bruce George Jun 7 '11 at 2:04
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This is a very big question, and a lot of work has been done on making the Riemann mapping theorem more explicit. I have several comments:

  1. Böttcher coordinates provide explicit Riemann maps for the Fatou component containing a superattracting fixed point for a rational map. In particular, a Riemann map for the complement of the filled Julia set of a quadratic polynomial with connected Julia set can be computed fairly explicitly. Similarly, the Riemann map for the complement of the Mandelbrot set is fairly explicitly computable.

  2. Thurston and others have done some beautiful work involving approximating arbitrary Riemann maps using circle packings. See Circle Packing: A Mathematical Tale by Stephenson.

  3. To some extent, constructing a Riemann map is simply a matter of constructing a harmonic function on a given domain (as well as the associated harmonic conjugate), subject to certain boundary conditions. The solution to such problems is a huge topic of research in the study of PDE's, although the connection with Riemann maps is rarely mentioned.

Edit: Incidentally, if I wanted to construct a Riemann map explicitly on a given domain $D$, I would use the following PDE's approach. First, translate the domain so that it contains the origin. Next, use a numerical method to construct a harmonic function $F$ satisfying $$ F(z) \;=\; -\log |z| $$ for all $z\in\partial D$, and let $$ R(z) = |z|e^{F(z)}. $$ Then $R(0) = 0$, $R|_{\partial D} \equiv 1$, and $\log R$ is harmonic, so $R$ is the radial component (i.e. modulus) of a Riemann map on $D$. The angular component can now be determined by the fact that its level curves are perpendicular to the level curves of $R$, and have equal angular spacing near the origin.

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Many thanks for this fantastic answer! –  Bruce George Jun 7 '11 at 4:30
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