# Explicit Riemann mappings

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

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.