Has the 140-Year-Old Schwarz-Christoffel Math Problem Finally Been Solved?

[Diffy]

I am not entirely sure about the details but thought I would share...

Link to Press Release:

http://www.eurekalert.org/pub_releases/2008-03/icl-1yo030308.php

A problem, the Schwarz-Christoffel formula which has defeated mathematicians for almost 140 years has been solved by a researcher at Imperial College London.

More detail:
he Schwarz-Christoffel mapping maps a suitably "nice" (such as the interior of the unit circle, or the upper half plane) region of the complex plane (numbers of the form x+iy, mapped as (x,y) in cartesian coords, where i^2 = -1) to the interior of a general polygon (in the complex plane) in a conformal manner (angles are preserved). However, the polygon must be "simply connected" (any loop can be smoothly shrunk to a point, without getting "caught" around a hole); a few "multiply connected" polygons were possible, but not in the general case. Now it appears the formula can be extended to include polygons that include such a hole (such as perhaps the area outside a small square, but inside a larger one), to be mapped to some member well-described family of regions like the exterior of several unit disks
http://sinews.siam.org/old-issues/2008/januaryfebruary-2008/breakthrough-in-conformal-mapping

 

[fresh_42]

Here is the paper: http://wwwf.imperial.ac.uk/~dgcrowdy/PubFiles/Paper-20.pdf

Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.