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Representation of conformal maps by rational functions

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Abstract

The traditional view in numerical conformal mapping is that once the boundary correspondence function has been found, the map and its inverse can be evaluated by contour integrals. We propose that it is much simpler, and 10–1000 times faster, to represent the maps by rational functions computed by the AAA algorithm. To justify this claim, first we prove a theorem establishing root-exponential convergence of rational approximations near corners in a conformal map, generalizing a result of D. J. Newman in 1964. This leads to the new algorithm for approximating conformal maps of polygons. Then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part, there must be a singularity or loss of univalence exponentially close to the boundary, and polynomial approximations cannot be accurate unless of exponentially high degree. This motivates the application of the new algorithm to smooth domains, where it is again found to be highly effective.

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Notes

  1. The name comes from “adaptive Antoulas–Anderson” and is pronounced “triple-A”.

  2. We use the SC Toolbox commands w = [2+i 1+i 1+2i 2i 0 2];c = .7+.7i;p = polygon(w);opts = sctool.scmapopt(’Tolerance’,1e-12);f = diskmap(p,opts);f = center(f,c).

  3. Following Driscoll’s suggestion (private communication), we have also improved the accuracy of the inverse map by adding the line newton = false after the command [ode,newton,tol,maxiter] = sctool.scinvopt(options) in the Toolbox file @diskmap/private/dinvmap.m.

  4. One can prove that \(f^{-1}\) is analytic at a right-angle salient corner by analytically continuing the conformal map around the vertex with four applications of the Schwarz reflection principle. Such an argument shows that in general, a corner of a polygon is a nonsingular point of the inverse conformal map if and only if the interior angle is \(\pi \) divided by an integer. This is essentially the same as the observation about sharpness just after (1).

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Acknowledgements

This paper originated in stimulating discussions with Anne Greenbaum and Trevor Caldwell about their computations with the Kerzman–Stein integral equation, and Grady Wright gave key assistance in a Chebfun implementation. The heart of the paper is Schwarz–Christoffel mapping, which is made numerically possible by Toby Driscoll’s marvelous SC Toolbox. Driscoll, and Yuji Nakatsukasa offered helpful advice along the way, and the suggestions of Dmitry Belyaev, Chris Bishop, and Tom DeLillo were crucial for developing the theorems of Sect. 4. Among other things, Belyaev caught an error in an early version of Theorems 2 and 3 and Bishop pointed us to Theorem 6.1 of [16] and proposed the idea of Theorem 4. Much of this article was written during an extremely enjoyable 2017–2018 sabbatical visit by the second author to the Laboratoire de l’Informatique du Parallélisme at ENS Lyon hosted by Nicolas Brisebarre, Jean-Michel Muller, and Bruno Salvy.

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  1. Mathematical Institute University of Oxford, Oxford, OX2 6GG, UK

    Abinand Gopal & Lloyd N. Trefethen

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  1. Abinand Gopal
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  2. Lloyd N. Trefethen
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Correspondence to Lloyd N. Trefethen.

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Gopal, A., Trefethen, L.N. Representation of conformal maps by rational functions. Numer. Math. 142, 359–382 (2019). https://doi.org/10.1007/s00211-019-01023-z

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