Numerical conformal mapping of multiply connected regions by Fornberg-like methods (Q1806001)

Let \(\Omega\) be a domain outside of \(n\) disjoint Jordan curves \(\Gamma_k\),with \(\infty\in \Omega\). It is well-known (Koebe 1908) that there exists a circular domain \(D\), outside of \(n\) disjoint circles \(C_k\), with \(\infty\in D\), which can be mapped conformally onto \(\Omega\). The authors present a method to compute this mapping \(f\), in particular the positions of the circles \(C_k\) and the boundary correspondence from \(D\) to \(\Omega\) defined by \(f(z_k+ \rho_k e^{i\theta})= \gamma_k (S_k(\theta))\) where \(\gamma_k(S)\) is the representation of \(\Gamma_k\) in terms of arc length. This is done by a Newton like method, that is by linearisation, which is observed (but not proved) to be quadratically convergent. A substantial part of the paper deals with criteria, when a function defined on \(C= \cup C_k\) can be extended analytically into \(D\setminus \{\infty\}\). To approximate \(f\) numerically, the Fourier expansion of \(f(z_k+ \rho_k e^{i\theta})\) has to be truncated. Finally, a large linear system has to be solved. -- The amount of numerical labor seems enormous. Six experiments show good results, but in five of them the \(C_k\) are all analytic Jordan curves. The reviewer believes that no other method can beat Koebe's original method, based on an iteration of mappings of simply connected domains, in implicity and efficiency.