Schwarz-Christoffel formula for multiply connected domains
The author derives a Schwarz-Christoffel formula for the conformal map of an arbitrary \(n\)-connected domain \(\mathbb{D}\) bounded by \(n\) mutually disjoint circles \(|z-a_k|=r_k\), \(k=1,..,n\), onto the exterior of \(n\) mutually disjoint polygons. To construct the conformal map the author solves the Riemann-Hilbert problem \[ \Re[(t-a_k)\psi (t)]=-1,\,\,\,\,\,\,\,\,|t-a_k|=r_k,\,\,\,\,k=1,..,n, \] where the function \(\psi (z)\) is analytic in \(\mathbb{D}\) and continuous on \(\mathbb{D}\cup \partial\mathbb{D}\) except for certain singularities on the boundary of \(\partial \mathbb{D}\). The derivation utilizes the uniform convergence of the Poincaré series for the classical Schottky groups that was established in [\textit{V. Mityushev}, Proc. Am. Math. Soc. 126, No. 8, 2399--2406 (1998; Zbl 0899.30029)]. As a result the author is able to obtain the Schwarz-Christoffel formula for arbitrary \(n\)-connected domains \(\mathbb{D}\) without any geometric restriction on the location of the non-overlapping disks \(|z-a_k|\leq r_k\).
scientific article; zbMATH DE number 6124796