Uniform convergence of numerical conformal mappings of interior domains in the charge simulation method (Q1590700)

The author approximates a given conformal map \(g\) from the unit disk onto the interior of a Jordan curve \(C\) by expressions \(g_n(z)= e^{a_0}z \exp\{\sum^n_{k=1} a_k\log(1- {z\over z_{nk}})\}\) with \(z_{nk}=\rho \zeta_{nk}\), where \(\rho>1\) and \(\zeta_{nk}\) are equidistributed on the unit circle. The \(z_{nk}\) and \(\zeta_{nk}\) are called charge points and collocation points, respectively. The unknown charges \(a_k\) are determined by the condition \(\log|g_n(\zeta_{nk}) |=\log |g(\zeta_{nk}) |(k=1,2, \dots, n)\). If \(C\) is an analytic curve, \(|g_n(z)- g(z)|= O(q^n) (|z|<1)\) for some \(q<1\).