A concept for conformal mapping of multiply connected quasicircular domains (Q1360043)

The author presents a method for the conformal mapping \(f\) of an infinite domain \(D\) bounded by \(K\) nearly circular Jordan curves onto such a domain bounded by \(K\) circles with unknown centers \(\xi_j\) and radii \(a_j\). With \(g=f^{-1}\), it is assumed that \[ g(\xi)= \xi+\sum_{k=1}^K\;\sum_{n=1}^\infty G_{nk}\Biggl( \frac{a_k}{\xi-\xi_k} \Biggr)^n; \] the unknowns \(a_k\), \(\xi_k\), \(G_{kn}\) are computed indirectly by considering \[ \log\Biggl( \frac{g(\xi)}{\xi} \Biggr)= \sum_{k=1}^K\;\sum_{n=1}^\infty L_{kn}\Biggl( \frac{a_k}{\xi-\xi_k} \Biggr)^n \] with the unknowns \(a_k\), \(\xi_k\), \(L_{kn}\). The latter series is truncated, and the problem discretized. One arrives at a highly nonlinear system of \(2K\) equations, convergence is tested empirically. -- The problem could also be handled via Koebe's iteration method, which reduces the \(K\)-tuply connected case to a sequence of conformal mappings of simply connected domains.