Effective Discretization of the Energy Integral and Grunsky Coefficients in Annuli

Abstract

Let Γ be an analytic Jordan curve in the complex plane. We formulate a discrete minimal energy problem in a suitable class of functions whose solution provides a geometrically fast converging approximation to the equilibrium measure of Γ. For this purpose an extremal point system that was introduced by K. Menke in 1972 is applied. In particular, an explicit error bound for the discretization of the energy integral is computed. The key to this error estimate is a univalence criterion for Laurent series, proved by R. Kuhnau in 1972. Finally, an estimate for the discrepancy between the approximating measures and the equilibrium measure is derived from the discretization error of the energy integral.