On Computation of Capacities and Conformal Invariants

Abstract

We give a survey of the computation of the conformal capacity of planar condensers, generalized capacity, and logarithmic capacity with emphasis on our recent work 2020–2025. We also discuss some applications of our method based on the boundary integral equation with the generalized Neumann kernel to the computation of several other conformal invariants: harmonic measure, modulus of a quadrilateral, reduced modulus, hyperbolic capacity, and elliptic capacity. Here, the solution of the mixed Dirichlet-Neumann boundary value problem for the Laplace equation has a key role. At the end of the paper, we give a topic-wise structured list of our extensive bibliography on constructive complex analysis and potential theory.