Log-lightning computation of capacity and Green's function

See Video Abstract (click the "Video Abstract" button next to "PDF" button) A basic measure of size a set E in complex plane is logarithmic capacity cap(E). Capacities are known analytically for few simple shapes like ellipses, but most cases they must be computed numerically. We explore their computation by new "log-lightning'' method based on reciprocal-log approximations plane. For sequence 16 examples involving both connected and disconnected sets E, we compute capacities 8–15 digits accuracy at great speed MATLAB. The convergence almost-exponential with respect number poles employed, so it should possible many more if desired Maple or another extended-precision environment. This first systematic exploration applications log-lightning method, which opens up possibility solving Laplace problems an efficiency not achievable previous methods. computes just capacity, also Green's function its harmonic conjugate. It extends "domains negative measure" other Riemann surfaces.