Numerical computation of Stephenson's g-functions in multiply connected domains

There has been much recent attention on $h$-functions, so named since they describe the distribution of harmonic measure for a given multiply connected domain with respect to some basepoint. In this paper, we focus on a closely related function to the $h$-function, known as the $g$-function, which originally stemmed from questions posed by Stephenson in [3]. Computing the values of the $g$-function for a given planar domain and some basepoint in this domain requires solving a Dirichlet boundary value problem whose domain and boundary condition change depending on the input argument of the $g$-function. We use a well-established boundary integral equation method to solve the relevant Dirichlet boundary value problems and plot various graphs of the $g$-functions for different multiply connected circular and rectilinear slit domains.