Journal of the Ramanujan Mathematical Society
Volume 41, Issue 2, June 2026 pp. 125–142.
Computing new h-functions of several unbounded planar regions and
validation via different methods
Authors:
Arunmaran Mahenthiram and Piriyalucksan Paramesvarampillai
Author institution:Department of Mathematics and Statistics, University of Jaffna, Jaffna 40000, Sri Lanka.
Summary:
In a two-dimensional region Ω, a Brownian particle released from a
fixed location z{0} will travel randomly until it reaches the region's
boundary ∂Ω. For every value of r, we are interested in the
probability that it will strike the boundary somewhere within the distance r
from the starting point z{0}. The function h(r), also known as the
harmonic-measure distribution function or h-function of Ω with respect
to z{0}, is obtained by summing these probabilities for all values of r. This
h-function conveys details regarding the form of the boundary of Ω. In
this paper, we compute the h-functions of several new planar simply connected
unbounded regions using the method of conformal maps. In addition, we
describe the specific features of these h-functions, including the asymptotic
behavior. Moreover, we cross-check our h-function formulae through different
methods.
Contents
Full-Text PDF