Computing h-functions of Some Planar Simply Connected Two-dimensional Regions

The h-function h(r) represents the probability that a Brownian traveller released from a fixed point z0 in a region S2, first exits the region S2 somewhere within distance r of z0. We compute the h-functions of some planar simply connected regions, and investigate the asymptotics of these h-functions at certain values of r. Also, we analyse the derivatives of h(r) at a value of r where two regimes meet.