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.

Automated summary

The h-function represents the probability of a Brownian traveller exiting a fixed point in a region within a certain distance from the point of release. We calculate the h-functions for various planar simply connected regions and study their asymptotic behavior at specific values of r. Additionally, we analyze the derivatives of h(r) at the point where two different regimes meet.