Computing harmonic-measure distribution functions of some planar doubly connected regions

A harmonic-measure distribution function, or $h$-function, encodes the probability that a Brownian walker released from a fixed point in a region first hits the boundary of the region within a certain distance from the fixed point. We compute $h$-functions of several new planar doubly connected unbounded or bounded regions by using the $P$-function. Also, we cross-check our $h$-function graphs via numerical simulation.