The harmonic-measure distribution functions, or h -functions, associated with several classes of planar multiply connected domains Ω ⊂ C and basepoint z 0 ∈ Ω locations are the principal objects of consideration in this paper. The h -function with respect to Ω and z 0 encodes the probability that a particle undergoing Brownian motion in Ω first collides with the boundary ∂ Ω within a certain distance from the basepoint z 0 where it was initially released. Recently, Green et al. (Green et al. 2022 Proc. R. Soc. A 478 , 20210832. ( doi:10.1098/rspa.2021.0832 )) derived the first explicit formulae for the h -functions of multiply connected symmetrical rectilinear slit domains. In this paper, we generalize and extend the h -function calculations in Green et al. by considering various types of planar domains—those whose boundaries consist of either rectilinear slits or circles—as well as different locations of the basepoint. Throughout, we make judicious use of the Schottky-Klein prime function and its associated theory to derive analytical formulae for the h -functions. Our examples yield solutions to instances of a variant of the conformal Skorokhod embedding problem.
Green et al. (2025) studied this question.