Abstract:
The likelihood of a particle launched from a fixed point z_0 in a region Ω initially leaving the area within the distance r of z_0 is the harmonic-measure distribution function, or the h-function. This function is non-decreasing, right-continuous, and takes values on the unit interval [0,1]. The objective of this paper is to validate the h-function formula obtained via three different approaches for a simply connected region Ω formed by deleting a line segment [-i,i] from the complex plane with basepoint z_0=1. To evaluate the h-function, we employ various forms of conformal mappings, harmonic functions, and the prime function. These three approaches vary in their utilization of the prime function, both in terms of presence and methodology. In the first approach, we completely avoid employing the prime function. Rather, the conformal mapping from the unit disc is expressed through the combination of a Joukowski map and a Mobius transformation, with the h-function being determined by extracting the correct angle of view in the unit disc. In the second and third approaches, the prime function is utilized. In the second method, we do not employ the prime function in the conformal mapping from the disc to the region Ω, as it is essentially a Joukowski transformation. In the meantime, the main function is utilized in the Cayley-type map R(ζ) from the interior of the unit disc D_ζ to the lower half-plane, which is employed in the creation of the harmonic function Im[W(ζ)]. In the third approach, the prime function is applied twice: initially during the parallel-slit mapping from the unit disc D_ζ to the target region Ω, and subsequently in R(ζ) and consequently in Im[W(ζ)]. All three approaches yield identical h-function graphs. These methods provide a check for our h-function formula.
