Computing h- function of the complement of a slit via parallel-slit mapping and validation via different conformal mapping

Abstract

In potential theory, h-function is a concept intimately connected to Brownian motion and serves as a method for depicting the geometry of a domain. Consider a particle released from a fixed location z₀ in a region Ω and permitted to move randomly throughout Ω, exhibiting Brownian motion, until it first encounters the boundary. The distance from the starting point z₀ at the time of impact is recorded. Also, the process with millions of Brownian particles that were discharged from the same point z₀ is repeated. This information is formally expressed as a function known as the h-function of the given region Ω with respect to the specified basepoint z₀. The h-function is an increasing function and takes only the values in the interval [0,1]. This study forms the h-function of a simply connected region Ω formed by deleting a slit from the complex plane when the basepoint z₀ is fixed along the line of the slit, by two different methods. The first method starts with parallel-slit mapping, which is in terms of the prime function. This map transforms the interior of the unit disc D(ζ) to the region Ω. A Cayley-type map is used to transform the region D(ζ) to the lower half-plane and the boundary of the disc to the real line. A function W(ζ) is formulated whose imaginary part is harmonic, and the Im[W(ζ)] evaluated at ζ₀ produces the h-function of the region Ω, where ζ₀ is the preimage of the basepoint z₀. The second method is used to cross-check the formed h-function. In this method, a sequence of conformal maps whose composition transforms the given region Ω to the half-plane is used instead of the prime function. Subsequently, the intersection between the boundary of the region Ω and the closed ball of radius r centered at z₀ and its image in the half- plane is traced. Both methods produce the same h-function.