One-parameter families of conformal mappings of the half-plane onto polygonal domains with several slits

TL;DR

This paper extends Kufarev's Loewner differential equation method to handle multiple slits in conformal mappings of the half-plane onto polygonal domains, providing a system of ODEs and numerical validation.

Contribution

It introduces a modified Kufarev's method for multiple slits with interdependent lengths, deriving a system of ODEs for accessory parameters and validating it numerically.

Findings

Derived a system of ODEs for multiple slits

Validated the method through numerical calculations

Extended Kufarev's approach to more complex slit configurations

Abstract

Among various methods of finding accessory parameters in the Schwarz-Christoffel integrals, Kufarev's method, based on the Loewner differential equation, plays an important role. It is used for describing one-parameter families of functions that conformally map a canonical domain onto a polygon with a slit the endpoint of which moves along a polygonal line starting from a boundary point. We present a modification of Kufarev's method for the case of several slits, the lengths of which have depend of each other in a certain way. We justify the method and find a system of ODEs describing the dynamics of accessory parameters. We also present the results of numerical calculations which confirm the efficiency of our method.