Numerical computation of a preimage domain for an infinite strip with rectilinear slits

TL;DR

This paper introduces an iterative numerical method to compute a conformal preimage domain for an infinite strip with rectilinear slits, enabling applications like capacity estimation and flow analysis.

Contribution

The paper presents a novel iterative approach for numerically finding conformal maps of multiply connected domains with rectilinear slits, expanding computational tools in complex analysis.

Findings

Successfully computed conformal preimages for complex slit domains

Estimated capacities of condensers involving disjoint segments

Analyzed flow streamlines around obstacles in a strip

Abstract

Let $\Omega$ be the multiply connected domain in the extended complex plane $\overline{\C}$ obtained by removing $m$ non-overlapping rectilinear segments from the infinite strip $S=\{z\,:\, \left|\Im z\right|<\pi/2\}$. In this paper, we present an iterative method for numerical computation of a conformally equivalent bounded multiply connected domain $G$ in the interior of the unit disk $\D$ and the exterior of $m$ non-overlapping smooth Jordan curves. We demonstrate the utility of the proposed method through two applications. First, we estimate the capacity of condensers of the form $(S,E)$ where $E\subset S$ be a union of disjoint segments. Second, we determine the streamlines associated with uniform incompressible, inviscid and irrotational flow past disjoint segments in the strip $S$.