Moduli of quadrilaterals and quasiconformal reflection

TL;DR

This paper derives a formula for conformal mappings of the upper half plane onto the exterior of convex polygonal quadrilaterals, analyzing accessory parameters via hypergeometric functions and providing computational algorithms.

Contribution

It introduces a new explicit formula for conformal maps of polygonal quadrilaterals' exterior, connecting Schwarz-Christoffel transformations with Lauricella hypergeometric functions.

Findings

Derived a conformal mapping formula for convex quadrilaterals' exterior.

Analyzed accessory parameters using hypergeometric functions.

Developed a Mathematica algorithm for computations.

Abstract

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. In particular, we study the special case of an isosceles trapezoidal polygon $L$ and obtain some estimates for the coefficient of quasiconformal reflection over $L$ in terms of special functions and geometric parameters of~$L$.