Asymptotics of the exterior conformal modulus of a symmetric quadrilateral under stretching map

TL;DR

This paper investigates how the exterior conformal modulus of a symmetric quadrilateral behaves asymptotically under stretching, showing it approaches a universal form proportional to the logarithm of the stretching coefficient.

Contribution

It establishes the asymptotic behavior of the exterior conformal modulus under stretching, demonstrating independence from boundary shape using elliptic integral theory.

Findings

Asymptotic modulus approaches (1/π) log H as H→∞

Behavior is shape-independent for symmetric quadrilaterals

Uses elliptic integrals to confirm asymptotic formula

Abstract

In this work, we study the distortion of the exterior conformal modulus of a symmetric quadrilateral, when stretched in the direction of the abscissa axis with the coefficient $H\to \infty$. By using some facts from the theory of elliptic integrals, we confirm that the asymptotic behavior of this modulus does not depend on the shape of the boundary of the quadrilateral; moreover, it is equivalent to $(1/\pi)\log H$ as $H\to \infty$.