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

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 $H\to \infty$.