Asymptotics of the conformal modulus of a nonsymmetric unbounded doubly-connected domain under stretching

TL;DR

This paper investigates how the conformal modulus of a non-symmetric, unbounded doubly-connected domain behaves asymptotically when stretched infinitely along the x-axis, providing insights into a problem posed by Vourinen.

Contribution

It offers a partial solution to the asymptotic behavior of conformal modulus under stretching for arbitrary unbounded domains, extending previous understanding.

Findings

Describes asymptotic behavior of conformal modulus under stretching

Provides partial answer to Vourinen's problem for unbounded domains

Analyzes non-symmetric, unbounded doubly-connected domains

Abstract

We describe the asymptotic behavior of the conformal modulus of an unbounded doubly-connected domain, non-symmetric with respect to the coordinate axes, when stretched in the direction of the abscissa axis with coefficient $H\to +\infty$. Therefore, we give a partial answer to a problem, suggested by M.~Vourinen, for the case of an arbitrary unbounded domain.