Generalized elliptic integrals

TL;DR

This paper explores generalized elliptic integrals derived from conformal mappings of the upper half plane onto quadrilaterals, establishing key properties that extend classical results in conformal capacity and quasiconformal distortion.

Contribution

It introduces and analyzes generalized elliptic integrals associated with quadrilateral mappings, extending classical elliptic integral properties to more complex conformal shapes.

Findings

Established sharp monotonicity properties

Proved convexity of certain integral combinations

Generalized classical results to quadrilateral mappings

Abstract

Jacobi's elliptic integrals and elliptic functions arise naturally from the Schwarz-Christoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the analogous mapping of the upper half plane onto a quadrilateral and obtain sharp monotonicity and convexity properties for certain combinations of these integrals, thus generalizing analogous well-known results for classical conformal capacity and quasiconformal distortion functions.