Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

TL;DR

This paper develops numerical and theoretical methods for conformal mappings from a disk to symmetric circular-arc quadrilaterals, involving Weierstrass elliptic functions, and provides new formulas related to the Schwarzian derivative.

Contribution

It introduces a numerical approach to relate accessory parameters to geometric parameters in conformal mappings involving Weierstrass P-functions.

Findings

Numerical solutions for accessory parameters ensuring univalence.

New formulas for zeros and images of half-periods of Weierstrass P.

Mapping techniques for symmetric circular-arc quadrilaterals.

Abstract

Numerical and theoretical aspects of conformal mappings from a disk to a circular-arc quadrilateral, symmetric with respect to the coordinate axes, are developed. The problem of relating the accessory parameters (prevertices together with coefficients in the Schwarzian derivative) to the geometric parameters is solved numerically, including the determination of the parameters for univalence. The study involves the related mapping from an appropriate Euclidean rectangle to the circular-arc quadrilateral. Its Schwarzian derivative involves the Weierstrass P-function, and consideration of this related mapping problem leads to some new formulas concerning the zeroes and the images of the half-periods of P.