R. Michael Porter

Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

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 ℘-function, and consideration of this related mapping problem leads to some new formulas concerning the zeroes and the images of the half-periods of ℘.

Conformal mapping of right circular quadrilaterals

We study conformal mappings from the unit disc to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm–Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t, λ of the mapping problem. The parameter λ is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm–Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter λ. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series method for numerical solution. Rate of convergence, accuracy and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favourably with previously known methods for this type of problem.

Numerical Calculation of Conformal Mapping to a Disk Minus Finitely Many Horocycles

The Riemann mapping to the complement in a disk of a finite union of disjoint disks bounded by horocycles has a Schwarzian derivative in the form of a simple rational function R = R[{zk}, {rk}](z) with two accessory parameters zk, rk for each vertex ωk. It is shown that if the prevertices zk are presupposed (while the ωk are undetermined), there exists a unique set of values {rk} for which R is the Schwarzian derivative of such a horocyclic mapping. These values depend on the combinatorial structure of the adjacencies of horocycles. An algorithm is developed for calculating the correspondence, and numerical examples are presented.

General Solution of the Inhomogeneous Div-Curl System and Consequences

We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in

Gears, Pregears and Related Domains

\mathbb {R}^3

Contragenic functions on spheroidal domains

R3. An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation

Spectral parameter power series for Sturm–Liouville problems

DW=(Df/f)\overline{W}