Conformal mapping of right circular quadrilaterals

Abstract

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.