Doubly Connected Regions

Abstract

Some well-known numerical methods for approximating conformal mapping of doubly regions onto an annulus or the unit disk are presented. There is a definite need for a simple yet accurate method for mapping a general doubly connected region onto a circular annulus. According to Kantorovich and Krylov (1958, p. 362) the problem of finding the conformal modulus is ‘one of the difficult problems of the theory of conformal transformation’. As such, analytic solutions have been determined for a very restricted class of doubly connected regions, like those mentioned in Muskhelishvili (1963, §48). Numerical solutions are also confined to a limited class of regions where either one boundary is circular or axisymmetric. Most common methods use integral equations, iterations, polynomial approximations, and kernels. We shall develop Symm’s integral equations and the related orthonormal polynomial method. A dipole formulation that leads to the method of reduction of connectivity shall be presented. Another useful method for multiply connected regions, based on Mikhlin’s integral equation, that also works for simply and doubly connected regions as well will be discussed in Chapter 13.