Summary.We consider the integral equation method of Symm for the conformal mapping of simply-connected domains. For the numerical solution, we examine the use of spline functions of various degrees for the approximation of the source density c~. In particular, we consider ways for overcoming the difficulties associated with corner singularities. For this we modify the spline approximation and in the neighborhood of each corner, where a boundary singularity occurs, we approximate cr by a function which reflects the main singular behaviour of the source density. The singular functions are then blended with the splines, which approximate a on the remainder of the boundary, so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation. We show, by means of numerical examples, that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy.

This paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques. The paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration.

Summary.A numerical method, based on the integral equation formulation of Symm, is described for computing approximations to the mapping functions which accomplish the following conformal maps: (a) the mapping of a domain interior to a closed Jordan curve onto the interior of the unit disc, (b) the mapping of a domain exterior to a closed Jordan curve onto the exterior of the unit disc, (c) the mapping of a doubly-connected domain bounded by two closed Jordan curves onto a circular annulus. The numerical method is based on approximating the unknown source density by cubic splines and "singular" functions, and is particularly suited for the mapping of difficult domains having sharp corners.

Let f be the function which maps conformally a simply-connected d o ma i n Ω o n t o t h e u n i t d i s c . T h i s p a p e r i s c o n c e r n e d w i t h t h e problem of determining the dominant poles of f in comp1(Ω∩∂Ω), and of using this information in order to obtain accurate numerical approximations to f by means of the Bergman kernel method.

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