Summary: The paper is devoted to the development of an efficient numerical algorithm for computation of the conformal mapping of a double connected domain with the smooth boundary onto a circular annulus. The approach we present is the further development of the K. I. Babenko method of numerical conformal map of a simply connected domain onto the circle.
In the case of double connected domain there arises a serious difficulty connected with the following fact. The quotient of the radii of the annulus (so-called the conformal radius) is an invariant of the domain under investigation and have to be calculated also. Our approach is based on the double layer potential theory of the Dirichlet-Muskhelishvili problem in a multi-connected domain. Discretization of the system of boundary integral equations of the problem by means of Babenko’s method leads to the numerical algorithm which permits to compute simultaneously the conformal radius and the map. For the case of double connected domains bounded by circles and ellipses some test calculations are presented. The important case of the map of an annulus onto the given double connected domain will be treated in the forthcoming publication.