The approximate conformal mapping onto multiply connected domains

The method of boundary curve reparametrization is generalized to the case of multiply connected domains. We construct the approximate analytical conformal mapping of the unit disk with $N$ circular slits or an annulus with $(N-1)$ circular slits onto an arbitrary $(N+1)$ multiply connected finite domain with a smooth boundary. The method is based on the solution of the Fredholm equation. This solution is reduced to the solution of a linear system with unknown Fourier coefficients. The approximate mapping function has the form of a set of Laurent polynomials in the set of annular regions The method is easily computable.