Authors’ abstract: Denote by a ring domain with a smooth boundary and by its conformal image. It is known that the mapping satisfies the integral equation
where is the unit tangent vector, , , , . is the Kerzman-Stein kernel. It is smooth and skew-Hermitian. The equation (*) is separated into a system of two integral equations and another equation involving the modulus . The discritized integral equation leads to a system of nonlinear equations which is to be solved by an optimization method. An advantage of this approach is that it calculates the boundary correspondence functions and the modulus simultaneously. Some numerical examples are provided.