Justification of the collocation method for an integral equation of the exterior Dirichlet problem for the Laplace equation

A curvilinear integral equation of the exterior Dirichlet boundary value problem for the Laplace equation is considered. A new method is proposed to construct a quadrature formula for a singular integral. The method is used to derive a quadrature formula for the normal derivative of the double layer logarithmic potential. For specifically chosen control points, the equation is replaced by a system of algebraic equations, and the existence and uniqueness of a solution of this system are established. The convergence of the solution of this system to the exact solution of the integral equation is proved, and the rate of convergence of the method is deduced.