A new series solution method for the transmission problem

We derive analytic series representation for the Neumann-Poincare operator using the exterior conformal mapping for general shape domains. We derive the formula by using the Faber polynomial basis for arbitrary Lipschitz domains. With the proposed method we can approximate the spectrum of the smooth domains by finding the eigenvalues of matrices. We also find the connection of the boundary integral equations with the exterior conformal mapping, explicitly.