Boundary-Value Problems for the Helmholtz Equation in Domains of the Complex Plane

By using conformal mappings of a plane with elliptic hole and a plane with cross-shaped hole into the exterior of the circle, we construct systems of functions playing the role of bases in the spaces of functions analytic in these domains. The Faber polynomials are biorthogonal to the basis functions. We construct the solutions of the Helmholtz equation in the plane with holes whose boundary values coincide with the boundary values of analytic functions represented in the form of series in these bases.