By using the conformal mappings of plane with elliptic hole and a plane with cross-shaped hole into the outside of the circle, we construct systems of functions playing the role of bases in the spaces of the functions analytic in these domains.
The Faber polynomials are biorthogonal with 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.
Citation Example:Sukhorolskyi M. A. Boundary-value problems for the Helmholts equation in domains of the complex plane // Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 363-376.