Harmonic polynomials and their application in solving problems of the mathematical theory of cracks

We consider new spectral relations for harmonic polynomials of two variables. The basic relations of this type are the integral relations in a disk, which make it possible to use these polynomials to construct the solution of certain singular equations of Newtonian potential type, of both first and second kind, given on the disk. By use of these results we determine the eigenvalues of certain two-dimensional integral equations of second kind.