The Kontorovich-Lebedev transform in diffraction problems for conical surfaces

One powerful analytical method for solution of diffraction problems in regions with conical boundaries is the method of the Kontorovich-Lebedev integral transform. The inversion formula is preferable for solution of diffraction problems or for calculation of the far field. In the case of boundary conditions such as the Dirichlet or the Neumann condition for the Helmholz equation, algebraic equations for transforms are established and solutions are found easy by means of the inversion formula. Difficulties begin in problems with boundary conditions of the third kind which depend on radial coordinates. The situation takes place for conic surfaces conducting along spirals, for instance. In this case for transforms we can obtain linear nonhomogeneous fnnctional equations with entire differences and complex coefficients including the associated Legendre functions.