Series equations with associated Legendre functions on the boundary of conical and spherical regions and their application to a scalar problem of diffraction theory (Q2754691)

Dual series equations in Legendre functions with real lower indices arise in problems of diffraction in conical domains. Using formulae of re-expansion of the Legendre functions with different sequences of lower indices, dual equations are reduced to infinite systems of algebraic equations. It has been demonstrated that the series equations should be treated in limiting sense and a rule for the corresponding limiting process, that ensures a correct solution, has been derived. Analytic representations of inverse operators for scalar dynamic problems are constructed. The obtained operators are applied to the solution of a symmetric problem of electromagnetic TM-wave diffraction on coaxial sections of conical surfaces with different angles, that cannot be solved by the Wiener-Hopf method. Field distribution in the radiation zone is determined.