Formal power series and their applications to mathematical theory of diffraction

The formal power series (FPS) coefficients of which are smooth functions are considered. FPS form an algebra over the field $(\mathbb C)$ of complex numbers. It is possible to differentiate FPS. FPS are series having asymptotic character (in accordance with the definition by V. S. Buslaev and M. M. Scriganov). As an example of applications of FPS we consider the geometro-optical expansion for the scalar analog of Rayleigh waves.