On the relations connecting the scattering field with the scattering amplitude

Various functional relationships between the main parameters of the scattering process are considered in this paper: the scattering field and the scattering amplitude. These relations are either geometrically-optic type (decomposition in the form of the Atkinson-Wilcox series), or a connection through the representation of both functions in series in spherical functions(multipole representations), or as an integral representation (Devaney-Wolf representation). Such a variety of representations is also possible due to the analytic properties of both functions: the scattering field is an radiation solution of the Helmholtz equation, any two differentiable solutions of which are analytic functions of their arguments, and also because the scattering amplitude is an entire analytic function of its arguments. An analog of an Atkinson-Wilcox representation for the generalized scattering amplitude is also given, which is possible only in the case when the primary incident complex field is a radiating solution of the Helmholtz equation. The resulting scattering amplitude in this case also obeys the Helmholtz equation. It is shown that the Herglotz wave function coincides, up to a constant factor, with the Whittaker representation. The above results are very useful for applications, and in particular, in the problems of scientific instrument making.