Quasiclassical scattering of fast particles near singularities of the classical cross section

The quasiclassical asymptotic behavior of the eikonal amplitude of scattering by a potential satisfying the Laplace equation is investigated in the neighborhood of the zero and focal values of the momentum transfer. It is shown that the corresponding reference integrals describing a degeneracy of arbitrary order in the two-dimensional method of stationary phase admit separation of the variables in the complex space of the impact parameter. They are expressed in terms of special functions that generalize the Airy and Bessel functions. The basic properties of these functions and the transition to the ordinary quasiclassieal behavior with increasing distance from the point of degeneracy are studied.