Analog of the Watson transformation for angular variables and scattering on the potential $a/r^2$

A new representation for the angle-dependent amplitude of scattering on the spherically symmetrical potential is introduced which is complementary to the Watson complex angular momentum representation and includes the integration over the complex scattering angle. The representation proposed is particularly suitable for studying the problems related to the quasiclassical limit. The properties of the representation are illustrated by the example of scattering on the potential proportional to the inverse square of the distance. The role of orbiting and complex paths is discussed and a new closed formula for the amplitude in the eikonal approximation is derived. This formula is obtained also by means of approximate summing up the series over partial waves.