A non-integer Bessel uniform approximation for use in semiclassical collision theory

A new uniform approximation for use in semiclassical collision theory has been derived. It involves a Bessel function of non-integer order as the canonical integral. The integer Bessel uniform approximation is contained as a special case. The breakdown of the integer Bessel approximation for elastic transitions at high energies is analysed. The new approximation is derived with the help of Sommerfeld's contour integral representation of a Bessel function. Both classically allowed and classically forbidden transitions are treated. Limiting cases of the non-integer Bessel uniform approximation and generalizations to multidimensional integrals are also considered.