An asymptotic expansion for the hypergeometric function F12(a,b;c;x)

Abstract

Watson’s treatise expresses the ordinary Bessel function as a limit of a hypergeometric function, where the first two parameters go to infinity, while simultaneously the argument goes to zero. We have extended Watson’s method of proof to derive an asymptotic expression for the hypergeometric function to second order in the inverse of the first two parameters, with the ordinary Bessel function as its leading-order term. We show, as an example, that the use of this new result is pivotal in showing the correspondence between quantal and classical results of the differential cross section in Coulomb excitation.