Asymptotic methods in Rydberg collisions

Analytic expressions of some physical quantities for Rydberg transitions, like the first Born cross section for the n→n' transition, are found to contain terminating hypergeometric functions of large arguments which pose certain analytical and numerical problems that are well documented. We describe a simple method based on the use of the Jacobi polynomial to understand analytic properties and deduce an asymptotic expression of a general terminating hypergeometric function. To illustrate the utility, we apply the method to the n→n' Born cross section, for which situation the Tricomi asymptotic method had been used earlier in the literature. It is found that the Jacobi polynomial method gives strikingly good results for n→n' transitions over the entire range of the physical momentum transfer, in contrast to the Tricomi asymptotic method which gives unrealistically large results for large momentum transfers. (author). Letter-to-the-editor