On the evaluation of Bessel transformations with the oscillators via asymptotic series of Whittaker functions

Abstract

In this paper we are concerned with numerical evaluation of Bessel transformations with the oscillators. First, we rewrite the integrals by Whittaker functions. Then, based on the asymptotic series of Whittaker functions, the integrals are transformed into the problems involving the complex exponential functions, which can be efficiently computed by using the complex integration method. At the same time, we obtain the entirely different asymptotic expansions from the references. In particular, the asymptotic expansions can be denoted by the incomplete Gamma function. The error analysis shows that the decay of the error drastically improves as the frequency grows. The effectiveness and accuracy of the methods are tested for large arguments of Bessel functions.