Asymptotic expansion of the integral with two oscillations on an infinite interval

In this paper, we focus on constructing the asymptotic expansion for the highly oscillatory integral including of the product of exponential and Bessel oscillations with the stationary point. Based on the exact integral representation of Bessel function, the integral is transformed into a double oscillatory integral. For the resulting inner semi-infinite integral, we present a new way of a combination of the integration by parts and the Filon-type methods to produce the asymptotic expansion. Furthermore, the original oscillatory integral can be expanded in the sum of Gaussian hypergeometric function. The corresponding asymptotic property is also analysed. With increasing the oscillatory parameter, the error of the proposed asymptotic expansions decreases very fast. Numerical experiments are provided to illustrate the effectiveness of the expansion.