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.

在本文中，我们重点构建高振荡积分的渐近展开式，包括指数振荡和贝塞尔振荡与驻点的乘积。基于贝塞尔函数的精确积分表示，积分转化为双振荡积分。对于所得的内部半无限积分，我们提出了一种新的方法，将分部积分和 Filon 型方法相结合来产生渐近展开。此外，原始振荡积分可以在高斯超几何函数的和中展开。还分析了相应的渐近特性。随着振荡参数的增加，所提出的渐近展开式的误差减小得非常快。提供了数值实验来说明扩展的有效性。