In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral \(\int _{0}^{1} \frac{f(x)}{x-\tau } J_{m} (\omega x^{\gamma } )\textrm{d}x\) with the Cauchy type singular point, where \( 0< \tau < 1, m \ge 0, 2\gamma \in N^{+}. \) The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based on the special transformation of the integrand and the additivity of the integration interval, we convert the integral into three integrals. The explicit formula of the first one is expressed in terms of the Meijer G function. The second is computed by using the complex integration method and the Gauss– Laguerre quadrature rule. For the third, we adopt the Clenshaw– Curtis– Filon– type method to obtain the quadrature formula. In particular, the important recursive relationship of the required modified moments is derived by utilizing the Bessel equation and the properties of Chebyshev polynomials. Importantly, the strict error analysis is performed by a large amount of theoretical analysis. Our proposed methods only require a few nodes and interpolation multiplicities to achieve very high accuracy. Finally, numerical examples are provided to verify the validity of our theoretical analysis and the accuracy of the proposed methods.

在本文中，我们提出了一种有效的混合方法来计算高振荡贝塞尔积分\(\int _{0}^{1} \frac{f(x)}{x-\tau } J_{m} (\omega x^{\gamma } )\textrm{d}x\)具有柯西型奇异点，其中\( 0< \tau < 1, m \ge 0, 2\gamma \in N^{+}. \)混合方法是通过将复积分方法与Clenshaw-Curtis-Filon 型方法相结合而建立的。基于被积函数的特殊变换和积分区间的可加性，我们将积分转化为三积分。第一个的显式公式用 Meijer G 函数表示。第二个是使用复积分方法和高斯-拉盖尔求积法则计算的。对于第三种，我们采用Clenshaw-Curtis-Filon型方法来获得求积公式。特别是，利用贝塞尔方程和切比雪夫多项式的性质推导了所需修正力矩的重要递归关系。重要的是，严格的误差分析是通过大量的理论分析来进行的。我们提出的方法只需要几个节点和插值重数即可实现非常高的精度。最后，提供数值算例来验证我们理论分析的有效性和所提出方法的准确性。