In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by \(J_{m}(z)=\frac {1}{2}\left [H_{m}^{(1)}(z)+H_{m}^{(2)}(z)\right ]\), where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy’s residue theorem to convert these Fourier-type integrals into the infinite integrals on \([0,+\infty )\). Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method.

在本文中，我们重点研究具有端点奇异性的代数和对数类型的高振荡贝塞尔变换的计算和分析。基于数值最速下降法的改进，我们提出了一种新的有效的正交规则。首先，我们将考虑的积分分为\（J_ {m}（z）= \ frac {1} {2} \ left [H_ {m} ^ {（1）}（z）+ H_ {m}两部分^ {（（2）}（z）\ right] \），其中每个部分都可以转换为傅立叶型积分。然后，我们使用柯西残差定理将这些傅立叶型积分转换为\（[[0，+ \ infty）\）上的无穷积分。接下来，通过构造一些适当的高斯正交规则，可以有效地计算所得的无限积分。另外，我们以频率参数的反幂进行误差分析。最后，提供了几个数值示例，说明了该方法的有效性和准确性。