The objective of this paper, is to apply a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals ${\int }_c^{d}G\left(x\right)e^{i\mu {\left(x-c\right)}^k}dx,-\mathrm{\infty }<c<d<\mathrm{\infty },$ and ${\int }_{-1}^{1}f\left(x\right)H_l\left(x\right)e^{i\mu x}dx,l=1,2,3$. Where G and f are non-oscillatory sufficiently smooth functions on the interval of integration. $H_l$ is a product of singular factors and $\mu \gg 1$ is an oscillatory parameter. The devised scheme is known to have rapid asymptotic order. Nevertheless, when a stationary point appears in $\left[c,d\right]$ or $\left[-1,1\right]$ the construction of Gauss rules becomes imperative in this approach. The computation of these integrals requires f and G to be analytic in a large complex region C accommodating the interval of integration. The integrals are changed into a problem of integrals on $\left[0,\mathrm{\infty }\right)$; which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function $e^{-x^k}$ with k positive. Symbolic powers of MATHEMATICA programming code and algorithms are provided for automatic computation of oscillatory integrals, (mentioned above) to test the efficiency of the presented experiments. Tangible and illustrative numerical examples are given to support our analysis.

本文的目的是应用一种基于线积分之和的方法来快速计算奇异和高振荡积分 ${\int }_C^{d}G\left(X\right){Ë}^{\mathrm{一世}\mu {\left(X-C\right)}^{ķ}}dX，-\mathrm{\infty }<C<d<\mathrm{\infty }，$ 和 ${\int }_{-\mathrm{1个}}^{\mathrm{1个}}F\left(X\right)H_{升}\left(X\right){Ë}^{\mathrm{一世}\mu X}dX，升=\mathrm{1个}，2，3$。其中G和f是在积分区间上的非振荡足够平滑的函数。$H_{升}$ 是奇异因素的产物， $\mu \gg \mathrm{1个}$是一个振荡参数。已知设计的方案具有快速渐近阶。不过，当静止点出现在$\left[C，d\right]$ 要么 $\left[-\mathrm{1个}，\mathrm{1个}\right]$高斯规则​​的构建在这种方法中变得势在必行。这些积分的计算需要在容纳积分间隔的大复数区域C中对f和G进行分析。积分变成一个积分问题$\left[0，\mathrm{\infty }\right)$; 稍后使用广义Gauss-Laguerre规则或相对于Freud加权函数的Gauss规则的构造来计算${Ë}^{-X^{ķ}}$与k正。提供了MATHEMATICA编程代码和算法的符号能力，用于自动计算振荡积分（如上所述），以测试所提出实验的效率。给出了具体的示例性数值示例以支持我们的分析。