The Clenshaw-Curtis (C-C) rule is a quadrature formula for integrals on an interval [− 1,1] and efficient for smooth integrands f(x). Analogous rules exist: Fejér’s first and second kind, Basu and corrected C-C rules. We attempt to extend these five rules to integrals over a semi-infinite interval \([0,\infty )\) to develop corresponding formulae. Developing contour integration representations of the errors of the formulae, we prove that for f(z) analytic in a region containing \([0,\infty )\) in the complex plane z, the errors are of O(h^je^−c/h) (\(j=1,2,\dots ,5\)), respectively, with a constant c > 0 as step size h → + 0. The extension of Fejér’s second rule (the case j = 1) agrees with a formula based on the Sinc interpolation. Numerical experiments show that new formulae inherit nice features of the C-C rule and its four analogs. For large h, the convergence rates are twice as fast as the asymptotic rates for small h. The kink phenomenon that is explained in the C-C rule and Fejér’s first rule appears in the convergence curves.

Clenshaw-Curtis (CC) 规则是区间 [− 1,1] 上积分的求积公式，对于平滑被积函数f ( x ) 是有效的。存在类似的规则：Fejér 的第一类和第二类、Basu 和更正的 CC 规则。我们尝试将这五个规则扩展到半无限区间\([0,\infty )\)上的积分，以开发相应的公式。开发公式误差的轮廓积分表示，我们证明对于复平面z 中包含\([0,\infty )\)的区域中的f ( z ) 解析，误差为O ( h ^j e ^{− ç /h} ) ( \(j=1,2,\dots ,5\) )，分别以常数c > 0 作为步长h → + 0。Fejér 的第二条规则的扩展（情况j = 1）与基于 Sinc 插值的公式。数值实验表明，新公式继承了CC规则及其四个类似物的优良特性。对于大h，收敛速度是小h的渐近速度的两倍。CC 规则和 Fejér 第一规则中解释的扭结现象出现在收敛曲线中。