Extensions of Clenshaw-Curtis-type rules to integrals over a semi-infinite interval

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: Fej ' er'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, infinity) 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, infinity) in the complex plane z, the errors are of O(h(j)e(-c/h)) (j = 1, 2, ..., 5), respectively, with a constant c > 0 as step size h -> +0. The extension of Fej ' er'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 Fejer's first rule appears in the convergence curves.