DQAINF: An algorithm for automatic integration of infinite oscillating tails
The paper describes a quadrature routine designed to integrate a (scalar or vector) function with a certain type of infinite oscillating, decaying tails over an infinite interval. The algorithm is based on the assumption that the oscillating behavior is due to the superposition of periodic functions which change sign when evaluated at points of distance half a period. Hence, following the basic ideas of \textit{J. N. Lyness} [J. Comp. Appl. Math. 12/13, 109-117 (1985; Zbl 0574.65013)], partitioning the original integral into an infinite series of integrals all of same interval length yields close relations to alternating series, and the Euler transformation (together with some modifications) implies very stable schemes for accelerated approximations. A FORTRAN subroutine for the algorithm is described in detail. Six examples show the efficiency of the method. Reviewer's remark: The explicit (exact) integration is possible for more examples than for the two indicated by the authors, in some cases even through the explicit representation of primitives with the help of higher transcendental functions. For instance: The integral \(I= \int^ \infty_ 1 p(x) dx/\sqrt x\) with \(p(x)= (-1)^ k\) for \(k- 1\leq x< k\) (Example 1) is related to the Riemann \(\zeta\)-function; we find \(I= 2- (2\sqrt 2- 1)\cdot \zeta(3/2)/\pi= 0.479580749561263933\). The integral of Example 2 may be represented by the Fresnel integrals, the special integral appearing here is \(\sqrt{2\pi} f(\sqrt{2/\pi})= 0.80952548174708844\) with a customary ``auxiliary function'' \(f(z)\), studied thoroughly by \textit{J. Boersma} [Math. Comput. 14, 380 (1960; Zbl 0091.293)] and tabulated in the handbook of \textit{M. Abramowitz} and \textit{I. A. Stegun} [Handbook of mathematical functions (1964; Zbl 0171.385)]. Similarly, the integral \(\int^ \infty_ 1 \sin(x+ 1/x)dx/\sqrt x\) of Example 3 is \(\sqrt{\pi/2}[\cos 2-S_ 1(2)\cos 2+ C_ 1(2)\sin 2]= 0.232948197094002526\), again with the Fresnel integrals. These are results due to H.-J. Fischer, Fac. of Math., Techn. Univ. Chemnitz.
scientific article; zbMATH DE number 698814
scientific article