Further extension of a class of periodizing variable transformations for numerical integration

Class SmSm variable transformations with integer mm, for accurate numerical computation of finite-range integrals via the trapezoidal rule, were introduced and studied by the author. A representative of this class is the sinmsinm-transformation. In a recent work of the author, this class was extended to arbitrary noninteger values of mm, and it was shown that exceptionally high accuracies are achieved by the trapezoidal rule in different circumstances with suitable values of mm. In another recent work by Monegato and Scuderi, the sinmsinm-transformation was generalized by introducing two integers pp and qq, instead of the single integer mm; we denote this generalization as the sinp,qsinp,q-transformation here. When p=q=mp=q=m, the sinp,qsinp,q-transformation becomes the sinmsinm-transformation. Unlike the sinmsinm-transformation which is symmetric, the sinp,qsinp,q-transformation is not symmetric when p≠qp≠q, and this offers an advantage when the behavior of the integrand at one endpoint is quite different from that at the other endpoint. In view of the developments above, in the present work, we generalize the class SmSm by introducing a new class of nonsymmetric variable transformations, which we denote as Sp,qSp,q, where pp and qq can assume arbitrary noninteger values, such that the sinp,qsinp,q-transformation is a representative of this class and Sm⊂Sm,mSm⊂Sm,m. We provide a detailed analysis of the trapezoidal rule approximation following a variable transformation from the class Sp,qSp,q, and show that, with suitable and not necessarily integer pp and qq, it achieves an unusually high accuracy when the integrand has algebraic endpoint singularities. We also illustrate our results with numerical examples via the sinp,qsinp,q-transformation. Finally, we discuss the computation of surface integrals in R3R3 containing point singularities with the help of class Sp,qSp,q transformations.