Approximate integration formulas for certain spherically symmetric regions

Abstract

where the As are constants and the Pi(Pil , Pin) are points in the, space. Each formula is exact for all polynomials up to and including a specified degree k where, as usual, k is called the degree of the formula. In Section 2 we give formulas of degrees 2, 3, 5 and 7 which are similar to previously developed formulas for other regions. The formulas of degree 2 are discussed by Stroud [14] for an arbitrary region; the formulas of degrees 3, 5 and 7 are similar to formulas for spheres given by Hammer and Stroud [4] and Ditkin [3]. In Sections 3, 4 and 5 we develop spherical product type formulas of arbitrarily high degree for U and V. These formulas have degree 2h 1 (h 1, 2, . ) and use hn points for even h and h' h -1+ 1 points for odd h. These formulas are obtained by products of one-dimensional formulas and are similar to formulas for the circle and 3-sphere given by Peirce [10, 11] and recently extended to the nsphere by Hetherington [6]. The spherical product formulas are most useful in 2 and 3 dimensions since in higher dimensions the numbers of points in the formulas become very large. In Section 4 we tabulate one-dimensional formulas, which are particular to the integrals U and V, from which product formulas for n = 2, 3, 4 may be constructed. From the formulas we give for U and V we can obtain formulas for any integrals of the form