Numerical Integration over the n-Dimensional Spherical Shell (original) (raw)

The n-dimensional generalisation of a theorem by W. H. Peirce [1] is given, providing a method for constructing product type integration rules of arbitrarily high polynomial precision over a hyperspherical shell region and using a weight function r. Table I lists orthogonal polynomials, coordinates and coefficients for integration points in the angular rules for 3rd and 7th degree precision and for ?i = 3(1)8. Table II gives the radial rules for a shell of internal radius R and outer radius 1 : (i) a formula for the coordinate and coefficient in the 3rd degree rule for arbitrary n, R; (ii) a formula for the coordinates and coefficients for the 7th degree rule for arbitrary n and R = 0 and (iii) a table of polynomials, coordinates and coefficients to 9D for n = 4, 5 and R = 0, |, \, f. 1. Introduction. W. H. Peirce has given a method for constructing product type integration rules of arbitrarily high precision for the circular annulus and the threedimensional spherical shell [1]. This paper generalises his theorem to n dimensions and contains his results as particular cases. The method used is similar to that used by Stroud and Secrest [5] for an infinite solid n-sphere and also to that used by Mysovskih [6] for the finite solid n-sphere. Let Xi, x2, ■ ■ ■ , xn be Cartesian coordinates in n-dimensional Euclidean space. The region of integration, D, is the spherical shell of inner radius R, outer radius 1 and with centre at the origin, defined by (1.1) R2 ^ ¿Xi2 ^ 1. ¿=i An integration rule (1.2) 1= f f(x)dr= T,aiSixi) J D ¿=1 is sought which is of precision k, i.e., it is exact for all functions/(xi, x2, ■ ■ ■ , xn) that are polynomials in Xi, x,, ■ ■ ■ , xn of at most kth degree and is not exact for some (k + l)st degree polynomial.