Numerical integration over the 𝑛-dimensional spherical shell

Abstract

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 s {r^s} . Table I lists orthogonal polynomials, coordinates and coefficients for integration points in the angular rules for 3rd and 7th degree precision and for n = 3 ( 1 ) 8 n = 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 , 1 4 , 1 2 , 3 4 R = 0,\tfrac {1}{4},\tfrac {1}{2},\tfrac {3}{4} .