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