Volumes of diced hyperspheres: resumming the Tam-Zardecki formula

摘要

A recursion relation is derived for the volume Gn(s) of the intersection of the n-dimensional hypercube {0 ⩽ xi ⩽ 1 for all i ⩽ n} and the hypersphere of radius r = √s centered at the origin. Laplace transformation is used to derive an expression for this function as a single contour integral of the Fourier type, thus decoupling the functions from the dimensional index and eliminating the error propagation involved in a successive application of the recursion formula. It is demonstrated that the corresponding hypersurface distribution function Fn(r) neither spreads indefinitely nor sharpens as a δ-sequence, but attains an asymptotically stable gaussian form about the moving point r=√n3. The result is inserted into the Tam-Zardecki formula for the total forward scattering amplitude of a laser beam, allowing the s-integral to be performed analytically and the series to be summed under the contour integral. Thus an infinite series, the nth term of which is an n-fold integral of a function depending on the n-dimensional radius, is reduced to a single integral of standard functions with exponential falloff, assuring rapid convergence.