Volumes of diced hyperspheres: Resumming the Tam-Zardecki formula (Q1112544)

A recursion relation is derived for the volume \(G_ n(s)\) of the intersection of the n-dimensional hypercube \(\{0\leq x_ i\leq 1\) for all \(i\leq n\}\) and the hypersphere of radius \(r=\sqrt{s}\) 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 \(F_ n(r)\) neither spreads indefinitely nor sharpens as a \(\delta\)-sequence, but attains an asymptotically stable Gaussian form about the moving point \(r=\sqrt{n/3}\). 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.