Rotationally invariant integrals of arbitrary dimensions

Abstract

In this note integrals over spherical volumes with rotationally invariant densities are computed. Exploiting the rotational invariance, and using identities in the integration over Gaussian functions, the general n-dimensional integral is solved up to a one-dimensional integral over the radial coordinate. The volume of an n-sphere with unit radius is computed analytically in terms of the Γ(z) special function, and its scaling properties that depend on the number of dimensions are discussed. The geometric properties of n-cubes with volumes equal to that of their corresponding n-spheres are also derived. In particular, one finds that the length of the side of such an n-cube asymptotes to zero as n increases, whereas the longest straight line that can fit within the cube asymptotes to a constant value. Finally, integrals over power-law form factors are computed for finite and infinite radial extent.