Integration of monomials over the unit spere and unit ball in $R^n$

TL;DR

This paper derives formulas for integrating monomials over the unit sphere and ball in high-dimensional space, explores their asymptotic behavior, and examines Fourier transforms of monomials on the sphere.

Contribution

It provides explicit integral formulas and asymptotic analysis for monomials over high-dimensional spheres and balls, including Fourier transforms of monomials on the sphere.

Findings

Explicit formulas for monomial integrals over spheres and balls in R^n.

Asymptotic behavior of these integrals as exponents grow large.

Analysis of Fourier transforms of monomials on the sphere.

Abstract

We compute the integral of monomials of the form $x^{2\beta}$ over the unit sphere and the unit ball in $R^n$ where $\beta = (\beta_1,...,\beta_n)$ is a multi-index with real components $\beta_k > -1/2$, $1 \le k \le n$, and discuss their asymptotic behavior as some, or all, $\beta_k \to\infty$. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in $ R^n$. We also consider the Fourier transform of monomials $x^\alpha$ restricted to the unit sphere in $R^n$, where the multi-indices $\alpha$ have integer components, and discuss their behaviour at the origin.