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

TL;DR

The paper derives a dimensionally robust generalization of the Wallis formula for monomial integrals over the unit sphere and unit ball in ${\mathbb{R}}^n$, proving exact expressions $\int_{\partial B(0,1)} x^{2\beta}\,d\sigma(x)=\frac{2}{\Gamma(|\beta|+n/2)} \prod_{k=1}^n \Gamma(\beta_k+1/2)$ and $\int_{B(0,1)} x^{2\beta}\,dx=\frac{1}{\Gamma(|\beta|+1+n/2)} \prod_{k=1}^n \Gamma(\beta_k+1/2)$ for $\beta_k>-1/2$, together with sharp asymptotics as subsets or all $\beta_k$ tend to infinity. The authors develop a Bessel-function framework, introducing $\Psi_\nu(t)=J_\nu(t)/t^\nu$, and derive Fourier-transform representations for monomials on the sphere, enabling connections to Wallis means via $\xi\to0$. They further apply these tools to compute integrals of trigonometric series on the sphere and ball, expressing results in terms of Bernoulli and Euler numbers. Overall, the work provides explicit higher-dimensional integral formulas, asymptotic regimes, and Fourier-analytic machinery relevant to harmonic analysis and physics applications in $n$-dimensional spaces.

Abstract

We compute the integral of monomials of the form $x^{2β}$ over the unit sphere and the unit ball in $R^n$ where $β= (β_1,...,β_n)$ is a multi-index with real components $β_k > -1/2$, $1 \le k \le n$, and discuss their asymptotic behavior as some, or all, $β_k \to\infty$. 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 $x^α$ restricted to the unit sphere in $R^n$, where the multi-indices $α$ have integer components, and discuss their behaviour at the origin.

Theorems & Definitions (8)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• proof
• Theorem 4.1
• proof