Integration of Monomials over the Unit Sphere and Unit Ball in Rn

Authors

  • Calixto P. Calderón Dept of Math, Stat & Comp Sci. University of Illinois at Chicago, Chicago IL 60607, USA.
  • Alberto Torchinsky Department of Mathematics, Indiana University, Bloomington IN 47405, USA.

DOI:

https://doi.org/10.17268/sel.mat.2025.01.01

Keywords:

Integration over the Unit Sphere in R^n, Integration over the Unit Ball in R^n

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 = (β1, . . . , βn) is a multi–index with real components βk > −1/2, 1 ≤ k ≤ n, and discuss their asymptotic behavior as some, or all, βk → ∞. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in Rn. We also consider the Fourier transform of monomials xα restricted to the unit sphere in Rn, where the multi–indices α have integer components, and discuss their behaviour at the origin.

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Published

2025-07-26

How to Cite

Calderón, C. P., & Torchinsky, A. (2025). Integration of Monomials over the Unit Sphere and Unit Ball in Rn. Selecciones Matemáticas, 12(01), 1 - 14. https://doi.org/10.17268/sel.mat.2025.01.01

Issue

Vol. 12 No. 01 (2025): January - July

Section

Articles

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