Integration of Monomials over the Unit Sphere and Unit Ball in Rn
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| Názov: | Integration of Monomials over the Unit Sphere and Unit Ball in Rn |
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| Autori: | Calixto P. Calderón, Alberto Torchinsky |
| Zdroj: | Selecciones Matemáticas, Vol 12, Iss 01, Pp 1-14 (2025) |
| Publication Status: | Preprint |
| Informácie o vydavateľovi: | Universidad Nacional de Trujillo, 2025. |
| Rok vydania: | 2025 |
| Predmety: | T57-57.97, Applied mathematics. Quantitative methods, Mathematics - Classical Analysis and ODEs, integration over the unit ball in r^n, QA1-939, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 26B25, 42B99, integration over the unit sphere in r^n, Mathematics |
| Popis: | We compute the integral of monomials of the form x2β over the unit sphere and the unit ball in Rn 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. |
| Druh dokumentu: | Article |
| ISSN: | 2411-1783 |
| DOI: | 10.17268/sel.mat.2025.01.01 |
| DOI: | 10.48550/arxiv.2501.08493 |
| Prístupová URL adresa: | http://arxiv.org/abs/2501.08493 https://doaj.org/article/367f7236d0b84965b0e7c2cbeae4fdca |
| Rights: | arXiv Non-Exclusive Distribution |
| Prístupové číslo: | edsair.doi.dedup.....725419689e1ab25434c6b203631a250b |
| Databáza: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | We compute the integral of monomials of the form x2β over the unit sphere and the unit ball in Rn 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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| ISSN: | 24111783 |
| DOI: | 10.17268/sel.mat.2025.01.01 |