Integration of Monomials over the Unit Sphere and Unit Ball in Rn
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 invol...
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| Format: | Article |
| Language: | Spanish |
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Universidad Nacional de Trujillo
2025-07-01
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| Series: | Selecciones Matemáticas |
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| Online Access: | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6616 |
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| author | Calixto P. Calderón Alberto Torchinsky |
| author_facet | Calixto P. Calderón Alberto Torchinsky |
| author_sort | Calixto P. Calderón |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-367f7236d0b84965b0e7c2cbeae4fdca |
| institution | Kabale University |
| issn | 2411-1783 |
| language | Spanish |
| publishDate | 2025-07-01 |
| publisher | Universidad Nacional de Trujillo |
| record_format | Article |
| series | Selecciones Matemáticas |
| spelling | doaj-art-367f7236d0b84965b0e7c2cbeae4fdca2025-08-20T03:31:59ZspaUniversidad Nacional de TrujilloSelecciones Matemáticas2411-17832025-07-01120111410.17268/sel.mat.2025.01.01Integration of Monomials over the Unit Sphere and Unit Ball in RnCalixto P. Calderón0https://orcid.org/0000-0002-4211-2110Alberto Torchinsky1https://orcid.org/0000-0001-8325-3617Dept of Math, Stat & Comp Sci. University of Illinois at Chicago, Chicago IL 60607, USA.Department of Mathematics, Indiana University, Bloomington IN 47405, USA.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.https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6616integration over the unit sphere in r^nintegration over the unit ball in r^n |
| spellingShingle | Calixto P. Calderón Alberto Torchinsky Integration of Monomials over the Unit Sphere and Unit Ball in Rn Selecciones Matemáticas integration over the unit sphere in r^n integration over the unit ball in r^n |
| title | Integration of Monomials over the Unit Sphere and Unit Ball in Rn |
| title_full | Integration of Monomials over the Unit Sphere and Unit Ball in Rn |
| title_fullStr | Integration of Monomials over the Unit Sphere and Unit Ball in Rn |
| title_full_unstemmed | Integration of Monomials over the Unit Sphere and Unit Ball in Rn |
| title_short | Integration of Monomials over the Unit Sphere and Unit Ball in Rn |
| title_sort | integration of monomials over the unit sphere and unit ball in rn |
| topic | integration over the unit sphere in r^n integration over the unit ball in r^n |
| url | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6616 |
| work_keys_str_mv | AT calixtopcalderon integrationofmonomialsovertheunitsphereandunitballinrn AT albertotorchinsky integrationofmonomialsovertheunitsphereandunitballinrn |