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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.08493 |
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| _version_ | 1866915104296009728 |
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| author | Calderon, Calixto P. Torchinsky, Alberto |
| author_facet | Calderon, Calixto P. Torchinsky, Alberto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integration of monomials over the unit spere and unit ball in $R^n$ Calderon, Calixto P. Torchinsky, Alberto Classical Analysis and ODEs 26B25, 42B99 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. |
| title | Integration of monomials over the unit spere and unit ball in $R^n$ |
| topic | Classical Analysis and ODEs 26B25, 42B99 |
| url | https://arxiv.org/abs/2501.08493 |