Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hype...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2015 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2015
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147003 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl, R.M. Palmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862680404447723520 |
|---|---|
| author | Cohl, H.S. Palmer, R.M. |
| author_facet | Cohl, H.S. Palmer, R.M. |
| citation_txt | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl, R.M. Palmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.
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| first_indexed | 2025-12-07T15:46:36Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147003 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:46:36Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cohl, H.S. Palmer, R.M. 2019-02-12T18:16:31Z 2019-02-12T18:16:31Z 2015 Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl, R.M. Palmer // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 31C12; 32Q10; 33C05; 33C45; 33C55; 35J05; 35A08; 42A16 DOI:10.3842/SIGMA.2015.015 https://nasplib.isofts.kiev.ua/handle/123456789/147003 For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials. This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of
 Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html.
 Much thanks to Willard Miller and George Pogosyan for valuable discussions. We would also
 like to express our gratitude to the anonymous referees and the editors for this special issue in
 honour of Luc Vinet, for their significant contributions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Article published earlier |
| spellingShingle | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Cohl, H.S. Palmer, R.M. |
| title | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_full | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_fullStr | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_full_unstemmed | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_short | Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry |
| title_sort | fourier and gegenbauer expansions for a fundamental solution of laplace's equation in hyperspherical geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147003 |
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