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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| Дата: | 2015 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2015
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.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| id |
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irk-123456789-1470032019-02-13T01:24:34Z Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Cohl, H.S. Palmer, R.M. 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. 2015 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147003 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Cohl, H.S. Palmer, R.M. |
| spellingShingle |
Cohl, H.S. Palmer, R.M. Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Cohl, H.S. Palmer, R.M. |
| author_sort |
Cohl, H.S. |
| title |
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_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_sort |
fourier and gegenbauer expansions for a fundamental solution of laplace's equation in hyperspherical geometry |
| publisher |
Інститут математики НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147003 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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