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Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
Інший ID:
2010 Mathematics Subject Classification: 35A08; 35J05; 32Q10; 31C12; 33C05
DOI: https://doi.org/10.3842/SIGMA.2011.108
DOI: https://doi.org/10.3842/SIGMA.2011.108
Посилання:
Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry / H.S. Cohl // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 39 назв. — англ.
Підтримка:
Much thanks to Ernie Kalnins, Willard Miller Jr., George Pogosyan, and Charles Clark for
valuable discussions. Much thanks as well to Richard Chapling for his comments in [4], in reference to an original version of the present paper and its implications. I would like to express my sincere gratitude to the anonymous referees and an editor at SIGMA whose helpful comments improved this paper. This work was partly conducted while H.S. Cohl was a National Research Council Research Postdoctoral Associate in the Information Technology Laboratory at the National Institute of Standards and Technology, Gaithersburg, Maryland, USA.
Дата:
2011
Переглядів:
439
Завантажень:
193
Анотація:
Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The R-radius hypersphere SdR with R>0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by d/2−1 and 1−d/2 respectively, with real argument x∈(−1,1).
