Geodesic
A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form of the fundamental solutions for any number of dimensions in terms of generalised hypergeometric functions, with different closed forms for even and odd-dimensional cases.
Keywords: hyperspherical geometry; fundamental solution; Laplace's equation; separation of variables; hypergeometric functions. 
@article{SIGMA_2016_12_a78,
     author = {Richard Chapling},
     title = {A {Hypergeometric} {Integral} with {Applications} to the {Fundamental} {Solution} of {Laplace's} {Equation} on {Hyperspheres}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a78/}
}
TY  - JOUR
AU  - Richard Chapling
TI  - A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2016
VL  - 12
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a78/
LA  - en
ID  - SIGMA_2016_12_a78
ER  - 
%0 Journal Article
%A Richard Chapling
%T A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres
%J Symmetry, integrability and geometry: methods and applications
%D 2016
%V 12
%U http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a78/
%G en
%F SIGMA_2016_12_a78
Richard Chapling. A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a78/

[1] Aubin T., Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998 | DOI | MR | Zbl

[2] Cohl H. S., “Fundamental solution of Laplace's equation in hyperspherical geometry”, SIGMA, 7 (2011), 108, 14 pp., arXiv: 1108.3679 | DOI | MR | Zbl

[3] Cohl H. S., Kalnins E. G., “Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry”, J. Phys. A: Math. Theor., 45 (2012), 145206, 32 pp., arXiv: 1105.0386 | DOI | MR | Zbl

[4] Cohl H. S., Palmer R. M., “Fourier and Gegenbauer expansions for a fundamental solution of Laplace's equation in hyperspherical geometry”, SIGMA, 11 (2015), 015, 23 pp., arXiv: 1405.4847 | DOI | MR | Zbl

[5] Courant R., Hilbert D., Methods of mathematical physics, v. I, Interscience Publishers, Inc., New York, N.Y., 1953 | MR

[6] Dutka J., “The early history of the hypergeometric function”, Arch. Hist. Exact Sci., 31 (1984), 15–34 | DOI | MR | Zbl

[7] Euler L., “Specimen transformationis singularis serierum [E710]”, Nova Acta Academiae Scientarum Imperialis Petropolitinae, 12 (1801), 58–70; reprinted in: Opera Omnia, Ser. 1, 16:2 (1935), 41–55

[8] Gauss C. F., “Disquisitiones generales circa seriem infinitam $1 + \frac {\alpha \beta} {1 \cdot \gamma}x + \frac {\alpha (\alpha+1) \beta (\beta+1)} {1 \cdot 2 \cdot \gamma (\gamma+1)} xx + \text{etc.}$”, Commentationes societatis regiae scientiarum Gottingensis recentiores, 2 (1813), 1–46; reprinted in: Gesammelte Werke, v. 3, K. Gesellschaft der Wissenschaften, Göttingen, 1876, 123–163, 207–229

[9] Iliev B. Z., Handbook of normal frames and coordinates, Progress in Mathematical Physics, 42, Birkhäuser Verlag, Basel, 2006 | DOI | MR | Zbl

[10] Lee J. M., Introduction to smooth manifolds, Graduate Texts in Mathematics, 218, 2nd ed., Springer, New York, 2013 | MR | Zbl

[11] Pringsheim A., Faber G., Molk J., Analyse Algébrique, v. II, Encyclopédie des sciences mathématiques, II, Gauthier-Villars, Paris, 1911

[12] Rainville E. D., “The contiguous function relations for ${}_pF_q$ with applications to Bateman's $J_n^{u,v}$ and Rice's $H_n(\zeta,p,v)$”, Bull. Amer. Math. Soc., 51 (1945), 714–723 | DOI | MR | Zbl

[13] Szmytkowski R., “Closed forms of the Green's function and the generalized Green's function for the Helmholtz operator on the $N$-dimensional unit sphere”, J. Phys. A: Math. Theor., 40 (2007), 995–1009 | DOI | MR | Zbl

[14] Vidūnas R., “Contiguous relations of hypergeometric series”, J. Comput. Appl. Math., 153 (2003), 507–519, arXiv: math.CA/0109222 | DOI | MR | Zbl

[15] Vorsselmann de Heer P. O. C., Specimen Inaugurale De Fractionibus Continuis, Altheer, 1833 http://eudml.org/doc/204259

[16] Yamasuge H., “Maximum principle for harmonic functions in Riemannian manifolds”, J. Inst. Polytech. Osaka City Univ. Ser. A, 8 (1957), 35–38 | MR | Zbl