Clifford analysis techniques for spherical PDE

Journal of Fourier Analysis and Applications

Volumn 8, Issue 6, 2002, Pages 535-563

  Liu, Hong ^a   Ryan, John ^b  

^a EMBRY RIDDLE AERONAUTICAL UNIVERSITY   (United States)

^b UNIVERSITY OF ARKANSAS   (United States)

Author keywords

Cayley transformation spherical Helmholtz equation; Conformal transformations; Green's formula; Higher order spherical Dirac operators; Spherical Dirac operator; Spherical Laplacian

Indexed keywords

EID: 0036449507     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-002-0026-1     Document Type: Article

References (23)

1. Ahlfors, L. (1984). Old and new in Möbius groups, Ann. Acas. Sci. Fenn. Ser. A I Math., 9, 93-105.
2. Brackx, F., Delanghe, R. and Sommen, F. (1982). Clifford Analysis, Pitman, London.
3. Cnops, J. and Malonek, H. (1995). An Introduction to Clifford Analysis, Univ. Coimbra, Coimbra.
4. Delanghe, R., Sommen, F., and Soucek, V. (1992). Clifford Analysis and Spinor Valued Functions, Kluwer Acad. Publ., Dordrecht.
5. Durand, L., Fishbane, P.M., and Simmons, Jr., L.M. (1976). Expansion formulas and addition theorems for Gegenbauer functions, J. Math. Phys., 17, 1933-1948.
6. Fegan, H.D. (1976). Conformally invariant first order differential operators, Quart. J. Math. Oxford, 27(2), 371-378.
7. Hua, L.-K. (1981). Starting with the Unit Circle, Background to Higher Analysis, Springer-Verlag, translated by Kuniko Weltin.
8. Van Lancker, P. (1996). Clifford analysis on the unit sphere, Ph.D. paper, University of Gent.
9. Van Lancker, P. (1998). Clifford analysis on the sphere, Dietrich, V., et al., Eds., Clifford Algebra and Their Applications in Mathematical Physics, Kluwer Acad. Publ., 201-215.
10. Van Lancker, P. (1999). Approximation theorems for spherical monogenics of complex degree, Bull. Belg. Math. Soc., 6, 193-279.
11. MacRobert, T.M. (1967). Spherical Harmonics, An Elementary Treatise on Harmonic Functions with Applications, Pergamon Press, 1967.
12. Mitrea, M. (1999). Generalized Dirac operators on non-smooth manifolds and Maxwell's equations, preprint.
13. Peetre, J. and Qian, T. (1994). M obius covariance of iterated Dirac operators, J. Austral. Math. Soc., (Series A), 56, 403-414.
14. Ryan, J. (1996). Dirac operators on spheres and hyperbolae, Bol. Soc. Mat. Mexicana, 3(3), 255-269.
15. Ryan, J. (1997). Clifford Analysis on Spheres and Hyperbolae, Mathematical Methods in the Applied Sciences, Vol. 20, 1617-1624.
16. Ryan, J. (1995). Cauchy-Green type formulae in Clifford analysis, Trans. of the Am. Math. Soc., 347(4), 1331-1341, April.
17. Ryan, J., editor. (1996). Clifford Algebras in Analysis and Related Topics, CRC Press, Inc.
18. Ryan, J. (1995). Conformally covariant operators in Clifford analysis, ZAA, 14, 677-704.
19. n, Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics, World Scientific, 390-399.
20. Seeley, R.T. Spherical Harmonics, papers in analysis, 115-121.
21. Sommen, F. (1981). Spherical monogenic functions and analytic functions on the unit sphere, Tokyo J. Math., 4(2).
22. Sudbery, A. (1979). Quaternionic analysis, Math. Proc. of Cambridge Philosophical Soc., 85, 199-225.
23. Trautman, A. (1995). The Dirac Operator on Hypersurfaces, Vol. 26, 1283-1310, Acta Physica Polonica B, No. 7.