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Symmetry and separation of variables for the Helmholtz and Laplace equations

  • C. P. Boyer (a1), E. G. Kalnins (a2) and W. Miller (a3)
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This paper is one of a series relating the symmetry groups of the principal linear partial differential equations of mathematical physics and the coordinate systems in which variables separate for these equations. In particular, we mention [1] and paper [2] which is a survey of and introduction to the series. Here we apply group-theoretic methods to study the separable coordinate systems for the Helmholtz equation.

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References
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[1] Boyer, C., Kalnins, E. G., and Miller, W. Jr., Lie theory and separation of variables. 6. The equation iUt + Δ2U=0 , J. Math. Phys., 16 (1975), 499511.
[2] Miller, W. Jr., Symmetry, separation of variables and special functions, in Proceedings of the Advanced Seminar on Special Functions, Madison, Wisconsin, 1975, Academic Press, New York, 1975.
[3] Morse, P. and Feshbach, H., Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 1953.
[4] Makarov, A. A., Smorodinsky, J. a., Valiev, K., and Winternitz, P., A systematic search for nonrelativistic systems with dynamical symmetries. Part I: The integrals of motion, Nuovo Cimento, 52A (1967), pp. 1061-1084.
[5] Bôcher, M., Die Reihenentwickelungen der Potentialtheorie, Leipzig, 1894.
[6] Bateman, H., Partial Differential Equations of Mathematical Physics, Cambridge University Press (First Edition, 1932), Reprinted 1969.
[7] Bateman, H., Electrical and Optical Wave-Motion, Dover 1955 (Reprint of 1914 Edition).
[8] Vilenkin, N. Y., Special Functions and the Theory of Group Representations, AMS Trans., Providence, Rhode Island, 1968 (English Transl.).
[9] Miller, W. Jr., Symmetry Groups and Their Applications, Academic Press, New York, 1972.
[10] Arscott, F., Periodic Differential Equations, Pergamon, Oxford, England, 1964.
[11] Erdélyi, A. et al., Higher Transcendental Functions. Vols. 1 and 2, McGraw-Hill, New York, 1953.
[12] Urwin, K. and Arscott, F., Theory of the Whittaker-Hill Equation, Proc. Roy. Soc. Edinb., A69 (1970), 2844.
[13] Miller, W. Jr., Lie Theory and Separation of Variables II: Parabolic Coordinates, SIAM J. Math. Anal., 5 (1974), 822836.
[14] Miller, W. Jr., Lie Theory and Special Functions, Academic Press, New York, 1968.
[15] Patera, J. and Winternitz, P., A new basis for the representations of the rotation group. Lamé and Heun polynomials, J. Math. Phys., 14 (1973), 1130-1139.
[16] Kalnins, E. G. and Miller, W. Jr., Lie theory and separation of variables. 4. The groups SO (2,1) and SO (3), J. Math. Phys., 15 (1974), 1263-1274.
[17] Miller, W. Jr., Symmetry and Separation of Variables for Linear Differential Equations, Addison-Wesley, Reading, Mass. (to appear).
[18] Miller, W. Jr., Lie Theory and Separation of Variables. 1. Parabolic Cylinder Coordinates, SIAM J. Math. Anal., 5 (1974), 626643.
[19] Bucholz, H., The Confluent Hypergeometric Function, Springer-Verlag, New York, 1969.
[20] Weisner, L., Group-theoretic origin of certain generating functions, Pacific J. Math., 5 (1955), 1033-1039.
[21] Kalnins, E. G. and Miller, W. Jr., Lie theory and separation of variables. 9. Orthogonal inseparable coordinate systems for the wave equation Ψtt—Δ2Ψ=0 , J. Math. Phys. (to appear).
[22] Kalnins, E. G. and Miller, W. Jr., Lie theory and separation of variables. 10. Non-orthogonal UN-separable solutions of the wave equation ttΨ=Δ2Ψ , J. Math. Phys. (to appear).
[23] Schafke, F. W., Einfuhrung in die Théorie der Speziellen Funktion der Mathematischen Physik, Springer-Verlag, Berlin, 1963.
[24] Miller, W. Jr., Special Functions and the Complex Euclidean Group in 3-Space. I, J. Math. Phys., 9 (1968), 11631175.
[25] Winternitz, P., Lukǎc, I., and Smorodinskiǐ, Y., Quantum numbers in the little groups of the Poincaré group, Soviet Physics JNP, 7 (1968), 139145.
[26] Kalnins, E. G. and Miller, W. Jr., Lie theory and separation of variables. 8. Semi-subgroup coordinates for Ψtt—Δ2Ψ=0 , J. Math. Phys. (to appear).
[27] Kalnins, E. G. and Miller, W. Jr., Lie theory and separation of variables. 11. The Euler-Poisson-Darboux equation, J. Math. Phys (to appear).
[28] Miller, W. Jr., Special Functions and the Complex Euclidean Group in 3-Space. III, J. Math. Phys., 9 (1968), 14341444.
[29] Viswanathan, B., Generating functions for ultraspherical functions, Can J. Math., 20 (1968), 120134.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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