Generalized spheroidal wave equation and limiting cases
Abstract
We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a doubleconfluent Heun equation by a takinglimit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called WhittakerInce limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the WhittakerHill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and doubleconfluent Heun equations. In particular, we find that each of the LindemannStieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Pressmore »
 Authors:
 Instituto de Cosmologia, Relatividade e Astrofisica (ICRABR), Centro Brasileiro de Pesquisas Fisicas (CBPF), Rua Dr. Xavier Sigaud, CEP 22290180, Rio de Janeiro 150 (Brazil)
 Publication Date:
 OSTI Identifier:
 20929619
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2406057; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMMETRY; BESSEL FUNCTIONS; GEOMETRY; HYPERGEOMETRIC FUNCTIONS; MATHEMATICAL SOLUTIONS; MATHIEU EQUATION; MORSE POTENTIAL; SCHROEDINGER EQUATION; SINGULARITY
Citation Formats
Bonorino Figueiredo, B. D. Generalized spheroidal wave equation and limiting cases. United States: N. p., 2007.
Web. doi:10.1063/1.2406057.
Bonorino Figueiredo, B. D. Generalized spheroidal wave equation and limiting cases. United States. doi:10.1063/1.2406057.
Bonorino Figueiredo, B. D. Mon .
"Generalized spheroidal wave equation and limiting cases". United States.
doi:10.1063/1.2406057.
@article{osti_20929619,
title = {Generalized spheroidal wave equation and limiting cases},
author = {Bonorino Figueiredo, B. D.},
abstractNote = {We find sets of solutions to the generalized spheroidal wave equation (GSWE) or, equivalently, to the confluent Heun equation. Each set is constituted by three solutions, one given by a series of ascending powers of the independent variable, and the others by series of regular and irregular confluent hypergeometric functions. For a fixed set, the solutions converge over different regions of the complex plane but present series coefficients proportional to each other. These solutions for the GSWE afford solutions to a doubleconfluent Heun equation by a takinglimit process due to Leaver. [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. Another procedure, called WhittakerInce limit [B. D. Figueiredo, J. Math. Phys. 46, 113503 (2005)], provides solutions in series of powers and Bessel functions for two other equations with a different type of singularity at infinity. In addition, new solutions are obtained for the WhittakerHill and Mathieu equations [F. M. Arscott, Proc. R. Soc. Edinburg A67, 265 (1967)] by considering these as special cases of both the confluent and doubleconfluent Heun equations. In particular, we find that each of the LindemannStieltjes solutions for the Mathieu equation [E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press (1945)] is associated with two expansions in series of Bessel functions. We also discuss a set of solutions in series of hypergeometric and confluent hypergeometric functions for the GSWE and use their Leaver limits to obtain infiniteseries solutions for the Schroedinger equation with an asymmetric doubleMorse potential. Finally, the possibility of extending the solutions of the GSWE to the general Heun equation is briefly discussed.},
doi = {10.1063/1.2406057},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 48,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}

The differential equation, x(xx/sub 0/)(d/sup 2/y/dx/sup 2/)+(B/sub 1/+B/sub 2/x) (dy/dx)+(..omega../sup 2/x(xx/sub 0/) (2eta..omega..(xx/sub 0/)+B/sub 3/)y = 0, arises both in the quantum scattering theory of nonrelativistic electrons from polar molecules and ions, and, in the guise of Teukolsky's equations, in the theory of radiation processes involving black holes. This article discusses analytic representations of solutions to this equation. Previous results of Hylleraas (E. Hylleraas, Z. Phys. 71, 739 (1931)), Jaffe (G. Jaffe, Z. Phys. 87, 535 (1934)), Baber and Hasse (W. G. Baber and H. R. Hasse, Proc. Cambridge Philos. Soc. 25, 564 (1935)), and Chu and Stratton (L. J.more »

Solutions to a generalized spheroidal wave equation in molecular physics and general relativity, and an analysis of the quasinormal modes of Kerr black holes
The purpose of this dissertation is to present important new analytic representations for solutions to the generalized spheroidal wave equation x(xx/sub 0/)d/sup 2/y/dx/sup 2/ + (B/sub 1/ + B/sub 2/x)dy/dx + (omega/sup 2/x(xx/sub 0/)  2(eta)(omega)(x  x/sub 0/) + B/sub 3/)y = 0. Discussion of solutions on the angular interval (0 less than or equal to x less than or equal to x/sub 0/) is included, but the major emphasis is on the radial functions for which (x/sub 0/ less than or equal to x < infinity). The study starts with the 1930s work of E. Hylleraas, G. Jaffe,more » 
THE CLOSED SOLUTION OF THE STATIONARY MULTIGROUP DIFFUSION EQUATION OF A REFLECTING SPHEROIDAL REACTOR AND THE ADAPTATION OF THE CYLINDRICAL REACTOR TO THE SPHEROIDAL REACTOR (in German)
As is known the problem of a reactor homogeneously reflecting on all sides, whose flux distribution depends on more than a position coordinate, was solved earlier by approximation methods. For the practically important case of the cylindrical reactor reflecting on blanket and casing. Marchuk, Ziegler, et al. present various approximation methods which satisfy in good approximation the calculation of critical volumes. However, with these methods it is not possible to make reliable assumptions on the flux distribution of single neutron groups. By introduction of rotational elliptical coordinates, it is easy to solve rigorously the multigroup diffusion problems of a reflectingmore » 
The generalized SHwave equation
The authors present a generalization of the SHwave equation for anisotropic and dissipative media. The most general case in which SHwaves are decoupled from P and SVwaves at all propagation angles is that of propagation in the plane of symmetry of a monoclinic medium. In the isotropic case, the SH constitutive equation involves only one elastic coefficient (the rigidity); here, three elastic coefficients are needed. Moreover, dissipation is introduced by using Boltzmann`s law based on several relaxation mechanisms. Anisotropic attenuation and velocity dispersion are guaranteed by choosing different relaxation functions for the principal axes. The wave equation, in the displacementmore »