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1: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
►§14.30(i) Definitions
… ► … ►Most mathematical properties of can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. … ►2: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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SPHEREPACK 2.0: A Model Development Facility.
NCAR Technical Note
Technical Report TN-436-STR, National Center for Atmospheric Research.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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A harmonic mean inequality for the gamma function.
J. Comput. Appl. Math. 87 (2), pp. 195–198.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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3: Bibliography M
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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications.
3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
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New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems.
J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..
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Connection between quantum systems involving the fourth Painlevé transcendent and -step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial.
J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..
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On the evaluation of Bessel functions.
Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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4: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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The constrained quantum mechanical harmonic oscillator.
Proc. Cambridge Philos. Soc. 62, pp. 277–286.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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5: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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6: Bibliography G
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A harmonic mean inequality for the gamma function.
SIAM J. Math. Anal. 5 (2), pp. 278–281.
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A code to evaluate prolate and oblate spheroidal harmonics.
Comput. Phys. Comm. 108 (2-3), pp. 267–278.
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Evaluation of toroidal harmonics.
Comput. Phys. Comm. 124 (1), pp. 104–122.
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DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics.
Comput. Phys. Comm. 139 (2), pp. 186–191.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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7: 10.73 Physical Applications
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►With the spherical harmonic
defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions.
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§10.73(iii) Kelvin Functions
… ►§10.73(iv) Bickley Functions
…8: Bibliography V
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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions.
CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en
Informatica, Amsterdam.
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Expansion of vacuum magnetic fields in toroidal harmonics.
Comput. Phys. Comm. 81 (1-2), pp. 74–90.
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An infinite series of Weber’s parabolic cylinder functions.
Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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9: Bibliography W
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Evaluating elliptic functions and their inverses.
Comput. Math. Appl. 39 (3-4), pp. 131–136.
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Infinitely differentiable generalized logarithmic and exponential functions.
Math. Comp. 57 (196), pp. 723–733.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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Harmonic Analysis.
In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.),
Studies in Mathematics, Vol. 3, pp. 124–178.
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On the functions associated with the parabolic cylinder in harmonic analysis.
Proc. London Math. Soc. 35, pp. 417–427.
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