ellipsoidal%20harmonics

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1: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

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§14.30(i) Definitions

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§14.30(iii) Sums

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2: 29.18 Mathematical Applications

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§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates

\alpha,\beta,\gamma

: … ►

29.18.10

u(\alpha,\beta,\gamma)=u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma),

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Symbols:: $\alpha$ : ellipsoidal coordinate, $\beta$ : ellipsoidal coordinate and $\gamma$ : ellipsoidal coordinate
Permalink:: http://dlmf.nist.gov/29.18.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(ii), §29.18 and Ch.29

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§29.18(iii) Spherical and Ellipsoidal Harmonics

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4: 23.21 Physical Applications

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§23.21(iii) Ellipsoidal Coordinates

►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation …The Laplacian operator

\nabla^{2}

(§1.5(ii)) is given by ►

23.21.2

(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

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5: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by …

T. M. MacRobert (1967) 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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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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J. Meixner (1944) Die Laméschen Wellenfunktionen des Drehellipsoids. Forschungsbericht No. 1952 ZWB (German).

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Notes:: English translation: Lamé wave functions of the ellipsoid of revolution appeared as NACA Technical Memorandum No. 1224, 1949
External Links:: MathReview
Referenced by:: §30.9(iii).
Encodings:: BibTeX

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H. J. W. Müller (1966a) Asymptotic expansions of ellipsoidal wave functions and their characteristic numbers. Math. Nachr. 31, pp. 89–101.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.11, §29.7(i), §29.7(ii).
Encodings:: BibTeX

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H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

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H. J. W. Müller (1966c) On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

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7: Bibliography

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M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:: For corrigendum see same journal v. 90 (1998), no. 2, p. 265
External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §5.6(i).
Encodings:: BibTeX

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F. M. Arscott (1956) Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.

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External Links:: ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

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F. M. Arscott (1959) A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

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External Links:: ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §29.11.
Encodings:: BibTeX

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8: Bibliography W

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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G. N. Watson (1935b) The surface of an ellipsoid. Quart. J. Math., Oxford Ser. 6, pp. 280–287.

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External Links:: Document, ZentralBlatt
Referenced by:: §19.33(i).
Encodings:: BibTeX

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G. Weiss (1965) 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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External Links:: MathReview, ZentralBlatt
Referenced by:: §1.15(iii), §1.15(iv), §1.15(v).
Encodings:: BibTeX

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E. T. Whittaker (1902) On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

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External Links:: Document, ZentralBlatt
Referenced by:: §12.1, §12.5(i), §12.5(ii), §12.9(i).
Encodings:: BibTeX

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9: 19.15 Advantages of Symmetry

… ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). For example, the computation of depolarization factors for solid ellipsoids is simplified considerably; compare (19.33.7) with Cronemeyer (1991). …

10: Bibliography H

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F. E. Harris (2002) Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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External Links:: Document
Referenced by:: §8.24(iii).
Encodings:: BibTeX

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E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.

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Notes:: Reprinted by Chelsea Publishing Company, New York, 1955 and 1965.
External Links:: ISBN 0-8284-0104-7, MathReview, ZentralBlatt
Referenced by:: §14.1, §14.11, §14.16(ii), §14.16(iii), §14.20(x), §14.27, §14.27, §14.3(i), §14.3(i), Ch.14, §29.18(iii), Ch.29.
Encodings:: BibTeX

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L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:: Reprinted in 1979.
External Links:: ISBN 0-8218-1556-3, MathReview, ZentralBlatt
Referenced by:: §35.4(i).
Encodings:: BibTeX

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