ellipsoidal%20coordinates
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1: 29.18 Mathematical Applications
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§29.18(i) Sphero-Conal Coordinates
… ►when transformed to sphero-conal coordinates : … ►§29.18(ii) Ellipsoidal Coordinates
►The wave equation (29.18.1), when transformed to ellipsoidal coordinates : … ►§29.18(iii) Spherical and Ellipsoidal Harmonics
…2: 23.21 Physical Applications
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§23.21(iii) Ellipsoidal Coordinates
►Ellipsoidal coordinates may be defined as the three roots of the equation …where are the corresponding Cartesian coordinates and , , are constants. The Laplacian operator (§1.5(ii)) is given by ►
23.21.2
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3: 30.14 Wave Equation in Oblate Spheroidal Coordinates
§30.14 Wave Equation in Oblate Spheroidal Coordinates
►§30.14(i) Oblate Spheroidal Coordinates
… ►The coordinate surfaces are oblate ellipsoids of revolution with focal circle , . … ►§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids
►Equation (30.13.7) for together with the boundary condition on the ellipsoid given by , poses an eigenvalue problem with as spectral parameter. …4: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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§30.13(i) Prolate Spheroidal Coordinates
… ►The coordinate surfaces are prolate ellipsoids of revolution with foci at , . … ►§30.13(v) The Interior Dirichlet Problem for Prolate Ellipsoids
… ►For the Dirichlet boundary-value problem of the region between two ellipsoids, the eigenvalues are determined from … ►5: 19.33 Triaxial Ellipsoids
§19.33 Triaxial Ellipsoids
►§19.33(i) Surface Area
… ►§19.33(ii) Potential of a Charged Conducting Ellipsoid
… ►§19.33(iii) Depolarization Factors
… ►§19.33(iv) Self-Energy of an Ellipsoidal Distribution
…6: 29.11 Lamé Wave Equation
7: Bibliography M
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Die Laméschen Wellenfunktionen des Drehellipsoids.
Forschungsbericht No. 1952
ZWB (German).
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Lie theory and separation of variables. I: Parabolic cylinder coordinates.
SIAM J. Math. Anal. 5 (4), pp. 626–643.
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Asymptotic expansions of ellipsoidal wave functions and their characteristic numbers.
Math. Nachr. 31, pp. 89–101.
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Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions.
Math. Nachr. 32, pp. 49–62.
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On asymptotic expansions of ellipsoidal wave functions.
Math. Nachr. 32, pp. 157–172.
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8: 19.15 Advantages of Symmetry
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►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).
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9: 36.5 Stokes Sets
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►For , the Stokes set is expressed in terms of scaled coordinates
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36.5.7
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36.5.10
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►With coordinates
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36.5.17
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10: Bibliography H
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The combination of -matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr.
J. Phys. B 26 (12), pp. 1775–1790.
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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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The Theory of Spherical and Ellipsoidal Harmonics.
Cambridge University Press, London-New York.
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