relation to eigenfunctions
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11—15 of 15 matching pages
11: Bibliography T
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Eigenfunction Expansions Associated with Second-Order Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction expansions associated with second-order differential equations. Part I.
Second edition, Clarendon Press, Oxford.
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Hyperspherical elliptic harmonics and their relation to the Heun equation.
Phys. Rev. A 63 (032510), pp. 1–8.
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On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes.
Phys. A 244 (1-4), pp. 402–413.
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12: Bibliography R
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Normal limit theorems for symmetric random matrices.
Probab. Theory Related Fields 112 (3), pp. 411–423.
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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators.
Z. Angew. Math. Mech. 78 (1), pp. 66–72.
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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent.
Proc. Roy. Soc. London Ser. A 361, pp. 265–275.
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Elliptic and modular functions from Gauss to Dedekind to Hecke.
Cambridge University Press, Cambridge.
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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media.
In Differential Operators and Related Topics, Vol. I (Odessa,
1997),
Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
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13: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►Positive-energy functions correspond to processes such as Rutherford scattering and Coulomb excitation of nuclei (Alder et al. (1956)), and atomic photo-ionization and electron-ion collisions (Bethe and Salpeter (1977)).
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►The solutions to this equation are closely related to the Coulomb functions; see Greiner et al. (1985).
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►The WKBJ approximations of §33.23(vii) may also be used to estimate the penetrability.
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►Examples of applications to noninteger and/or complex variables are as follows.
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Eigenstates using complex-rotated coordinates , so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).
14: Bibliography K
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Bernstein, Pick, Poisson and related integral expressions for Lambert
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Integral Transforms Spec. Funct. 23 (11), pp. 817–829.
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Cyclic identities for Jacobi elliptic and related functions.
J. Math. Phys. 44 (4), pp. 1822–1841.
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Theta relations and projective normality of Abelian varieties.
Amer. J. Math. 98 (4), pp. 865–889.
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The structure relation for Askey-Wilson polynomials.
J. Comput. Appl. Math. 207 (2), pp. 214–226.
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Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions.
J. Math. Anal. Appl. 324 (1), pp. 285–303.
15: 30.14 Wave Equation in Oblate Spheroidal Coordinates
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►Oblate spheroidal coordinates are related to Cartesian coordinates by
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►The wave equation (30.13.7), transformed to oblate spheroidal coordinates , admits solutions of the form (30.13.8), where satisfies the differential equation
…Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution .
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►Moreover, the solution has to be bounded along the -axis: this requires
to be bounded when .
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►The corresponding eigenfunctions are then given by (30.13.8), (30.14.8), (30.13.13), (30.13.12), with .
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