harmonic trapping potentials

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11: 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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M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.

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External Links:: ISSN 0036-1399, Document, MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §29.7(ii).
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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12: 10.73 Physical Applications

… ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … ►With the spherical harmonic

Y_{\ell,m}\left(\theta,\phi\right)

defined as in §14.30(i), the solutions are of the form

f=g_{\ell}(k\rho)Y_{\ell,m}\left(\theta,\phi\right)

with

g_{\ell}=\mathsf{j}_{\ell}

\mathsf{y}_{\ell}

{\mathsf{h}^{(1)}_{\ell}}

, or

{\mathsf{h}^{(2)}_{\ell}}

, depending on the boundary conditions. …In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …

13: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

14: 29.18 Mathematical Applications

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§29.18(iii) Spherical and Ellipsoidal Harmonics

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15: 34.3 Basic Properties: $\mathit{3j}$ Symbol

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§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

►For the polynomials

P_{l}

see §18.3, and for the function

Y_{{l},{m}}

see §14.30. … ►

34.3.20

Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{l_{2}},{m_{2}}}\left(\theta,% \phi\right)=\sum_{l,m}\left(\frac{(2l_{1}+1)(2l_{2}+1)(2l+1)}{4\pi}\right)^{% \frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{pmatrix}\overline{Y_{{l},{m}}\left(\theta,\phi\right)}\begin% {pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix},

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.3.E20
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the spherical harmonic in the sum over $l, m$ has been explicity marked up as a complex conjugate.
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

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34.3.22

\int_{0}^{2\pi}\!\int_{0}^{\pi}Y_{{l_{1}},{m_{1}}}\left(\theta,\phi\right)Y_{{% l_{2}},{m_{2}}}\left(\theta,\phi\right)Y_{{l_{3}},{m_{3}}}\left(\theta,\phi% \right)\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi=\left(\frac{(2l_{1}+1)(2l_% {2}+1)(2l_{3}+1)}{4\pi}\right)^{\frac{1}{2}}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.30(ii), §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

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16: Bibliography M

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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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I. Marquette and C. Quesne (2013) 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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External Links:: MathReview Entry
Referenced by:: §18.36(vi).
Encodings:: BibTeX

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I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and

k

-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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External Links:: ISSN 0022-2488, Document, Link, MathReview Entry
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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G. Matviyenko (1993) On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:: ISSN 1063-5203, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(iii).
Encodings:: BibTeX

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N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.

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External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §15.19(i), §15.19(i).
Encodings:: BibTeX

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17: 25.16 Mathematical Applications

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25.16.5

H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},

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Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer, $s$ : complex variable and $h(n)$ : harmonic number
Keywords:: Euler sum, infinite series, series representation
Source:: Apostol and Vu (1984, p. 85)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E5
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►where

H_{n}

is given by (25.11.33). … ►

25.16.13

\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\zeta\left(4% \right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer and $h(n)$ : harmonic number
Keywords:: harmonic number, infinite series, special value
Source:: Flajolet and Salvy (1998, (4-1), p. 24)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E13
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

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18: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

19: Bibliography

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J. C. Adams and P. N. Swarztrauber (1997) SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

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Notes:: SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.
External Links:: FullText, SPHEREPACK 3.1
Referenced by:: 1st item.
Encodings:: BibTeX

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H. H. Aly, H. J. W. Müller-Kirsten, and N. Vahedi-Faridi (1975) Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

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Notes:: Erratum: same journal 17 (1975), no. 7, p. 1361
External Links:: Document, MathReview (K. Veselic), ZentralBlatt
Referenced by:: 4th item.
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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20: Bibliography D

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P. Dean (1966) The constrained quantum mechanical harmonic oscillator. Proc. Cambridge Philos. Soc. 62, pp. 277–286.

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External Links:: Document, MathReview (T. Erber)
Referenced by:: §12.11(i), §12.11(i), §12.17.
Encodings:: BibTeX

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L. Dekar, L. Chetouani, and T. F. Hammann (1999) Wave function for smooth potential and mass step. Phys. Rev. A 59 (1), pp. 107–112.

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External Links:: Document
Referenced by:: §15.18.
Encodings:: BibTeX

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R. Dutt, A. Khare, and U. P. Sukhatme (1988) Supersymmetry, shape invariance, and exactly solvable potentials. Amer. J. Phys. 56, pp. 163–168.

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Referenced by:: §18.38(iii).
Encodings:: BibTeX

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