zonal spherical harmonics

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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: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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Normalization

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Orthogonal Invariance

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Summation

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Mean-Value

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3: 18.38 Mathematical Applications

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Zonal Spherical Harmonics

►Ultraspherical polynomials are zonal spherical harmonics. …

4: 35.11 Tables

§35.11 Tables

►Tables of zonal polynomials are given in James (1964) for

|\kappa|\leq 6

, Parkhurst and James (1974) for

|\kappa|\leq 12

, and Muirhead (1982, p. 238) for

|\kappa|\leq 5

. Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

5: 35.12 Software

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Demmel and Koev (2006). Computation of zonal polynomials in MATLAB.

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Stembridge (1995). Maple software for zonal polynomials.

►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).

6: Bibliography T

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A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

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External Links:: ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt
Referenced by:: §35.4(i), §35.9.
Encodings:: BibTeX

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J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

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A. Terras (1988) Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt
Referenced by:: §35.1, §35.5(ii), Notations K.
Encodings:: BibTeX

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O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:: Document
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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T. Ton-That, K. I. Gross, D. St. P. Richards, and P. J. Sally (Eds.) (1995) Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

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Notes:: Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994
External Links:: ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Donald St. P. Richards.
Encodings:: BibTeX

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7: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL

(2,\mathbb{R})

, and spherical functions on certain nonsymmetric Gelfand pairs. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

8: Bibliography G

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W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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External Links:: Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §5.6(i).
Encodings:: BibTeX

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A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.

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Notes:: Double-precision Fortran, maximum accuracy 18S
External Links:: Document, Code, ZentralBlatt
Referenced by:: 5th item, 1st item.
Encodings:: BibTeX

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A. Gil and J. Segura (2000) Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.

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External Links:: Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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A. Gil and J. Segura (2001) 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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External Links:: Document, Code, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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K. I. Gross and D. St. P. Richards (1987) Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.

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External Links:: ISSN 0002-9947, FullText, MathReview (Wayne J. Holman, III), ZentralBlatt
Referenced by:: §35.8(i), §35.8(ii), §35.8(iv), Ch.35.
Encodings:: BibTeX

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9: 35.10 Methods of Computation

… ►For small values of

\|\mathbf{T}\|

the zonal polynomial expansion given by (35.8.1) can be summed numerically. … ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

10: 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. ►

34.3.19

P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)=\sum_{l}(2l+1% )\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix}^{2}P_{l}\left(\cos\theta\right),

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\cos\NVar{z}$ : cosine function, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §14.18(iv), §14.30(iii), §34.3(vii)
Permalink:: http://dlmf.nist.gov/34.3.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §34.3(vii), §34.3 and Ch.34

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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 sphericalharmonic 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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