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21: 10.73 Physical Applications

… ►Bessel functions of the first kind, $J_n\left(x\right)$, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation ${\nabla }^{2}V=0$, or by the Helmholtz equation $({\nabla }^2+k^2)\psi =0$. … ►Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …

22: 36.7 Zeros

… ►Near $z=z_n$, and for small $x$ and $y$, the modulus $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …In the symmetry planes (e. …

23: Bibliography Z

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“Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

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External Links:

MathReview (Vadim B. Kuznetsov), ZentralBlatt

Referenced by:

§18.38(iii).

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24: Bibliography C

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Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.

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Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Mohammad Ahmad), ZentralBlatt

Referenced by:

§20.11(v), §20.7(i), §20.7(i), §20.7(ii), §20.7(ii), §20.7(ii), §20.7(iii), §20.7(iii), §20.7(vii), §20.7(vii), Profile Bille C. Carlson.

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Elliptic Integrals: Symmetry and Symbolic Integration. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti dei Convegni Lincei, Vol. 147, pp. 161–181.

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External Links:

MathReview (S. L. Kalla), ZentralBlatt

Referenced by:

§19.26(iii), §19.29(i).

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Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.

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External Links:

MathReview

Referenced by:

§29.19(ii).

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Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).

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External Links:

Link, ISSN 2073-8994, Document, ZentralBlatt

Referenced by:

§14.6(ii), §14.6(ii), Erratum (V1.1.1) for Subsection 14.6(ii).

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25: 36.2 Catastrophes and Canonical Integrals

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§36.2(iii) Symmetries

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26: 20.9 Relations to Other Functions

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27: 36.11 Leading-Order Asymptotics

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Asymptotics along Symmetry Lines

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28: 18.3 Definitions

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29: Bibliography O

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Spectral decomposition of tent maps using symmetry considerations. J. Statist. Phys. 84 (1-2), pp. 269–276.

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External Links:

ISSN 0022-4715, Document, MathReview (Makoto Mori), ZentralBlatt

Referenced by:

§24.18.

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30: Bibliography M

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Symmetry and Separation of Variables. Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam.

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Notes:

Encyclopedia of Mathematics and its Applications, Vol. 4

External Links:

ISBN 0-201-13503-5, MathReview (Ernest G. Kalnins), ZentralBlatt

Referenced by:

§31.17(i).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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A connection formula for the $q$-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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External Links:

ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt

Referenced by:

§17.6(ii), §17.6(ii).

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