with toroidal symmetry

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11: 4.3 Graphics

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. Parallel tangent lines at $(1,0)$ and $(0,1)$ make evident the mirror symmetry across the line $y=x$, demonstrating the inverse relationship between the two functions. Magnify

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12: 14.34 Software

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§14.34(iv) Conical (Mehler) and/or Toroidal Functions

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13: 34.7 Basic Properties: $\mathit{9}j$ Symbol

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§34.7(ii) Symmetry

►The $\mathit{9}j$ symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent $\mathit{9}j$ symbols. … ►For further symmetry properties of the $\mathit{9}j$ symbol see Edmonds (1974, pp. 102–103) and Varshalovich et al. (1988, §10.4.1). …

14: 20.11 Generalizations and Analogs

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§20.11(v) Permutation Symmetry

… ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. …

15: 34.5 Basic Properties: $\mathit{6}j$ Symbol

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§34.5(ii) Symmetry

… ►Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). …

16: 14.1 Special Notation

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17: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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§34.3(ii) Symmetry

… ►Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). …

18: Bibliography V

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

BibTeX

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19: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

BibTeX

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Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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Painlevé Equations through Symmetry. Translations of Mathematical Monographs, Vol. 223, American Mathematical Society, Providence, RI.

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Notes:

Translated from the 2000 Japanese original by the author

External Links:

ISBN 0-8218-3221-2, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.2(v).

Encodings:

BibTeX

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20: 24.4 Basic Properties

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§24.4(ii) Symmetry

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