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11: 34.3 Basic Properties: 3 j Symbol
§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). …
12: 34.5 Basic Properties: 6 j Symbol
§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). …
13: Bibliography N
  • G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
  • M. Noumi and Y. Yamada (1999) Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.
  • M. Noumi (2004) Painlevé Equations through Symmetry. Translations of Mathematical Monographs, Vol. 223, American Mathematical Society, Providence, RI.
  • 14: 24.4 Basic Properties
    §24.4(ii) Symmetry
    15: 10.73 Physical Applications
    Bessel functions of the first kind, J n ( x ) , arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation 2 V = 0 , or by the Helmholtz equation ( 2 + k 2 ) ψ = 0 . … Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …
    16: Bibliography C
  • B. C. Carlson (2004) Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.
  • B. C. Carlson (2011) Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.
  • B. C. Carlson (1998) 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.
  • P. A. Clarkson (1991) 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.
  • H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).
  • 17: 36.2 Catastrophes and Canonical Integrals
    §36.2(iii) Symmetries
    18: 36.7 Zeros
    Near z = z n , and for small x and y , the modulus | Ψ ( E ) ( x ) | 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. …
    19: 20.9 Relations to Other Functions
    20: 36.11 Leading-Order Asymptotics
    Asymptotics along Symmetry Lines