permutation%20symmetry
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21: 10.73 Physical Applications
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►Bessel functions of the first kind, , arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation , or by the Helmholtz equation .
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation.
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22: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.
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23: 18.6 Symmetry, Special Values, and Limits to Monomials
§18.6 Symmetry, Special Values, and Limits to Monomials
►§18.6(i) Symmetry and Special Values
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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On the distribution of the length of the longest increasing subsequence of random permutations.
J. Amer. Math. Soc. 12 (4), pp. 1119–1178.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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25: 7.4 Symmetry
26: 19.36 Methods of Computation
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19.36.2
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►If , , and are permuted so that , then the computation of is fastest if we make by choosing when or when .
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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27: 19.21 Connection Formulas
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►Let , , and be positive and distinct, and permute
and to ensure that does not lie between and .
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►Because is completely symmetric, can be permuted on the right-hand side of (19.21.10) so that if the variables are real, thereby avoiding cancellations when is calculated from and (see §19.36(i)).
…where both summations extend over the three cyclic permutations of .
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►and may be permuted.
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►For each value of , permutation of produces three values of , one of which lies in the same region as and two lie in the other region of the same type.
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28: 14.31 Other Applications
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►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)).
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29: 36.2 Catastrophes and Canonical Integrals
30: 8 Incomplete Gamma and Related
Functions
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