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1: 19.15 Advantages of Symmetry
§19.15 Advantages of Symmetry
Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … … Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. …
2: Bille C. Carlson
The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. …In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …
3: Peter A. Clarkson
Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …
4: 7.4 Symmetry
§7.4 Symmetry
g ( - z ) = 2 sin ( 1 4 π + 1 2 π z 2 ) - g ( z ) .
5: 14.31 Other Applications
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)). …
6: 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
Table 18.6.1: Classical OP’s: symmetry and special values.
p n ( x ) p n ( - x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
7: 21.3 Symmetry and Quasi-Periodicity
§21.3 Symmetry and Quasi-Periodicity
§21.3(i) Riemann Theta Functions
For Riemann theta functions with half-period characteristics, …
8: 34.7 Basic Properties: 9 j Symbol
§34.7(ii) Symmetry
The 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 9 j symbols. … For further symmetry properties of the 9 j symbol see Edmonds (1974, pp. 102–103) and Varshalovich et al. (1988, §10.4.1). …
9: 4.3 Graphics
See accompanying text
Figure 4.3.1: ln x and 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
10: 20.11 Generalizations and Analogs
§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. …