# symmetry

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## 1—10 of 52 matching pages

##### 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

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►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.
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►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.
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##### 3: Peter A. Clarkson

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►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.
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##### 4: 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

… ► …##### 5: 7.4 Symmetry

###### §7.4 Symmetry

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$\mathrm{g}\left(-z\right)=\sqrt{2}\mathrm{sin}\left(\frac{1}{4}\pi +\frac{1}{2}\pi {z}^{2}\right)-\mathrm{g}\left(z\right).$

##### 6: 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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##### 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: 4.3 Graphics

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

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