transformations%20replaced%20by%20symmetry
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1: 1.14 Integral Transforms
§1.14 Integral Transforms
►§1.14(i) Fourier Transform
… ►§1.14(iii) Laplace Transform
… ►Fourier Transform
… ►Laplace Transform
…2: 20.11 Generalizations and Analogs
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►If both are positive, then allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
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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. …3: 20 Theta Functions
Chapter 20 Theta Functions
…4: 19.15 Advantages of Symmetry
§19.15 Advantages of Symmetry
… ►Symmetry in of , , and replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). 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). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …5: 20.10 Integrals
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§20.10(i) Mellin Transforms with respect to the Lattice Parameter
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20.10.1
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20.10.2
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20.10.3
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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
…6: Bibliography N
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Applicable Anal. 39 (4), pp. 249–263.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Methods Appl. Anal. 3 (1), pp. 98–108.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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7: 32.8 Rational Solutions
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– possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants.
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►Rational solutions of exist for and are generated using the seed solution and the Bäcklund transformations (32.7.1) and (32.7.2).
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32.8.3
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32.8.4
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8: 20.7 Identities
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§20.7(iv) Reduction Formulas for Products
… ►§20.7(vi) Landen Transformations
… ►See Lawden (1989, pp. 19–20). … ►§20.7(viii) Transformations of Lattice Parameter
… ►§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products
…9: 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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§14.31(ii) Conical Functions
… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …10: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung.
Birkhäuser Verlag, Basel und Stuttgart (German).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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