lattice walks
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1: 16.24 Physical Applications
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§16.24(i) Random Walks
►Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148). …2: 23.7 Quarter Periods
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23.7.1
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23.7.2
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23.7.3
►where and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.
3: 23.3 Differential Equations
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►The lattice invariants are defined by
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►The lattice roots satisfy the cubic equation
…and are denoted by .
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►Let , or equivalently be nonzero, or be distinct.
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►Conversely, , , and the set are determined uniquely by the lattice
independently of the choice of generators.
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4: 23.14 Integrals
5: 23.10 Addition Theorems and Other Identities
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23.10.4
►For further addition-type identities for the -function see Lawden (1989, §6.4).
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23.10.8
►(23.10.8) continues to hold when , , are permuted cyclically.
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►Also, when is replaced by the lattice invariants and are divided by and , respectively.
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6: 23.2 Definitions and Periodic Properties
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§23.2(i) Lattices
… ► … ► and are meromorphic functions with poles at the lattice points. is even and is odd. …The function is entire and odd, with simple zeros at the lattice points. …7: 23.21 Physical Applications
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►The Weierstrass function plays a similar role for cubic potentials in canonical form .
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►Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations.
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►where are the corresponding Cartesian coordinates and , , are constants.
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23.21.3
►Another form is obtained by identifying , , as lattice roots (§23.3(i)), and setting
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8: 23.4 Graphics
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§23.4(i) Real Variables
►Line graphs of the Weierstrass functions , , and , illustrating the lemniscatic and equianharmonic cases. … ► … ►Surfaces for the Weierstrass functions , , and . … ► …9: 23.9 Laurent and Other Power Series
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►Let be the nearest lattice point to the origin, and define
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►For , and with as in §23.3(i),
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23.9.6
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►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as .
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