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11: 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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12: 20.2 Definitions and Periodic Properties
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►The four points
are the vertices of the fundamental parallelogram in the -plane; see Figure 20.2.1.
The points
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20.2.5
,
►are the lattice points.
The theta functions are quasi-periodic on the lattice:
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13: 26.3 Lattice Paths: Binomial Coefficients
§26.3 Lattice Paths: Binomial Coefficients
►§26.3(i) Definitions
… ► is the number of lattice paths from to . …The number of lattice paths from to , , that stay on or above the line is … ► …14: 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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15: 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 . … ► …16: 26.5 Lattice Paths: Catalan Numbers
§26.5 Lattice Paths: Catalan Numbers
►§26.5(i) Definitions
… ►It counts the number of lattice paths from to that stay on or above the line . … ► …17: 23.5 Special Lattices
§23.5 Special Lattices
… ►§23.5(ii) Rectangular Lattice
… ►§23.5(iii) Lemniscatic Lattice
… ►§23.5(iv) Rhombic Lattice
… ►§23.5(v) Equianharmonic Lattice
…18: 23.23 Tables
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►2 in Abramowitz and Stegun (1964) gives values of , , and to 7 or 8D in the rectangular and rhombic cases, normalized so that and (rectangular case), or and (rhombic case), for = 1.
…05, and in the case of the user may deduce values for complex by application of the addition theorem (23.10.1).
►Abramowitz and Stegun (1964) also includes other tables to assist the computation of the Weierstrass functions, for example, the generators as functions of the lattice invariants and .
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19: 29.2 Differential Equations
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►This equation has regular singularities at the points
, where , and , are the complete elliptic integrals of the first kind with moduli , , respectively; see §19.2(ii).
In general, at each singularity each solution of (29.2.1) has a branch point (§2.7(i)).
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29.2.9
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29.2.11
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