lattice%20points
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1: 36.4 Bifurcation Sets
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§36.4(i) Formulas
βΊCritical Points for Cuspoids
… βΊCritical Points for Umbilics
… βΊThis is the codimension-one surface in space where critical points coalesce, satisfying (36.4.1) and … βΊThis is the codimension-one surface in space where critical points coalesce, satisfying (36.4.2) and …2: 23.14 Integrals
3: 23.9 Laurent and Other Power Series
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βΊLet be the nearest lattice point to the origin, and define
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βΊFor , and with as in §23.3(i),
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23.9.6
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βΊAlso, 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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4: 23.2 Definitions and Periodic Properties
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βΊ
§23.2(i) Lattices
… βΊ … βΊThe double series and double product are absolutely and uniformly convergent in compact sets in that do not include lattice points. … βΊ and are meromorphic functions with poles at the lattice points. …The function is entire and odd, with simple zeros at the lattice points. …5: 23.6 Relations to Other Functions
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βΊIn this subsection , are any pair of generators of the lattice
, and the lattice roots , , are given by (23.3.9).
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βΊFor further results for the -function see Lawden (1989, §6.2).
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βΊ
Rectangular Lattice
… βΊGeneral Lattice
βΊLet be a point of different from , and define by …6: 26.2 Basic Definitions
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Lattice Path
βΊA lattice path is a directed path on the plane integer lattice . …For an example see Figure 26.9.2. βΊA k-dimensional lattice path is a directed path composed of segments that connect vertices in so that each segment increases one coordinate by exactly one unit. … βΊ7: 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
βΊwhere and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.
8: 23.3 Differential Equations
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βΊThe lattice invariants are defined by
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βΊThe lattice roots satisfy the cubic equation
…and are denoted by .
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βΊLet , or equivalently be nonzero, or be distinct.
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βΊConversely, , , and the set are determined uniquely by the lattice
independently of the choice of generators.
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9: 23.20 Mathematical Applications
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βΊ
Rectangular Lattice
… βΊRhombic Lattice
… βΊFor each pair of edges there is a unique point such that . … βΊPoints on the curve can be parametrized by , , where and : in this case we write . … βΊThese cases correspond to rhombic and rectangular lattices, respectively. …10: 23.22 Methods of Computation
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