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1: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
If ω 1 and ω 3 are nonzero real or complex numbers such that ( ω 3 / ω 1 ) > 0 , then the set of points 2 m ω 1 + 2 n ω 3 , with m , n , constitutes a lattice 𝕃 with 2 ω 1 and 2 ω 3 lattice generators. … The double series and double product are absolutely and uniformly convergent in compact sets in that do not include lattice points. … ( z ) and ζ ( z ) are meromorphic functions with poles at the lattice points. …The function σ ( z ) is entire and odd, with simple zeros at the lattice points. …
2: 20.2 Definitions and Periodic Properties
are the lattice points. The theta functions are quasi-periodic on the lattice: …
3: 23.9 Laurent and Other Power Series
Let z 0 ( 0 ) be the nearest lattice point to the origin, and define …
4: 26.12 Plane Partitions
It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point ( h , j , k ) π . …
5: 23.20 Mathematical Applications
There is a unique point z 0 [ ω 1 , ω 1 + ω 3 ] [ ω 1 + ω 3 , ω 3 ] such that ( z 0 ) = 0 . … For each pair of edges there is a unique point z 0 such that ( z 0 ) = 0 . … Points P = ( x , y ) on the curve can be parametrized by x = ( z ; g 2 , g 3 ) , 2 y = ( z ; g 2 , g 3 ) , where g 2 = - 4 a and g 3 = - 4 b : in this case we write P = P ( z ) . …
6: 23.6 Relations to Other Functions
Let z be a point of different from e 1 , e 2 , e 3 , and define w by …
7: 21.3 Symmetry and Quasi-Periodicity
The set of points m 1 + Ω m 2 form a g -dimensional lattice, the period lattice of the Riemann theta function. …
8: 22.4 Periods, Poles, and Zeros
The set of points z = m K + n i K , m , n , comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by m K + n i K , where again m , n . …
9: 23.22 Methods of Computation
§23.22(ii) Lattice Calculations
Starting from Lattice
Suppose that the lattice 𝕃 is given. …The corresponding values of e 1 , e 2 , e 3 are calculated from (23.6.2)–(23.6.4), then g 2 and g 3 are obtained from (23.3.6) and (23.3.7). … Suppose that the invariants g 2 = c , g 3 = d , are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). …
10: 9.16 Physical Applications
The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. … Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …