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11: 23.6 Relations to Other Functions
In this subsection 2 ω 1 , 2 ω 3 are any pair of generators of the lattice 𝕃 , and the lattice roots e 1 , e 2 , e 3 are given by (23.3.9). … For further results for the σ -function see Lawden (1989, §6.2). … Again, in Equations (23.6.16)–(23.6.26), 2 ω 1 , 2 ω 3 are any pair of generators of the lattice 𝕃 and e 1 , e 2 , e 3 are given by (23.3.9). …
Rectangular Lattice
General Lattice
12: 23.19 Interrelations
23.19.3 J ( τ ) = g 2 3 g 2 3 27 g 3 2 ,
where g 2 , g 3 are the invariants of the lattice 𝕃 with generators 1 and τ ; see §23.3(i). …
13: 23.1 Special Notation
𝕃 lattice in .
2 ω 1 , 2 ω 3 lattice generators ( ( ω 3 / ω 1 ) > 0 ).
Δ discriminant g 2 3 27 g 3 2 .
The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . … Abramowitz and Stegun (1964, Chapter 18) considers only rectangular and rhombic lattices23.5); ω 1 , ω 3 are replaced by ω , ω for the former and by ω 2 , ω for the latter. …
14: 23.13 Zeros
For information on the zeros of ( z ) see Eichler and Zagier (1982).
15: 23.11 Integral Representations
23.11.2 ( z ) = 1 z 2 + 8 0 s ( e s sinh 2 ( 1 2 z s ) f 1 ( s , τ ) + e i τ s sin 2 ( 1 2 z s ) f 2 ( s , τ ) ) d s ,
23.11.3 ζ ( z ) = 1 z + 0 ( e s ( z s sinh ( z s ) ) f 1 ( s , τ ) e i τ s ( z s sin ( z s ) ) f 2 ( s , τ ) ) d s ,
16: 23.12 Asymptotic Approximations
23.12.2 ζ ( z ) = π 2 4 ω 1 2 ( z 3 + 2 ω 1 π cot ( π z 2 ω 1 ) 8 ( z ω 1 π sin ( π z ω 1 ) ) q 2 + O ( q 4 ) ) ,
23.12.3 σ ( z ) = 2 ω 1 π exp ( π 2 z 2 24 ω 1 2 ) sin ( π z 2 ω 1 ) ( 1 ( π 2 z 2 ω 1 2 4 sin 2 ( π z 2 ω 1 ) ) q 2 + O ( q 4 ) ) ,
23.12.4 η 1 = π 2 4 ω 1 ( 1 3 8 q 2 + O ( q 4 ) ) ,
17: 31.2 Differential Equations
k 2 = ( e 2 e 3 ) / ( e 1 e 3 ) ,
e 1 = ( ω 1 ) ,
e 2 = ( ω 2 ) ,
e 1 + e 2 + e 3 = 0 ,
where 2 ω 1 and 2 ω 3 with ( ω 3 / ω 1 ) > 0 are generators of the lattice 𝕃 for ( z | 𝕃 ) . …
18: 23.8 Trigonometric Series and Products
23.8.1 ( z ) + η 1 ω 1 π 2 4 ω 1 2 csc 2 ( π z 2 ω 1 ) = 2 π 2 ω 1 2 n = 1 n q 2 n 1 q 2 n cos ( n π z ω 1 ) ,
23.8.2 ζ ( z ) η 1 z ω 1 π 2 ω 1 cot ( π z 2 ω 1 ) = 2 π ω 1 n = 1 q 2 n 1 q 2 n sin ( n π z ω 1 ) .
23.8.6 σ ( z ) = 2 ω 1 π exp ( η 1 z 2 2 ω 1 ) sin ( π z 2 ω 1 ) n = 1 1 2 q 2 n cos ( π z / ω 1 ) + q 4 n ( 1 q 2 n ) 2 ,
19: 26.2 Basic Definitions
Lattice Path
A lattice path is a directed path on the plane integer lattice { 0 , 1 , 2 , } × { 0 , 1 , 2 , } . …For an example see Figure 26.9.2. A k-dimensional lattice path is a directed path composed of segments that connect vertices in { 0 , 1 , 2 , } k so that each segment increases one coordinate by exactly one unit. …
20: 16.24 Physical Applications
§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). …