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1: 23.7 Quarter Periods
23.7.1 ( 1 2 ω 1 ) = e 1 + ( e 1 e 3 ) ( e 1 e 2 ) = e 1 + ω 1 2 ( K ( k ) ) 2 k ,
23.7.2 ( 1 2 ω 2 ) = e 2 i ( e 1 e 2 ) ( e 2 e 3 ) = e 2 i ω 1 2 ( K ( k ) ) 2 k k ,
23.7.3 ( 1 2 ω 3 ) = e 3 ( e 1 e 3 ) ( e 2 e 3 ) = e 3 ω 1 2 ( K ( k ) ) 2 k ,
where k , k and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.
2: 23.3 Differential Equations
The lattice invariants are defined by … The lattice roots satisfy the cubic equation …and are denoted by e 1 , e 2 , e 3 . … Let g 2 3 27 g 3 2 , or equivalently Δ be nonzero, or e 1 , e 2 , e 3 be distinct. … Conversely, g 2 , g 3 , and the set { e 1 , e 2 , e 3 } are determined uniquely by the lattice 𝕃 independently of the choice of generators. …
3: 23.14 Integrals
23.14.2 2 ( z ) d z = 1 6 ( z ) + 1 12 g 2 z ,
4: 23.10 Addition Theorems and Other Identities
23.10.4 σ ( u + v ) σ ( u v ) σ ( x + y ) σ ( x y ) + σ ( v + x ) σ ( v x ) σ ( u + y ) σ ( u y ) + σ ( x + u ) σ ( x u ) σ ( v + y ) σ ( v y ) = 0 .
For further addition-type identities for the σ -function see Lawden (1989, §6.4). …
23.10.8 ( ( 2 z ) e 1 ) 2 ( z ) = ( ( ( z ) e 1 ) 2 ( e 1 e 2 ) ( e 1 e 3 ) ) 2 .
(23.10.8) continues to hold when e 1 , e 2 , e 3 are permuted cyclically. … Also, when 𝕃 is replaced by c 𝕃 the lattice invariants g 2 and g 3 are divided by c 4 and c 6 , respectively. …
5: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
( z ) and ζ ( z ) are meromorphic functions with poles at the lattice points. ( z ) is even and ζ ( z ) is odd. …The function σ ( z ) is entire and odd, with simple zeros at the lattice points. …
6: 23.21 Physical Applications
The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x 4 x 3 . … Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … where x , y , z are the corresponding Cartesian coordinates and e 1 , e 2 , e 3 are constants. …
23.21.3 f ( ρ ) = 2 ( ( ρ e 1 ) ( ρ e 2 ) ( ρ e 3 ) ) 1 / 2 .
Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
7: 23.4 Graphics
§23.4(i) Real Variables
Line graphs of the Weierstrass functions ( x ) , ζ ( x ) , and σ ( x ) , illustrating the lemniscatic and equianharmonic cases. …
See accompanying text
Figure 23.4.7: ( x ) with ω 1 = K ( k ) , ω 3 = i K ( k ) for 0 x 9 , k 2 = 0. … Magnify
Surfaces for the Weierstrass functions ( z ) , ζ ( z ) , and σ ( z ) . …
See accompanying text
Figure 23.4.8: ( x + i y ) with ω 1 = K ( k ) , ω 3 = i K ( k ) for 2 K ( k ) x 2 K ( k ) , 0 y 6 K ( k ) , k 2 = 0.9 . (The scaling makes the lattice appear to be square.) Magnify 3D Help
8: 23.9 Laurent and Other Power Series
Let z 0 ( 0 ) be the nearest lattice point to the origin, and define …
c 2 = 1 20 g 2 ,
For j = 1 , 2 , 3 , and with e j as in §23.3(i),
23.9.6 ( ω j + t ) = e j + ( 3 e j 2 5 c 2 ) t 2 + ( 10 c 2 e j + 21 c 3 ) t 4 + ( 7 c 2 e j 2 + 21 c 3 e j + 5 c 2 2 ) t 6 + O ( t 8 ) ,
Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as 1 / ( z ) 0 . …
9: 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
10: 23.23 Tables
2 in Abramowitz and Stegun (1964) gives values of ( z ) , ( z ) , and ζ ( z ) to 7 or 8D in the rectangular and rhombic cases, normalized so that ω 1 = 1 and ω 3 = i a (rectangular case), or ω 1 = 1 and ω 3 = 1 2 + i a (rhombic case), for a = 1. …05, and in the case of ( z ) the user may deduce values for complex z 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 g 2 and g 3 . …