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1: 23.3 Differential Equations
§23.3(i) Invariants, Roots, and Discriminant
The lattice invariants are defined by … Given g 2 and g 3 there is a unique lattice 𝕃 such that (23.3.1) and (23.3.2) are satisfied. …Similarly for ζ ( z ; g 2 , g 3 ) and σ ( z ; g 2 , g 3 ) . As functions of g 2 and g 3 , ( z ; g 2 , g 3 ) and ζ ( z ; g 2 , g 3 ) are meromorphic and σ ( z ; g 2 , g 3 ) is entire. …
2: 23.4 Graphics
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Figure 23.4.1: ( x ; g 2 , 0 ) for 0 x 9 , g 2 = 0. … Magnify
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Figure 23.4.2: ( x ; 0 , g 3 ) for 0 x 9 , g 3 = 0. … Magnify
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Figure 23.4.3: ζ ( x ; g 2 , 0 ) for 0 x 8 , g 2 = 0. … Magnify
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Figure 23.4.4: ζ ( x ; 0 , g 3 ) for 0 x 8 , g 3 = 0. … Magnify
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Figure 23.4.5: σ ( x ; g 2 , 0 ) for - 5 x 5 , g 2 = 0. … Magnify
3: 23.14 Integrals
23.14.2 2 ( z ) d z = 1 6 ( z ) + 1 12 g 2 z ,
4: 23.19 Interrelations
23.19.2 J ( τ ) = 4 27 ( 1 - λ ( τ ) + λ 2 ( τ ) ) 3 ( λ ( τ ) ( 1 - λ ( τ ) ) ) 2 ,
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). …
5: 23.2 Definitions and Periodic Properties
23.2.4 ( z ) = 1 z 2 + w 𝕃 { 0 } ( 1 ( z - w ) 2 - 1 w 2 ) ,
23.2.5 ζ ( z ) = 1 z + w 𝕃 { 0 } ( 1 z - w + 1 w + z w 2 ) ,
23.2.6 σ ( z ) = z w 𝕃 { 0 } ( ( 1 - z w ) exp ( z w + z 2 2 w 2 ) ) .
6: 23.10 Addition Theorems and Other Identities
23.10.1 ( u + v ) = 1 4 ( ( u ) - ( v ) ( u ) - ( v ) ) 2 - ( u ) - ( v ) ,
23.10.2 ζ ( u + v ) = ζ ( u ) + ζ ( v ) + 1 2 ζ ′′ ( u ) - ζ ′′ ( v ) ζ ( u ) - ζ ( v ) ,
23.10.3 σ ( u + v ) σ ( u - v ) σ 2 ( u ) σ 2 ( v ) = ( v ) - ( u ) ,
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. …
7: 23.9 Laurent and Other Power Series
23.9.2 ( z ) = 1 z 2 + n = 2 c n z 2 n - 2 , 0 < | z | < | z 0 | ,
23.9.3 ζ ( z ) = 1 z - n = 2 c n 2 n - 1 z 2 n - 1 , 0 < | z | < | z 0 | .
c 2 = 1 20 g 2 ,
c 3 = 1 28 g 3 ,
23.9.7 σ ( z ) = m , n = 0 a m , n ( 10 c 2 ) m ( 56 c 3 ) n z 4 m + 6 n + 1 ( 4 m + 6 n + 1 ) ! ,
8: 23.23 Tables
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 . …
9: 23.1 Special Notation
𝕃

lattice in .

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 ) . …
10: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions λ , J , and η . …
See accompanying text
Figure 23.16.1: Modular functions λ ( i y ) , J ( i y ) , η ( i y ) for 0 y 3 . … Magnify