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1: 31.2 Differential Equations
Weierstrasss Form
2: 23.19 Interrelations
23.19.3 J ( τ ) = g 2 3 g 2 3 - 27 g 3 2 ,
3: 23.1 Special Notation
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 ) . …
4: 29.2 Differential Equations
we have …For the Weierstrass function see §23.2(ii). …
5: 23.7 Quarter Periods
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 ,
6: 1.9 Calculus of a Complex Variable
DeMoivre’s Theorem
Cauchy’s Theorem
Cauchy’s Integral Formula
Liouville’s Theorem
Weierstrass M -test
7: 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 ,
8: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
9: 23.20 Mathematical Applications
§23.20(ii) Elliptic Curves
10: 23.6 Relations to Other Functions
23.6.20 e 3 = - K 2 3 ω 1 2 ( 1 + k 2 ) .