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Weierstrass elliptic functions

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1: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
2: 23.13 Zeros
§23.13 Zeros
3: 23.1 Special Notation
𝕃 lattice in .
= e i π τ nome.
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: 23.21 Physical Applications
§23.21 Physical Applications
§23.21(ii) Nonlinear Evolution Equations
§23.21(iii) Ellipsoidal Coordinates
  • String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

  • 5: 23.4 Graphics
    §23.4(i) Real Variables
    See accompanying text
    Figure 23.4.6: σ ( x ; 0 , g 3 ) for - 5 x 5 , g 3 = 0. … Magnify
    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
    §23.4(ii) Complex Variables
    See accompanying text
    Figure 23.4.12: ( 3.7 ; a + i b , 0 ) for - 5 a 3 , - 4 b 4 . … Magnify 3D Help
    6: 23.5 Special Lattices
    §23.5(ii) Rectangular Lattice
    §23.5(iii) Lemniscatic Lattice
    §23.5(iv) Rhombic Lattice
    For the case ω 3 = e π i / 3 ω 1 see §23.5(v).
    §23.5(v) Equianharmonic Lattice
    7: 23.23 Tables
    §23.23 Tables
    8: 23.20 Mathematical Applications
    §23.20 Mathematical Applications
    §23.20(i) Conformal Mappings
    §23.20(iii) Factorization
    §23.20(v) Modular Functions and Number Theory
    9: Peter L. Walker
    10: 23 Weierstrass Elliptic and Modular
    Functions
    Chapter 23 Weierstrass Elliptic and Modular Functions