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Weierstrass elliptic-function form

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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: 23.1 Special Notation
𝕃 lattice in .
= e i π τ nome.
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 ) . …
3: 23.9 Laurent and Other Power Series
§23.9 Laurent and Other Power Series
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 . …
4: 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 . …
§23.21(ii) Nonlinear Evolution Equations
§23.21(iii) Ellipsoidal Coordinates
where x , y , z are the corresponding Cartesian coordinates and e 1 , e 2 , e 3 are constants. … Another form is obtained by identifying e 1 , e 2 , e 3 as lattice roots (§23.3(i)), and setting …
5: 31.2 Differential Equations
§31.2(ii) Normal Form of Heun’s Equation
§31.2(iii) Trigonometric Form
§31.2(iv) Doubly-Periodic Forms
Jacobi’s Elliptic Form
Weierstrass’s Form
6: 23.3 Differential Equations
The lattice invariants are defined by … and are denoted by e 1 , e 2 , e 3 . … 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. …
§23.3(ii) Differential Equations and Derivatives
7: 23.14 Integrals
§23.14 Integrals
23.14.2 2 ( z ) d z = 1 6 ( z ) + 1 12 g 2 z ,
8: 23.13 Zeros
§23.13 Zeros
For information on the zeros of ( z ) see Eichler and Zagier (1982).
9: 19.25 Relations to Other Functions
§19.25(vi) Weierstrass Elliptic Functions
Let 𝕃 be a lattice for the Weierstrass elliptic function ( z ) . …The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which ( z ) - e j < 0 , for some j . … for some 2 ω j 𝕃 and ( ω j ) = e j . … in which 2 ω 1 and 2 ω 3 are generators for the lattice 𝕃 , ω 2 = - ω 1 - ω 3 , and η j = ζ ( ω j ) (see (23.2.12)). …
10: 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