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23 Weierstrass Elliptic and Modular FunctionsNotation

§23.1 Special Notation

(For other notation see Notation for the Special Functions.)

𝕃

lattice in .

,n

integers.

m

integer, except in §23.20(ii).

z=x+iy

complex variable, except in §§23.20(ii), 23.21(iii).

[a,b] or (a,b)

closed, or open, straight-line segment joining a and b, whether or not a and b are real.

primes

derivatives with respect to the variable, except where indicated otherwise.

K(k), K(k)

complete elliptic integrals (§19.2(i)).

2ω1,2ω3

lattice generators ((ω3/ω1)>0).

ω2

-ω1-ω3.

τ=ω3/ω1

lattice parameter (τ>0).

q=eiπω3/ω1
=eiπτ

nome.

Δ

discriminant g23-27g32.

n

set of all integer multiples of n.

S1/S2

set of all elements of S1, modulo elements of S2. Thus two elements of S1/S2 are equivalent if they are both in S1 and their difference is in S2. (For an example see §20.12(ii).)

G×H

Cartesian product of groups G and H, that is, the set of all pairs of elements (g,h) with group operation (g1,h1)+(g2,h2)=(g1+g2,h1+h2).

The main functions treated in this chapter are the Weierstrass -function (z)=(z|𝕃)=(z;g2,g3); the Weierstrass zeta function ζ(z)=ζ(z|𝕃)=ζ(z;g2,g3); the Weierstrass sigma function σ(z)=σ(z|𝕃)=σ(z;g2,g3); the elliptic modular function λ(τ); Klein’s complete invariant J(τ); Dedekind’s eta function η(τ).

Other Notations

Whittaker and Watson (1927) requires only (ω3/ω1)0, instead of (ω3/ω1)>0. Abramowitz and Stegun (1964, Chapter 18) considers only rectangular and rhombic lattices (§23.5); ω1, ω3 are replaced by ω, ω for the former and by ω2, ω for the latter. Silverman and Tate (1992) and Koblitz (1993) replace 2ω1 and 2ω3 by ω1 and ω3, respectively. Walker (1996) normalizes 2ω1=1, 2ω3=τ, and uses homogeneity (§23.10(iv)). McKean and Moll (1999) replaces 2ω1 and 2ω3 by ω1 and ω2, respectively.