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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.2 Definitions and Periodic Properties


§23.2(i) Lattices

If ω1 and ω3 are nonzero real or complex numbers such that (ω3/ω1)>0, then the set of points 2mω1+2nω3, with m,n, constitutes a lattice 𝕃 with 2ω1 and 2ω3 lattice generators.

The generators of a given lattice 𝕃 are not unique. For example, if

23.2.1 ω1+ω2+ω3=0,

then 2ω2, 2ω3 are generators, as are 2ω2, 2ω1. In general, if

23.2.2 χ1 =aω1+bω3,
χ3 =cω1+dω3,

where a,b,c,d are integers, then 2χ1, 2χ3 are generators of 𝕃 iff

23.2.3 ad-bc=1.

§23.2(ii) Weierstrass Elliptic Functions

23.2.4 (z)=1z2+w𝕃\{0}(1(z-w)2-1w2),
23.2.5 ζ(z)=1z+w𝕃\{0}(1z-w+1w+zw2),
23.2.6 σ(z)=zw𝕃\{0}((1-zw)exp(zw+z22w2)).

The double series and double product are absolutely and uniformly convergent in compact sets in that do not include lattice points. Hence the order of the terms or factors is immaterial.

(z) and ζ(z) are meromorphic functions with poles at the lattice points. (z) is even and ζ(z) is odd. The poles of (z) are double with residue 0; the poles of ζ(z) are simple with residue 1. The function σ(z) is entire and odd, with simple zeros at the lattice points. When it is important to display the lattice with the functions they are denoted by (z|𝕃), ζ(z|𝕃), and σ(z|𝕃), respectively.

§23.2(iii) Periodicity

If 2ω1, 2ω3 is any pair of generators of 𝕃, and ω2 is defined by (23.2.1), then

23.2.9 (z+2ωj)=(z),

Hence (z) is an elliptic function, that is, (z) is meromorphic and periodic on a lattice; equivalently, (z) is meromorphic and has two periods whose ratio is not real. We also have

23.2.10 (ωj)=0,

The function ζ(z) is quasi-periodic: for j=1,2,3,

23.2.11 ζ(z+2ωj)=ζ(z)+2ηj,


23.2.12 ηj=ζ(ωj).


23.2.13 η1+η2+η3=0,
23.2.14 η3ω2-η2ω3=η2ω1-η1ω2=η1ω3-η3ω1=12π.

For j=1,2,3, the function σ(z) satisfies

23.2.15 σ(z+2ωj)=-2ηj(z+ωj)σ(z),
23.2.16 σ(2ωj)=-2ηjωj.

More generally, if j=1,2,3, k=1,2,3, jk, and m,n, then

23.2.17 σ(z+2mωj+2nωk)/σ(z)=(-1)m+n+mnexp((2mηj+2nηk)(mωj+nωk+z)).

For further quasi-periodic properties of the σ-function see Lawden (1989, §6.2).