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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.10 Addition Theorems and Other Identities

Contents

§23.10(i) Addition Theorems

23.10.1 (u+v)=14((u)-(v)(u)-(v))2-(u)-(v),
23.10.2 ζ(u+v)=ζ(u)+ζ(v)+12ζ′′(u)-ζ′′(v)ζ(u)-ζ(v),
23.10.3 σ(u+v)σ(u-v)σ2(u)σ2(v)=(v)-(u),
23.10.4 σ(u+v)σ(u-v)σ(x+y)σ(x-y)+σ(v+x)σ(v-x)σ(u+y)σ(u-y)+σ(x+u)σ(x-u)σ(v+y)σ(v-y)=0.

For further addition-type identities for the σ-function see Lawden (1989, §6.4).

If u+v+w=0, then

23.10.5 |1(u)(u)1(v)(v)1(w)(w)|=0,

and

23.10.6 (ζ(u)+ζ(v)+ζ(w))2+ζ(u)+ζ(v)+ζ(w)=0.

§23.10(ii) Duplication Formulas

23.10.7 (2z)=-2(z)+14(′′(z)(z))2,
23.10.8 ((2z)-e1)2(z)=(((z)-e1)2-(e1-e2)(e1-e3))2.

(23.10.8) continues to hold when e1, e2, e3 are permuted cyclically.

23.10.9 ζ(2z)=2ζ(z)+12ζ′′′(z)ζ′′(z),
23.10.10 σ(2z)=-(z)σ4(z).

§23.10(iii) n-Tuple Formulas

For n=2,3,,

23.10.11 n2(nz)=j=0n-1=0n-1(z+2jnω1+2nω3),
23.10.12 nζ(nz)=-n(n-1)(η1+η3)+j=0n-1=0n-1ζ(z+2jnω1+2nω3),
23.10.13 σ(nz)=An-n(n-1)(η1+η3)zj=0n-1=0n-1σ(z+2jnω1+2nω3),

where

23.10.14 An=nj=0n-1=0jn-11σ((2jω1+2ω3)/n).

Equivalently,

23.10.15 An=(π2G2ω1)n2-1qn(n-1)/2n-1exp(-(n-1)η13ω1((2n-1)(ω12+ω32)+3(n-1)ω1ω3)),

where

23.10.16 q =πω3/ω1,
G =n=1(1-q2n).

§23.10(iv) Homogeneity

For any nonzero real or complex constant c

23.10.17 (cz|c𝕃) =c-2(z|𝕃),
23.10.18 ζ(cz|c𝕃) =c-1ζ(z|𝕃),
23.10.19 σ(cz|c𝕃) =cσ(z|𝕃).

Also, when 𝕃 is replaced by c𝕃 the lattice invariants g2 and g3 are divided by c4 and c6, respectively.

For these results and further identities see Lawden (1989, §6.6) and Apostol (1990, p. 14).