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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.6 Relations to Other Functions


§23.6(i) Theta Functions

In this subsection 2ω1, 2ω3 are any pair of generators of the lattice 𝕃, and the lattice roots e1, e2, e3 are given by (23.3.9).

23.6.1 q =eiπτ,
τ =ω3/ω1.
23.6.2 e1 =π212ω12(θ24(0,q)+2θ44(0,q)),
23.6.3 e2 =π212ω12(θ24(0,q)-θ44(0,q)),
23.6.4 e3 =-π212ω12(2θ24(0,q)+θ44(0,q)).
23.6.5 (z)-e1 =(πθ3(0,q)θ4(0,q)θ2(πz/(2ω1),q)2ω1θ1(πz/(2ω1),q))2,
23.6.6 (z)-e2 =(πθ2(0,q)θ4(0,q)θ3(πz/(2ω1),q)2ω1θ1(πz/(2ω1),q))2,
23.6.7 (z)-e3 =(πθ2(0,q)θ3(0,q)θ4(πz/(2ω1),q)2ω1θ1(πz/(2ω1),q))2.
23.6.8 η1=-π212ω1θ1′′′(0,q)θ1(0,q).
23.6.9 σ(z)=2ω1exp(η1z22ω1)θ1(πz/(2ω1),q)πθ1(0,q),
23.6.10 σ(ω1) =2ω1exp(12η1ω1)θ2(0,q)πθ1(0,q),
23.6.11 σ(ω2) =2ω1iexp(12η1ω1τ2)θ3(0,q)πq1/4θ1(0,q),
23.6.12 σ(ω3) =-2ω1exp(12η1ω1)θ4(0,q)πq1/4θ1(0,q).

With z=πu/(2ω1),

23.6.13 ζ(u)=η1ω1u+π2ω1ddzlnθ1(z,q),
23.6.14 (u)=(π2ω1)2(θ1′′′(0,q)3θ1(0,q)-d2dz2lnθ1(z,q)),
23.6.15 σ(u+ωj)σ(ωj)=exp(ηju+ηju22ω1)θj+1(z,q)θj+1(0,q),

For further results for the σ-function see Lawden (1989, §6.2).

§23.6(ii) Jacobian Elliptic Functions

Again, in Equations (23.6.16)–(23.6.26), 2ω1,2ω3 are any pair of generators of the lattice 𝕃 and e1,e2,e3 are given by (23.3.9).

23.6.16 k2 =e2-e3e1-e3,
k2 =e1-e2e1-e3,
23.6.17 K2 =(K(k))2=ω12(e1-e3),
K2 =(K(k))2=ω32(e3-e1).
23.6.18 e1 =K23ω12(1+k2),
23.6.19 e2 =K23ω12(k2-k2),
23.6.20 e3 =-K23ω12(1+k2).
23.6.21 (z)-e1 =K2ω12cs2(Kzω1,k),
23.6.22 (z)-e2 =K2ω12ds2(Kzω1,k),
23.6.23 (z)-e3 =K2ω12ns2(Kzω1,k).
23.6.24 (z+ω1)-e1 =(Kkω1)2sc2(Kzω1,k),
23.6.25 (z+ω2)-e2 =-(Kkkω1)2sd2(Kzω1,k),
23.6.26 (z+ω3)-e3 =(Kkω1)2sn2(Kzω1,k).

In (23.6.27)–(23.6.29) the modulus k is given and K=K(k), K=K(k) are the corresponding complete elliptic integrals (§19.2(ii)). Also, 𝕃1, 𝕃2, 𝕃3 are the lattices with generators (4K,2iK), (2K-2iK,2K+2iK), (2K,4iK), respectively.

23.6.27 ζ(z|𝕃1)-ζ(z+2K|𝕃1)+ζ(2K|𝕃1)=ns(z,k),
23.6.28 ζ(z|𝕃2)-ζ(z+2K|𝕃2)+ζ(2K|𝕃2)=ds(z,k),
23.6.29 ζ(z|𝕃3)-ζ(z+2iK|𝕃3)-ζ(2iK|𝕃3)=cs(z,k).

Similar results for the other nine Jacobi functions can be constructed with the aid of the transformations given by Table 22.4.3.

For representations of the Jacobi functions sn, cn, and dn as quotients of σ-functions see Lawden (1989, §§6.2, 6.3).

§23.6(iii) General Elliptic Functions

For representations of general elliptic functions (§23.2(iii)) in terms of σ(z) and (z) see Lawden (1989, §§8.9, 8.10), and for expansions in terms of ζ(z) see Lawden (1989, §8.11).

§23.6(iv) Elliptic Integrals

Rectangular Lattice

Let z be on the perimeter of the rectangle with vertices 0,2ω1,2ω1+2ω3,2ω3. Then t=(z) is real (§§23.5(i)23.5(ii)), and

23.6.30 z=12tdu(u-e1)(u-e2)(u-e3),
te1, z(0,ω1],
23.6.31 z-ω1 =i2te1du(e1-u)(u-e2)(u-e3),
e2te1, z[ω1,ω1+ω3],
23.6.32 z-ω3 =12e3tdu(e1-u)(e2-u)(u-e3),
e3te2, z[ω3,ω1+ω3],
23.6.33 z=i2-tdu(e1-u)(e2-u)(e3-u),
te3, z(0,ω3].
23.6.34 2ω1=e1du(u-e1)(u-e2)(u-e3)=e3e2du(e1-u)(e2-u)(u-e3),
23.6.35 2ω3=ie2e1du(e1-u)(u-e2)(u-e3)=i-e3du(e1-u)(e2-u)(e3-u).

For (23.6.30)–(23.6.35) and further identities see Lawden (1989, §6.12).

See also §§19.2(i), 19.14, and Erdélyi et al. (1953b, §13.14).

For relations to symmetric elliptic integrals see §19.25(vi).

General Lattice

Let z be a point of different from e1,e2,e3, and define w by

23.6.36 w=zdu4u3-g2u-g3=12zdu(u-e1)(u-e2)(u-e3),

where the integral is taken along any path from z to that does not pass through any of e1,e2,e3. Then z=(w), where the value of w depends on the choice of path and determination of the square root; see McKean and Moll (1999, pp. 87–88 and §2.5).