Digital Library of Mathematical Functions
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22 Jacobian Elliptic FunctionsProperties

§22.2 Definitions

The nome q is given in terms of the modulus k by

22.2.1 q=exp(-πK(k)/K(k)),

where K(k), K(k) are defined in §19.2(ii). Inversely,

22.2.2 k =θ22(0,q)θ32(0,q),
k =θ42(0,q)θ32(0,q),
K(k) =π2θ32(0,q),

where k=1-k2 and the theta functions are defined in §20.2(i).

With

22.2.3 ζ=πz2K(k),
22.2.4 sn(z,k)=θ3(0,q)θ2(0,q)θ1(ζ,q)θ4(ζ,q)=1ns(z,k),
22.2.5 cn(z,k)=θ4(0,q)θ2(0,q)θ2(ζ,q)θ4(ζ,q)=1nc(z,k),
22.2.6 dn(z,k)=θ4(0,q)θ3(0,q)θ3(ζ,q)θ4(ζ,q)=1nd(z,k),
22.2.7 sd(z,k)=θ32(0,q)θ2(0,q)θ4(0,q)θ1(ζ,q)θ3(ζ,q)=1ds(z,k),
22.2.8 cd(z,k)=θ3(0,q)θ2(0,q)θ2(ζ,q)θ3(ζ,q)=1dc(z,k),
22.2.9 sc(z,k)=θ3(0,q)θ4(0,q)θ1(ζ,q)θ2(ζ,q)=1cs(z,k).

As a function of z, with fixed k, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. Each is meromorphic in z for fixed k, with simple poles and simple zeros, and each is meromorphic in k for fixed z. For k[0,1], all functions are real for z.

Glaisher’s Notation

The Jacobian functions are related in the following way. Let p, q, r be any three of the letters s, c, d, n. Then

22.2.10 pq(z,k)=pr(z,k)qr(z,k)=1qp(z,k),

with the convention that functions with the same two letters are replaced by unity; e.g. ss(z,k)=1.

The six functions containing the letter s in their two-letter name are odd in z; the other six are even in z.

In terms of Neville’s theta functions (§20.1)

22.2.11 pq(z,k)=θp(z|τ)/θq(z|τ),

where

22.2.12 τ=K(k)/K(k),

and on the left-hand side of (22.2.11) p, q are any pair of the letters s, c, d, n, and on the right-hand side they correspond to the integers 1,2,3,4.