10.2 The Elliptic Functions
133
= k2tau2k(ktau, what) either τ from k or vice versa. Here, “k,”
respectively, “tau” is restricted to complex scalars.
10.2 The Elliptic Functions
The evaluation of the elliptic functions is based on the ϑ functions. Therefore, in a
first step, we have to compute the parameter τ .
10.2.1 Equations and Computational Aspects
The nome q, Eq. (9.2), is given by
q =
exp(−πK (k))
exp(−πK(k))
(10.4a)
with
K
(k) = K(k
) and k
=
1 − k 2 ,
(10.4b)
and thus,
τ = i
K (k)
K(k)
.
(10.4c)
The Jacobi elliptic functions are related to the ϑ functions via
sn(z, k) =
ϑ 3 (0, τ )ϑ 1 (ζ, τ )
ϑ 2 (0, τ )ϑ 4 (ζ, τ )
=
1
ns(z, k)
(10.5a)
cn(z, k) =
ϑ 4 (0, τ )ϑ 2 (ζ, τ )
ϑ 2 (0, τ )ϑ 4 (ζ, τ )
=
1
nc(z, k)
(10.5b)
dn(z, k) =
ϑ 4 (0, τ )ϑ 3 (ζ, τ )
ϑ 3 (0, τ )ϑ 4 (ζ, τ )
=
1
nd(z, k)
(10.5c)
sd(z, k) =
ϑ 3 (0, τ ) 2 ϑ 1 (ζ, τ )
ϑ 2 (0, τ )ϑ 4 (0, τ )ϑ 3 (ζ, τ )
=
1
ds(z, k)
(10.5d)
cd(z, k) =
ϑ 3 (0, τ )ϑ 2 (ζ, τ )
ϑ 2 (0, τ )ϑ 3 (ζ, τ )
=
1
dc(z, k)
(10.5e)
sc(z, k) =
ϑ 3 (0, τ )ϑ 1 (ζ, τ )
ϑ 4 (0, τ )ϑ 2 (ζ, τ )
=
1
cs(z, k)
(10.5f)