134
10 Jacobi Elliptic Functions
with
ζ =
πz
2K(k)
(10.5g)
and
k =
ϑ 2 (0, τ ) 2
ϑ 3 (0, τ ) 2 ,
k
=
ϑ 4 (0, τ ) 2
ϑ 3 (0, τ ) 2 , and K(k) =
π
2
ϑ 3 (0, τ )
2 .
(10.5h)
The Jacobi elliptic functions are double periodic meromorphic functions. An
example that uncovers this double periodic behavior for sn(z, k) is shown in
Fig. 10.1. When the argument z and the parameter k are real, with 0 < k 2 < 1,
all twelve Jacobi elliptic functions will be real. From Eq. (10.3a), it follows that
cn, sn, and dn parameterize an elliptic curve. Table 10.1 lists the periods, zeros,
and poles of the fundamental elliptic functions.
The derivatives of the three elliptic functions sn(z, k), cn(z, k), and dn(z, k) are
d
dz
sn(z, k) = cn(z, k) dn(z, k)
(10.6)
d
dz
cn(z, k) = −sn(z, k) dn(z, k)
(10.7)
d
dz
dn(z, k) = −k
2 sn(z, k) cn(z, k),
(10.8)
Fig. 10.1 Jacobi elliptic
function sn(z, k);
k = 0.5990 . . .. The hight is
given by the absolute value,
and the phase is color coded:
z-value in units of the
complete elliptic integral
K(k). The figure uncovers the
double periodic behavior
4*K
2*K
4*K
2*K
Im(z)
0
Re(z)
0
-2*K
-2*K
-4*K -4*K
5
10
Table 10.1 Periods, zeros, and poles of the elementary elliptic functions. (n and m are integer
numbers.)
Function
Periods
Zeros
Poles
sn(z, k)
4K, 2iK
2mK + 2niK
2mK + (2n + 1)iK
cn(z, k)
4K, 2(K + iK )
(2m + 1)K + 2niK
2mK + (2n + 1)iK
dn(z, k)
2K, 4iK
(2m + 1)K + (2n + 1)iK
2mK + (2n + 1)iK