11.2 Equations and Computation
147
With
λ k =
√
x k
√
y k +
√
x k
√
z k +
√
y k
√
z k ,
we iterate
A n =
1
4
(A n−1 + λ n−1 ), x n =
1
4
(x n−1 + λ n−1 ), · · · , p n =
1
4
(p n−1 + λ n−1 ),
till x n , y n , z n , and p n are sufficiently close to each other. The result is then given by
Carlson [1]
RJ (x, y, z, p) = 4
−n A
−
3
2
n [1 −
3
14
E 2 +
1
6
E 3 +
9
88
E
2
2 −
3
22
E 4
(11.9c)
−
9
52
E 2 E 3 +
3
26
E 5 ] +
n−1
m=0
4 −m
d m
RC(1, 1 + e m ),
with
d m = (
√
p m +
√
x m )(
√
p m +
√
y m )(
√
p m +
√
z m ) e m =
4 −3m δ
d 2
m
dx =
A 0 − x
4 n A n
, dy =
A 0 − y
4 n A n
, dz =
A 0 − z
4 n A n
, and dp = (−dx − dy − dz)/2,
and
E 2 = dxdy + dxdz + dydz − 3dp
2 , E 3 = dxdydz − 2E 2 dp + 4dp
3 ,
E 4 = (2dxdydz + E 2 dp + 3dp
3 )dp, E 5 = dxdydxdp
2 .
Jacobi’s Zeta Function
Jacobi’s zeta function is defined as
Z(x, k) = E(x, k) − F (x, k) ·
E(k)
K(k)
,
(11.10a)
or alternatively, via the logarithmic derivative of Jacoby’s θ 4 function and can be
evaluated [1] by
Z(x, k) =
k 2
3K(k)
sin x cos x
1 − k 2 sin
2 x RJ (0, 1 − k
2 , 1, 1 − k
2 sin
2 x).
(11.10b)