11.2 Equations and Computation
145
we compute
x k+1 =
1
4
(x k + λ k )
(11.8c)
and similar y k+1 , z k+1 and iterate this series till x n ≈ y n ≈ z n . With
A =
1
5
(x n + y n + 3z n )
(11.8d)
and
Δx =
A − x
A
, Δy =
A − y
A
, Δz =
A − z
A
,
(11.8e)
we get the correction values
E 2 = ΔxΔy, E 3 = (Δz)
2 , E 4 = E 2 − E 3 , E 5 = E 2 − 6E 3 , E 6 = E 5 + 2E 4 ,
(11.8f)
and finally
RD(x, y, z) = 4
−n A
−
3
2
(11.8g)
·
1 −
3
14
E 2 +
1
6
E 3 +
9
88
E
2
2 −
3
22
E 4 −
9
52
E 2 E 3 +
3
26
E 5
+3
n−1
m=0
4 −m
√ z m (z m + λ m )
.
Carlson’s Elliptic Integral of Third Kind RJ
The integral RJ , Fig. 11.1, symmetric in (x, y, z) is given by
RJ (x, y, z, p) =
3
2
∞
0
[(t + x)(t + y)(t + z)]
−
1
2 (t + p)
−1 ,
(11.9a)
with p = 0. Special values are RJ (x, x, x, x) =
1
√
x
3 , RJ (x, 0, 0, p) = ∞,
RJ (0, y, y, p) =
3π
2(y
√
p+p
√
y) : p > 0
−
3π
2
√
y(y−p) : p < 0,
RJ (x, y, z, z) = RD(x, y, z), RJ (x, x, x, p) = RD(p, p, x),
if y = p and y = z RJ (x, y, y, p) =
3
p − y
(RC(x, y) − RC(x, p)),