144
11 Elliptic Integrals
and
Δx =
A − x
A
, Δy =
A − y
A
, Δz =
A − z
A
,
(11.6e)
we get the correction values
E 2 = ΔxΔy − (Δz)
2 , E 3 = ΔxΔyΔz,
(11.6f)
and finally
RF (x, y, z) =
1
√
A
1 −
1
10
E 2 +
1
14
E 3 +
1
24
E
2
2 −
3
44
E 2 E 3
.
(11.6g)
Carlson’s Elliptic Integral RG
The symmetric elliptic integral of second kind RG is
RG(x, y, z) =
1
4
∞
0
[(t + x)(t + y)(t + z)]
−
1
2
x
t + x
+
y
t + y
+
z
t + z
,
(11.7a)
=
1
2
zRF (x, y, z) −
1
3
(x − z)(y − z)RD(x, y, z) +
xy
z
.
(11.7b)
Except for the special values, this last equation will be used for evaluating
RG(x, y, z). Due to the symmetry, we select the coordinates such that z is unequal
zero.
Special values are RG(x, 0, 0) =
1
2
√
x, RG(x, x, x) =
1
√
x
, and RG(0, y, y) =
1
4 π
√ y, and its coordinate permutations.
Carlson’s Elliptic Integral of Second Kind RD
An elliptic integral of second kind, symmetric with respect to the first two
coordinates, is
RD(x, y, z) =
3
2
∞
0
[(t + x)(t + y)]
−
1
2 (t + z)
−
3
2 = RJ (x, y, z, z).
(11.8a)
Special values are RD(0, 0, z) = ∞, RD(x, x, x) = 1/
√
x
3 , and
RD(0, y, y) =
3
4 πy −3/2 . The evaluation in all other cases will be based on
Carlson’s duplication formula [1].
With
λ k =
√
x k
√
y k +
√
x k
√
z k +
√
y k
√
z k ,
(11.8b)