11.2 Equations and Computation
141
=
φ
0
1 − k 2 sin
2 θdθ
(11.2b)
and the complete elliptic integral of second kind by E(k) = E(π/2, k). The
complete one holds E(1) = 1, E(0) = π/2, and
E(k) = 2 · RG(0, 1 − k
2 , 1).
(11.2c)
The computation of the incomplete elliptic integral of second kind is based on the
special values E(0, k) = 0, E(φ, 0) = φ, and E(φ, 1) =
φ
0
1 − sin
2 θdθ is
evaluated via direct integration.
E(φ, k) = RF (c − 1, c − k
2 , c) −
k 2
3
RD(c − 1, c − k
2 , c) with c =
1
sin 2 φ
.
(11.2d)
Legendre’s Elliptic Integral of Third Kind Π
The elliptic integral of third kind Π is given by
Π(φ, n, k) =
sin φ
0
dt
(1 − t 2 )(1 − k 2 t 2 )(1 − nt 2 )
(11.3a)
=
φ
0
dθ
1 − k 2 sin 2 φ (1 − n sin 2 φ)
,
(11.3b)
and the complete integral by Π(n, k) = Π(π/2, n, k). Unfortunately, there are
some variations in the defining equation: Some authors call n α 2 and some are using
(1 + n · · · ) instead of the minus sign. Special values of the complete integral are
Π(n, 0) = 0 if n ∈ R and n > 1, otherwise Π(n, 0) =
π
2
√
1−n
. The computation is
based on the Carlson functions RF and RJ for arbitrary values:
Π(n, k) = RF (0, 1 − k
2 , 1) +
n
3
RJ (0, 1 − k
2 , 1, 1 − n).
(11.3c)
Special values for the incomplete integral are Π(0, n, k) = 0, Π(φ, 0, 0) = φ, and
Π(φ, 1, 0) = tan φ, and
Π(φ, 0, k) = sin φ RF (cos
2 φ, 1 − k
2 sin
2 φ, 1),
(11.3d)
with c =
1
sin 2 φ
Π(φ, n, 0) = RC(c − 1, c − n).
(11.3e)