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11 Elliptic Integrals
Heumann’s Λ Function
Heumann’s Λ function can be computed and is defined by Carlson [1]
Λ 0 (x, k) =
2
π
(1 − k 2 ) sin x cos x
Δ
×
RF (0, 1 − k
2 , 1) +
k 2
3Δ 2 RJ (0, 1 − k
2 , 1, 1 −
k 2
Δ 2
(11.11)
with Δ =
1 − (1 − k 2 ) sin
2 x.
Bulirsch’s Elliptic Integral el1
An alternative set of elliptic functions was introduced by Bulirsch. Definitions can
be found in [2], and the evaluations are based on Carlson’s paper [1].
el1(x, k c ) is an incomplete integral of first kind given by
el1(x, k c ) = xRF (1, 1 + k
2
c x
2 , 1 + x
2 ).
(11.12)
Bulirsch’s Elliptic Integral el2
el2(x, k c ) is an incomplete integral of second kind defined by
el2(x, k c , a, b) =
arctan x
0
a + b tan 2 θ
(1 + tan 2 θ)(1 + k 2
c tan 2 θ)
dθ
(11.13a)
and can be evaluated with
el2(x, k c , a, b) = a x RF (1, 1 + k
2
c x
2 , 1 + x
2 )
+
1
3
(b − a)x
3 RD(1, 1 + k
2
c x
2 , 1 + x
2 ).
(11.13b)
Bulirsch’s Elliptic Integral el3
el3(x, k c ) is an incomplete integral of third kind
el3(x, k c , p) =
arctan x
0
dθ
(cos 2 θ + p sin 2 θ)
cos 2 θ + k 2
c sin 2 θ
,
(11.14a)
related to Carlson’s elliptic integrals by
el3(x, k c , p) = x RF (1, 1 + k
2
c x
2 , 1 + x
2 )
+
1
3
(1 − p) x
3 RJ (1, 1 + k
2
c x
2 , 1 + x
2 , 1 + p x
2 ).
(11.14b)