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19
Elliptic Integrals
Legendre’s Integrals
19.3
Graphics
19.3
Graphics
19.4
Derivatives and Differential Equations
Figure 19.3.5
(See
in context
.)
Figure 19.3.5:
Π
(
α
2
,
k
)
as a function of
k
2
and
α
2
for
−
2
≤
k
2
<
1
,
−
2
≤
α
2
≤
2
. Cauchy principal values are shown when
α
2
>
1
. The function is unbounded as
α
2
→
1
−
, and also (with the same sign as
1
−
α
2
) as
k
2
→
1
−
. As
α
2
→
1
+
it has the limit
K
(
k
)
−
(
E
(
k
)
/
k
′
2
)
. If
α
2
=
0
, then it reduces to
K
(
k
)
. If
k
2
=
0
, then it has the value
1
2
π
/
1
−
α
2
when
α
2
<
1
, and 0 when
α
2
>
1
. See §
19.6(i)
.
ⓘ
Annotations:
Symbols:
π
: the ratio of the circumference of a circle to its diameter
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
E
(
k
)
: Legendre’s complete elliptic integral of the second kind
,
Π
(
α
2
,
k
)
: Legendre’s complete elliptic integral of the third kind
,
k
: real or complex modulus
,
k
′
: complementary modulus
and
α
2
: real or complex parameter
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