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About the Project
19
Elliptic Integrals
Legendre’s Integrals
19.3
Graphics
19.3
Graphics
19.4
Derivatives and Differential Equations
Figure 19.3.4
(See
in context
.)
Figure 19.3.4:
E
(
ϕ
,
k
)
as a function of
k
2
and
sin
2
ϕ
for
−
1
≤
k
2
≤
2
,
0
≤
sin
2
ϕ
≤
1
. If
sin
2
ϕ
=
1
(
≥
k
2
), then the function reduces to
E
(
k
)
, with value 1 at
k
2
=
1
. If
sin
2
ϕ
=
1
/
k
2
(
<
1
), then it has the value
k
E
(
1
/
k
)
+
(
k
′
2
/
k
)
K
(
1
/
k
)
, with limit 1 as
k
2
→
1
+
: put
c
=
k
2
in (
19.25.7
) and use (
19.25.1
).
ⓘ
Annotations:
Symbols:
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
E
(
k
)
: Legendre’s complete elliptic integral of the second kind
,
E
(
ϕ
,
k
)
: Legendre’s incomplete elliptic integral of the second kind
,
sin
z
: sine function
,
ϕ
: real or complex argument
,
k
: real or complex modulus
and
k
′
: complementary modulus
Permalink:
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