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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 19.3.11
(See
in context
.)
Figure 19.3.11:
ℜ
(
E
(
k
)
)
as a function of complex
k
2
for
−
2
≤
ℜ
(
k
2
)
≤
2
,
−
2
≤
ℑ
(
k
2
)
≤
2
. The real part is symmetric under reflection in the real axis. On the branch cut (
k
2
>
1
) it has the value
k
E
(
1
/
k
)
+
(
k
′
2
/
k
)
K
(
1
/
k
)
, with limit 1 as
k
2
→
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
,
ℑ
: imaginary part
,
ℜ
: real part
,
k
: real or complex modulus
and
k
′
: complementary modulus
Permalink:
http://dlmf.nist.gov/19.3.F11.viz
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