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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 19.3.3
(See
in context
.)
3D
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Figure 19.3.3:
F
(
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
K
(
k
)
, becoming infinite when
k
2
=
1
. If
sin
2
ϕ
=
1
/
k
2
(
<
1
), then it has the value
K
(
1
/
k
)
/
k
: put
c
=
k
2
in (
19.25.5
) and use (
19.25.1
).
3D
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ⓘ
Annotations:
Symbols:
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
F
(
ϕ
,
k
)
: Legendre’s incomplete elliptic integral of the first kind
,
sin
z
: sine function
,
ϕ
: real or complex argument
and
k
: real or complex modulus
Permalink:
http://dlmf.nist.gov/19.3.F3.mag
Encodings:
Magnified png
,
Vizualization
,
pdf
See also:
Annotations for
§19.3
and
Ch.19
