DLMF
Index
Notations
Search
Help?
Citing
Customize
Annotate
UnAnnotate
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
.)
3D
Help
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
).
3D
Help
ⓘ
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:
http://dlmf.nist.gov/19.3.F4.mag
Encodings:
Magnified png
,
Vizualization
,
pdf
See also:
Annotations for
§19.3
and
Ch.19
