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NIST
29
Lamé Functions
Lamé Polynomials
29.12
Definitions
29.14
Orthogonality
§29.13
Graphics
ⓘ
Permalink:
http://dlmf.nist.gov/29.13
See also:
Annotations for
Ch.29
Contents
§29.13(i)
Eigenvalues for Lamé Polynomials
§29.13(ii)
Lamé Polynomials: Real Variable
§29.13(iii)
Lamé Polynomials: Complex Variable
§29.13(i)
Eigenvalues for Lamé Polynomials
ⓘ
Keywords:
Lamé polynomials
,
eigenvalues
,
graphics
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.13.i
See also:
Annotations for
§29.13
and
Ch.29
Figure 29.13.1:
a
2
m
(
k
2
)
,
b
2
m
(
k
2
)
as functions of
k
2
for
m
=
0
,
1
,
2
(
a
’s),
m
=
1
,
2
(
b
’s).
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
m
: nonnegative integer
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F1
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(i)
,
§29.13
and
Ch.29
Figure 29.13.2:
a
1
m
(
k
2
)
,
b
1
m
(
k
2
)
as functions of
k
2
for
m
=
0
,
1
(
a
’s),
m
=
1
(
b
’s).
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
m
: nonnegative integer
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F2
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(i)
,
§29.13
and
Ch.29
Figure 29.13.3:
a
3
m
(
k
2
)
,
b
3
m
(
k
2
)
as functions of
k
2
for
m
=
0
,
1
,
2
,
3
(
a
’s),
m
=
1
,
2
,
3
(
b
’s).
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
m
: nonnegative integer
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F3
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(i)
,
§29.13
and
Ch.29
Figure 29.13.4:
a
4
m
(
k
2
)
,
b
4
m
(
k
2
)
as functions of
k
2
for
m
=
0
,
1
,
2
,
3
,
4
(
a
’s),
m
=
1
,
2
,
3
,
4
(
b
’s).
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
m
: nonnegative integer
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F4
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(i)
,
§29.13
and
Ch.29
§29.13(ii)
Lamé Polynomials: Real Variable
ⓘ
Keywords:
Lamé polynomials
,
graphics
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.13.ii
See also:
Annotations for
§29.13
and
Ch.29
Figure 29.13.5:
uE
4
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
uE
2
n
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F5
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.6:
uE
4
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
uE
2
n
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F6
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.7:
sE
5
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
sE
2
n
+
1
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F7
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.8:
sE
5
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
sE
2
n
+
1
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F8
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.9:
cE
5
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
cE
2
n
+
1
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F9
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.10:
cE
5
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
cE
2
n
+
1
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F10
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.11:
dE
5
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
dE
2
n
+
1
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F11
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.12:
dE
5
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
dE
2
n
+
1
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F12
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.13:
scE
4
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
scE
2
n
+
2
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F13
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.14:
scE
4
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
scE
2
n
+
2
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F14
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.15:
sdE
4
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
sdE
2
n
+
2
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F15
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.16:
sdE
4
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
sdE
2
n
+
2
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F16
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.17:
cdE
4
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
cdE
2
n
+
2
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F17
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.18:
cdE
4
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
cdE
2
n
+
2
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F18
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.19:
scdE
5
m
(
x
,
0.1
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
scdE
2
n
+
3
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F19
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
Figure 29.13.20:
scdE
5
m
(
x
,
0.9
)
for
-
2
K
≤
x
≤
2
K
,
m
=
0
,
1
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
scdE
2
n
+
3
m
(
z
,
k
2
)
: Lamé polynomial
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
m
: nonnegative integer
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F20
Encodings:
pdf
,
png
See also:
Annotations for
§29.13(ii)
,
§29.13
and
Ch.29
§29.13(iii)
Lamé Polynomials: Complex Variable
ⓘ
Keywords:
Lamé polynomials
,
graphics
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.13.iii
See also:
Annotations for
§29.13
and
Ch.29
Figure 29.13.21:
|
uE
4
1
(
x
+
i
y
,
0.1
)
|
for
-
3
K
≤
x
≤
3
K
,
0
≤
y
≤
2
K
′
.
K
=
1.61244
…
,
K
′
=
2.57809
…
.
Magnify
3D
Help
ⓘ
Symbols:
uE
2
n
m
(
z
,
k
2
)
: Lamé polynomial
,
K
′
(
k
)
: Legendre’s complementary complete elliptic integral of the first kind
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
i
: imaginary unit
,
x
: real variable
,
y
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F21
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.13(iii)
,
§29.13
and
Ch.29
Figure 29.13.22:
|
uE
4
1
(
x
+
i
y
,
0.5
)
|
for
-
3
K
≤
x
≤
3
K
,
0
≤
y
≤
2
K
′
.
K
=
K
′
=
1.85407
…
.
Magnify
3D
Help
ⓘ
Symbols:
uE
2
n
m
(
z
,
k
2
)
: Lamé polynomial
,
K
′
(
k
)
: Legendre’s complementary complete elliptic integral of the first kind
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
i
: imaginary unit
,
x
: real variable
,
y
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F22
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.13(iii)
,
§29.13
and
Ch.29
Figure 29.13.23:
|
uE
4
1
(
x
+
i
y
,
0.9
)
|
for
-
3
K
≤
x
≤
3
K
,
0
≤
y
≤
2
K
′
.
K
=
2.57809
…
,
K
′
=
1.61244
…
.
Magnify
3D
Help
ⓘ
Symbols:
uE
2
n
m
(
z
,
k
2
)
: Lamé polynomial
,
K
′
(
k
)
: Legendre’s complementary complete elliptic integral of the first kind
,
K
(
k
)
: Legendre’s complete elliptic integral of the first kind
,
i
: imaginary unit
,
x
: real variable
,
y
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F23
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.13(iii)
,
§29.13
and
Ch.29