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29
Lamé Functions
Lamé Functions
29.3
Definitions and Basic Properties
29.5
Special Cases and Limiting Forms
§29.4
Graphics
ⓘ
Permalink:
http://dlmf.nist.gov/29.4
See also:
Annotations for
Ch.29
Contents
§29.4(i)
Eigenvalues of Lamé’s Equation: Line Graphs
§29.4(ii)
Eigenvalues of Lamé’s Equation: Surfaces
§29.4(iii)
Lamé Functions: Line Graphs
§29.4(iv)
Lamé Functions: Surfaces
§29.4(i)
Eigenvalues of Lamé’s Equation: Line Graphs
ⓘ
Keywords:
Lamé functions
,
eigenvalues
,
graphics
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.4.i
See also:
Annotations for
§29.4
and
Ch.29
Figure 29.4.1:
a
ν
m
(
0.5
)
,
b
ν
m
+
1
(
0.5
)
as functions of
ν
for
m
=
0
,
1
,
2
,
3
.
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
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F1
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.2:
a
ν
3
(
0.5
)
−
b
ν
3
(
0.5
)
as a function of
ν
.
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F2
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.3:
a
1.5
m
(
k
2
)
,
b
1.5
m
+
1
(
k
2
)
as functions of
k
2
for
m
=
0
,
1
,
2
.
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/29.4.F3
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.4:
a
ν
m
(
0.1
)
,
b
ν
m
+
1
(
0.1
)
as functions of
ν
for
m
=
0
,
1
,
2
,
3
.
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
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F4
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.5:
a
ν
m
(
0.9
)
,
b
ν
m
+
1
(
0.9
)
as functions of
ν
for
m
=
0
,
1
,
2
,
3
.
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
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F5
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.6:
a
ν
2
(
0.5
)
−
b
ν
2
(
0.5
)
as a function of
ν
.
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F6
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.7:
a
ν
4
(
0.5
)
−
b
ν
4
(
0.5
)
as a function of
ν
.
Magnify
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F7
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
Figure 29.4.8:
a
2.5
m
(
k
2
)
,
b
2.5
m
+
1
(
k
2
)
as functions of
k
2
for
m
=
0
,
1
,
2
.
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/29.4.F8
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(i)
,
§29.4
and
Ch.29
§29.4(ii)
Eigenvalues of Lamé’s Equation: Surfaces
ⓘ
Keywords:
Lamé functions
,
eigenvalues
,
graphics
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.4.ii
See also:
Annotations for
§29.4
and
Ch.29
Figure 29.4.9:
a
ν
0
(
k
2
)
as a function of
ν
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
k
: real parameter
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F9
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(ii)
,
§29.4
and
Ch.29
Figure 29.4.10:
b
ν
1
(
k
2
)
as a function of
ν
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
k
: real parameter
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F10
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(ii)
,
§29.4
and
Ch.29
Figure 29.4.11:
a
ν
1
(
k
2
)
as a function of
ν
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
a
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
k
: real parameter
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F11
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(ii)
,
§29.4
and
Ch.29
Figure 29.4.12:
b
ν
2
(
k
2
)
as a function of
ν
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
b
ν
n
(
k
2
)
: eigenvalues of Lamé’s equation
,
k
: real parameter
and
ν
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F12
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(ii)
,
§29.4
and
Ch.29
§29.4(iii)
Lamé Functions: Line Graphs
ⓘ
Keywords:
Lamé functions
,
graphics
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.4.iii
See also:
Annotations for
§29.4
and
Ch.29
Figure 29.4.13:
𝐸𝑐
1.5
m
(
x
,
0.5
)
for
−
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.85407
…
.
Magnify
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F13
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.14:
𝐸𝑠
1.5
m
(
x
,
0.5
)
for
−
2
K
≤
x
≤
2
K
,
m
=
1
,
2
,
3
.
K
=
1.85407
…
.
Magnify
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F14
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.15:
𝐸𝑐
1.5
m
(
x
,
0.1
)
for
−
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F15
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.16:
𝐸𝑠
1.5
m
(
x
,
0.1
)
for
−
2
K
≤
x
≤
2
K
,
m
=
1
,
2
,
3
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F16
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.17:
𝐸𝑐
1.5
m
(
x
,
0.9
)
for
−
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F17
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.18:
𝐸𝑠
1.5
m
(
x
,
0.9
)
for
−
2
K
≤
x
≤
2
K
,
m
=
1
,
2
,
3
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F18
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.19:
𝐸𝑐
2.5
m
(
x
,
0.1
)
for
−
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F19
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.20:
𝐸𝑠
2.5
m
(
x
,
0.1
)
for
−
2
K
≤
x
≤
2
K
,
m
=
1
,
2
,
3
.
K
=
1.61244
…
.
Magnify
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F20
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.21:
𝐸𝑐
2.5
m
(
x
,
0.5
)
for
−
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
1.85407
…
.
Magnify
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F21
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.22:
𝐸𝑠
2.5
m
(
x
,
0.5
)
for
−
2
K
≤
x
≤
2
K
,
m
=
1
,
2
,
3
.
K
=
1.85407
…
.
Magnify
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F22
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.23:
𝐸𝑐
2.5
m
(
x
,
0.9
)
for
−
2
K
≤
x
≤
2
K
,
m
=
0
,
1
,
2
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F23
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
Figure 29.4.24:
𝐸𝑠
2.5
m
(
x
,
0.9
)
for
−
2
K
≤
x
≤
2
K
,
m
=
1
,
2
,
3
.
K
=
2.57809
…
.
Magnify
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
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/29.4.F24
Encodings:
pdf
,
png
See also:
Annotations for
§29.4(iii)
,
§29.4
and
Ch.29
§29.4(iv)
Lamé Functions: Surfaces
ⓘ
Keywords:
Lamé functions
,
graphics
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/29.4.iv
See also:
Annotations for
§29.4
and
Ch.29
Figure 29.4.25:
𝐸𝑐
1.5
0
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F25
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.26:
𝐸𝑠
1.5
1
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F26
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.27:
𝐸𝑐
1.5
1
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F27
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.28:
𝐸𝑠
1.5
2
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F28
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.29:
𝐸𝑐
2.5
0
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F29
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.30:
𝐸𝑠
2.5
1
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F30
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.31:
𝐸𝑐
2.5
1
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑐
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F31
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29
Figure 29.4.32:
𝐸𝑠
2.5
2
(
x
,
k
2
)
as a function of
x
and
k
2
.
Magnify
3D
Help
ⓘ
Symbols:
𝐸𝑠
ν
m
(
z
,
k
2
)
: Lamé function
,
x
: real variable
and
k
: real parameter
Permalink:
http://dlmf.nist.gov/29.4.F32
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§29.4(iv)
,
§29.4
and
Ch.29