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NIST
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:
${a}_{\nu}^{m}\left(0.5\right)$
,
${b}_{\nu}^{m+1}\left(0.5\right)$
as functions of
$\nu $
for
$m=0,1,2,3$
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$m$
: nonnegative integer
and
$\nu $
: 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:
${a}_{\nu}^{3}\left(0.5\right)-{b}_{\nu}^{3}\left(0.5\right)$
as a function of
$\nu $
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
and
$\nu $
: 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:
${a}_{1.5}^{m}\left({k}^{2}\right)$
,
${b}_{1.5}^{m+1}\left({k}^{2}\right)$
as functions of
${k}^{2}$
for
$m=0,1,2$
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: 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:
${a}_{\nu}^{m}\left(0.1\right)$
,
${b}_{\nu}^{m+1}\left(0.1\right)$
as functions of
$\nu $
for
$m=0,1,2,3$
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$m$
: nonnegative integer
and
$\nu $
: 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:
${a}_{\nu}^{m}\left(0.9\right)$
,
${b}_{\nu}^{m+1}\left(0.9\right)$
as functions of
$\nu $
for
$m=0,1,2,3$
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$m$
: nonnegative integer
and
$\nu $
: 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:
${a}_{\nu}^{2}\left(0.5\right)-{b}_{\nu}^{2}\left(0.5\right)$
as a function of
$\nu $
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
and
$\nu $
: 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:
${a}_{\nu}^{4}\left(0.5\right)-{b}_{\nu}^{4}\left(0.5\right)$
as a function of
$\nu $
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
and
$\nu $
: 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:
${a}_{2.5}^{m}\left({k}^{2}\right)$
,
${b}_{2.5}^{m+1}\left({k}^{2}\right)$
as functions of
${k}^{2}$
for
$m=0,1,2$
.
Magnify
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
${b}_{\nu}^{n}\left({k}^{2}\right)$
: 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:
${a}_{\nu}^{0}\left({k}^{2}\right)$
as a function of
$\nu $
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$k$
: real parameter
and
$\nu $
: 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:
${b}_{\nu}^{1}\left({k}^{2}\right)$
as a function of
$\nu $
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$k$
: real parameter
and
$\nu $
: 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:
${a}_{\nu}^{1}\left({k}^{2}\right)$
as a function of
$\nu $
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${a}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$k$
: real parameter
and
$\nu $
: 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:
${b}_{\nu}^{2}\left({k}^{2}\right)$
as a function of
$\nu $
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${b}_{\nu}^{n}\left({k}^{2}\right)$
: eigenvalues of Lamé’s equation
,
$k$
: real parameter
and
$\nu $
: 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:
${\mathit{Ec}}_{1.5}^{m}(x,0.5)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.85407\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Es}}_{1.5}^{m}(x,0.5)$
for
$-2K\le x\le 2K$
,
$m=1,2,3$
.
$K=1.85407\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Ec}}_{1.5}^{m}(x,0.1)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.61244\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Es}}_{1.5}^{m}(x,0.1)$
for
$-2K\le x\le 2K$
,
$m=1,2,3$
.
$K=1.61244\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Ec}}_{1.5}^{m}(x,0.9)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=2.57809\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Es}}_{1.5}^{m}(x,0.9)$
for
$-2K\le x\le 2K$
,
$m=1,2,3$
.
$K=2.57809\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Ec}}_{2.5}^{m}(x,0.1)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.61244\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Es}}_{2.5}^{m}(x,0.1)$
for
$-2K\le x\le 2K$
,
$m=1,2,3$
.
$K=1.61244\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Ec}}_{2.5}^{m}(x,0.5)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.85407\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Es}}_{2.5}^{m}(x,0.5)$
for
$-2K\le x\le 2K$
,
$m=1,2,3$
.
$K=1.85407\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Ec}}_{2.5}^{m}(x,0.9)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=2.57809\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Es}}_{2.5}^{m}(x,0.9)$
for
$-2K\le x\le 2K$
,
$m=1,2,3$
.
$K=2.57809\mathrm{\dots}$
.
Magnify
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: Lamé function
,
$K\left(k\right)$
: 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:
${\mathit{Ec}}_{1.5}^{0}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Es}}_{1.5}^{1}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Ec}}_{1.5}^{1}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Es}}_{1.5}^{2}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Ec}}_{2.5}^{0}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Es}}_{2.5}^{1}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Ec}}_{2.5}^{1}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$
: 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:
${\mathit{Es}}_{2.5}^{2}(x,{k}^{2})$
as a function of
$x$
and
${k}^{2}$
.
Magnify
3D
Help
ⓘ
Symbols:
${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$
: 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