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29
Lamé Functions
Lamé Polynomials
29.12
Definitions
29.14
Orthogonality
§29.13
Graphics
Permalink:
http://dlmf.nist.gov/29.13
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
Notes:
These graphs were produced at NIST.
Keywords:
Lamé polynomials
Permalink:
http://dlmf.nist.gov/29.13.i
Figure 29.13.1:
${a}_{2}^{m}\left({k}^{2}\right)$
,
${b}_{2}^{m}\left({k}^{2}\right)$
as functions of
${k}^{2}$
for
$m=0,1,2$
(
$a$
’s),
$m=1,2$
(
$b$
’s).
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/29.13.F1
Encodings:
pdf
,
png
Figure 29.13.2:
${a}_{1}^{m}\left({k}^{2}\right)$
,
${b}_{1}^{m}\left({k}^{2}\right)$
as functions of
${k}^{2}$
for
$m=0,1$
(
$a$
’s),
$m=1$
(
$b$
’s).
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/29.13.F2
Encodings:
pdf
,
png
Figure 29.13.3:
${a}_{3}^{m}\left({k}^{2}\right)$
,
${b}_{3}^{m}\left({k}^{2}\right)$
as functions of
${k}^{2}$
for
$m=0,1,2,3$
(
$a$
’s),
$m=1,2,3$
(
$b$
’s).
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/29.13.F3
Encodings:
pdf
,
png
Figure 29.13.4:
${a}_{4}^{m}\left({k}^{2}\right)$
,
${b}_{4}^{m}\left({k}^{2}\right)$
as functions of
${k}^{2}$
for
$m=0,1,2,3,4$
(
$a$
’s),
$m=1,2,3,4$
(
$b$
’s).
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/29.13.F4
Encodings:
pdf
,
png
§29.13(ii)
Lamé Polynomials: Real Variable
Notes:
These graphs were produced at NIST.
Keywords:
Lamé polynomials
Permalink:
http://dlmf.nist.gov/29.13.ii
Figure 29.13.5:
${\mathit{uE}}_{4}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{uE}}_{2n}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F5
Encodings:
pdf
,
png
Figure 29.13.6:
${\mathit{uE}}_{4}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{uE}}_{2n}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F6
Encodings:
pdf
,
png
Figure 29.13.7:
${\mathit{sE}}_{5}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{sE}}_{2n+1}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F7
Encodings:
pdf
,
png
Figure 29.13.8:
${\mathit{sE}}_{5}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{sE}}_{2n+1}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F8
Encodings:
pdf
,
png
Figure 29.13.9:
${\mathit{cE}}_{5}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{cE}}_{2n+1}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F9
Encodings:
pdf
,
png
Figure 29.13.10:
${\mathit{cE}}_{5}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{cE}}_{2n+1}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F10
Encodings:
pdf
,
png
Figure 29.13.11:
${\mathit{dE}}_{5}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{dE}}_{2n+1}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F11
Encodings:
pdf
,
png
Figure 29.13.12:
${\mathit{dE}}_{5}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1,2$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{dE}}_{2n+1}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F12
Encodings:
pdf
,
png
Figure 29.13.13:
${\mathit{scE}}_{4}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{scE}}_{2n+2}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F13
Encodings:
pdf
,
png
Figure 29.13.14:
${\mathit{scE}}_{4}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{scE}}_{2n+2}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F14
Encodings:
pdf
,
png
Figure 29.13.15:
${\mathit{sdE}}_{4}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{sdE}}_{2n+2}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F15
Encodings:
pdf
,
png
Figure 29.13.16:
${\mathit{sdE}}_{4}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{sdE}}_{2n+2}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F16
Encodings:
pdf
,
png
Figure 29.13.17:
${\mathit{cdE}}_{4}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{cdE}}_{2n+2}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F17
Encodings:
pdf
,
png
Figure 29.13.18:
${\mathit{cdE}}_{4}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{cdE}}_{2n+2}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F18
Encodings:
pdf
,
png
Figure 29.13.19:
${\mathit{scdE}}_{5}^{m}\left(x,0.1\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{scdE}}_{2n+3}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F19
Encodings:
pdf
,
png
Figure 29.13.20:
${\mathit{scdE}}_{5}^{m}\left(x,0.9\right)$
for
$-2K\le x\le 2K$
,
$m=0,1$
.
$K=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{scdE}}_{2n+3}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$m$
: nonnegative integer
,
$x$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F20
Encodings:
pdf
,
png
§29.13(iii)
Lamé Polynomials: Complex Variable
Notes:
These surfaces were produced at NIST.
Keywords:
Lamé polynomials
Permalink:
http://dlmf.nist.gov/29.13.iii
Visualization Help
Figure 29.13.21:
$\left|{\mathit{uE}}_{4}^{1}\left(x+\mathrm{i}y,0.1\right)\right|$
for
$-3K\le x\le 3K$
,
$0\le y\le 2{K}^{\prime}$
.
$K=1.61244\mathrm{\dots}$
,
${K}^{\prime}=2.57809\mathrm{\dots}$
.
Symbols:
${\mathit{uE}}_{2n}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
${K}^{\prime}\left(k\right)$
: Legendre’s complementary complete elliptic integral of the first kind
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$x$
: real variable
,
$y$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F21
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 29.13.22:
$\left|{\mathit{uE}}_{4}^{1}\left(x+\mathrm{i}y,0.5\right)\right|$
for
$-3K\le x\le 3K$
,
$0\le y\le 2{K}^{\prime}$
.
$K={K}^{\prime}=1.85407\mathrm{\dots}$
.
Symbols:
${\mathit{uE}}_{2n}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
${K}^{\prime}\left(k\right)$
: Legendre’s complementary complete elliptic integral of the first kind
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$x$
: real variable
,
$y$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F22
Encodings:
VRML
,
X3D
,
pdf
,
png
Visualization Help
Figure 29.13.23:
$\left|{\mathit{uE}}_{4}^{1}\left(x+\mathrm{i}y,0.9\right)\right|$
for
$-3K\le x\le 3K$
,
$0\le y\le 2{K}^{\prime}$
.
$K=2.57809\mathrm{\dots}$
,
${K}^{\prime}=1.61244\mathrm{\dots}$
.
Symbols:
${\mathit{uE}}_{2n}^{m}\left(z,{k}^{2}\right)$
: Lamé polynomial
,
${K}^{\prime}\left(k\right)$
: Legendre’s complementary complete elliptic integral of the first kind
,
$K\left(k\right)$
: Legendre’s complete elliptic integral of the first kind
,
$x$
: real variable
,
$y$
: real variable
and
$k$
: real parameter
Permalink:
http://dlmf.nist.gov/29.13.F23
Encodings:
VRML
,
X3D
,
pdf
,
png