29 Lamé Functions Lamé Polynomials 29.12 Definitions 29.14 Orthogonality

§29.13 Graphics

Permalink:: http://dlmf.nist.gov/29.13

§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: $\mathop{a^{{m}}_{{2}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{m}}_{{2}}\/}\nolimits\!\left(k^{2}\right)$ as functions of $k^{2}$ for

(

’s),

(

’s).

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F1
Encodings:: pdf, png

Figure 29.13.2: $\mathop{a^{{m}}_{{1}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{m}}_{{1}}\/}\nolimits\!\left(k^{2}\right)$ as functions of $k^{2}$ for

(

’s),

(

’s).

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F2
Encodings:: pdf, png

Figure 29.13.3: $\mathop{a^{{m}}_{{3}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{m}}_{{3}}\/}\nolimits\!\left(k^{2}\right)$ as functions of $k^{2}$ for

(

’s),

(

’s).

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F3
Encodings:: pdf, png

Figure 29.13.4: $\mathop{a^{{m}}_{{4}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{m}}_{{4}}\/}\nolimits\!\left(k^{2}\right)$ as functions of $k^{2}$ for

(

’s),

(

’s).

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer and : 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: $\mathop{\mathit{uE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F5
Encodings:: pdf, png

Figure 29.13.6: $\mathop{\mathit{uE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F6
Encodings:: pdf, png

Figure 29.13.7: $\mathop{\mathit{sE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F7
Encodings:: pdf, png

Figure 29.13.8: $\mathop{\mathit{sE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F8
Encodings:: pdf, png

Figure 29.13.9: $\mathop{\mathit{cE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F9
Encodings:: pdf, png

Figure 29.13.10: $\mathop{\mathit{cE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{cE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F10
Encodings:: pdf, png

Figure 29.13.11: $\mathop{\mathit{dE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{dE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F11
Encodings:: pdf, png

Figure 29.13.12: $\mathop{\mathit{dE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{dE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F12
Encodings:: pdf, png

Figure 29.13.13: $\mathop{\mathit{scE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F13
Encodings:: pdf, png

Figure 29.13.14: $\mathop{\mathit{scE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F14
Encodings:: pdf, png

Figure 29.13.15: $\mathop{\mathit{sdE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F15
Encodings:: pdf, png

Figure 29.13.16: $\mathop{\mathit{sdE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F16
Encodings:: pdf, png

Figure 29.13.17: $\mathop{\mathit{cdE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F17
Encodings:: pdf, png

Figure 29.13.18: $\mathop{\mathit{cdE}^{{m}}_{{4}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{cdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F18
Encodings:: pdf, png

Figure 29.13.19: $\mathop{\mathit{scdE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.1\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F19
Encodings:: pdf, png

Figure 29.13.20: $\mathop{\mathit{scdE}^{{m}}_{{5}}\/}\nolimits\!\left(x,0.9\right)$ for $-2\!\mathop{K\/}\nolimits\!\leq x\leq 2\!\mathop{K\/}\nolimits\!$ ,

. $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $|\mathop{\mathit{uE}^{{1}}_{{4}}\/}\nolimits\!\left(x+iy,0.1\right)|$ for $-3\!\mathop{K\/}\nolimits\!\leq x\leq 3\!\mathop{K\/}\nolimits\!$ , $0\leq y\leq 2\!\mathop{{K^{{\prime}}}\/}\nolimits\!$ . $\!\mathop{K\/}\nolimits\!=1.61244\ldots$ , $\!\mathop{{K^{{\prime}}}\/}\nolimits\!=2.57809\ldots$ .

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : real variable, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F21
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 29.13.22: $|\mathop{\mathit{uE}^{{1}}_{{4}}\/}\nolimits\!\left(x+iy,0.5\right)|$ for $-3\!\mathop{K\/}\nolimits\!\leq x\leq 3\!\mathop{K\/}\nolimits\!$ , $0\leq y\leq 2\!\mathop{{K^{{\prime}}}\/}\nolimits\!$ . $\!\mathop{K\/}\nolimits\!=\!\mathop{{K^{{\prime}}}\/}\nolimits\!=1.85407\ldots$ .

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : real variable, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F22
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 29.13.23: $|\mathop{\mathit{uE}^{{1}}_{{4}}\/}\nolimits\!\left(x+iy,0.9\right)|$ for $-3\!\mathop{K\/}\nolimits\!\leq x\leq 3\!\mathop{K\/}\nolimits\!$ , $0\leq y\leq 2\!\mathop{{K^{{\prime}}}\/}\nolimits\!$ . $\!\mathop{K\/}\nolimits\!=2.57809\ldots$ , $\!\mathop{{K^{{\prime}}}\/}\nolimits\!=1.61244\ldots$ .

Symbols:: $\mathop{\mathit{uE}^{{m}}_{{2n}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé polynomial, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : real variable, : real variable and : real parameter
Permalink:: http://dlmf.nist.gov/29.13.F23
Encodings:: VRML, X3D, pdf, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 29.12 Definitions 29.14 Orthogonality

§29.13 Graphics

Contents

§29.13(i) Eigenvalues for Lamé Polynomials

§29.13(ii) Lamé Polynomials: Real Variable

§29.13(iii) Lamé Polynomials: Complex Variable