29 Lamé Functions Lamé Functions 29.3 Definitions and Basic Properties 29.5 Special Cases and Limiting Forms

§29.4 Graphics

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Permalink:: http://dlmf.nist.gov/29.4
See also:: Annotations for Ch.29

§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

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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: $a^{m}_{\nu}\left(0.5\right)$ , $b^{m+1}_{\nu}\left(0.5\right)$ as functions of $\nu$ for

. Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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: $a^{3}_{\nu}\left(0.5\right)-b^{3}_{\nu}\left(0.5\right)$ as a function of $\nu$ . Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : 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: $a^{m}_{1.5}\left(k^{2}\right)$ , $b^{m+1}_{1.5}\left(k^{2}\right)$ as functions of $k^{2}$ for

. Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer and : 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: $a^{m}_{\nu}\left(0.1\right)$ , $b^{m+1}_{\nu}\left(0.1\right)$ as functions of $\nu$ for

. Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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: $a^{m}_{\nu}\left(0.9\right)$ , $b^{m+1}_{\nu}\left(0.9\right)$ as functions of $\nu$ for

. Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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: $a^{2}_{\nu}\left(0.5\right)-b^{2}_{\nu}\left(0.5\right)$ as a function of $\nu$ . Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : 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: $a^{4}_{\nu}\left(0.5\right)-b^{4}_{\nu}\left(0.5\right)$ as a function of $\nu$ . Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : 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: $a^{m}_{2.5}\left(k^{2}\right)$ , $b^{m+1}_{2.5}\left(k^{2}\right)$ as functions of $k^{2}$ for

. Magnify

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer and : 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

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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: $a^{0}_{\nu}\left(k^{2}\right)$ as a function of $\nu$ and $k^{2}$ . Magnify 3D Help

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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: $b^{1}_{\nu}\left(k^{2}\right)$ as a function of $\nu$ and $k^{2}$ . Magnify 3D Help

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Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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: $a^{1}_{\nu}\left(k^{2}\right)$ as a function of $\nu$ and $k^{2}$ . Magnify 3D Help

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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: $b^{2}_{\nu}\left(k^{2}\right)$ as a function of $\nu$ and $k^{2}$ . Magnify 3D Help

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Symbols:: $b^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, : 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

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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: $\mathit{Ec}^{m}_{1.5}\left(x,0.5\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.85407\dotsc$ . Magnify

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Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Es}^{m}_{1.5}\left(x,0.5\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.85407\dotsc$ . Magnify

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Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Ec}^{m}_{1.5}\left(x,0.1\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.61244\dotsc$ . Magnify

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Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Es}^{m}_{1.5}\left(x,0.1\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.61244\dotsc$ . Magnify

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Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Ec}^{m}_{1.5}\left(x,0.9\right)$ for $-2K\leq x\leq 2K$ ,

. $K=2.57809\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Es}^{m}_{1.5}\left(x,0.9\right)$ for $-2K\leq x\leq 2K$ ,

. $K=2.57809\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Ec}^{m}_{2.5}\left(x,0.1\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.61244\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Es}^{m}_{2.5}\left(x,0.1\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.61244\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Ec}^{m}_{2.5}\left(x,0.5\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.85407\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Es}^{m}_{2.5}\left(x,0.5\right)$ for $-2K\leq x\leq 2K$ ,

. $K=1.85407\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Ec}^{m}_{2.5}\left(x,0.9\right)$ for $-2K\leq x\leq 2K$ ,

. $K=2.57809\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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: $\mathit{Es}^{m}_{2.5}\left(x,0.9\right)$ for $-2K\leq x\leq 2K$ ,

. $K=2.57809\dotsc$ . Magnify

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real variable and : 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

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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: $\mathit{Ec}^{0}_{1.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Es}^{1}_{1.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Ec}^{1}_{1.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Es}^{2}_{1.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Ec}^{0}_{2.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Es}^{1}_{2.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Ec}^{1}_{2.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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: $\mathit{Es}^{2}_{2.5}\left(x,k^{2}\right)$ as a function of

and $k^{2}$ . Magnify 3D Help

ⓘ

Symbols:: $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, : real variable and : 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

29.3 Definitions and Basic Properties 29.5 Special Cases and Limiting Forms

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§29.4 Graphics

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