Lamé equation

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11: 31.8 Solutions via Quadratures

… ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …

12: 29.13 Graphics

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§29.13(i) Eigenvalues for Lamé Polynomials

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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). Magnify

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§29.13(ii) Lamé Polynomials: Real Variable

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§29.13(iii) Lamé Polynomials: Complex Variable

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Figure 29.13.23: $|{\mathrm{𝑢𝐸}}_4^1(x+\mathrm{i}y,0.9)|$ for $-3K\le x\le 3K$, $0\le y\le 2K^{\prime }$. … Magnify 3D Help

13: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). … ►where $u_1$, $u_2$, $u_3$ each satisfy the Lamé wave equation (29.11.1). …

14: 29.12 Definitions

§29.12 Definitions

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§29.12(i) Elliptic-Function Form

… ►There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of $z$. … ►

§29.12(ii) Algebraic Form

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15: 29.15 Fourier Series and Chebyshev Series

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29.15.7 $$a_{\nu }^{2m}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.12 $$a_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.17 $$b_{\nu }^{2m+1}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

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Symbols:

$b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.22 $$a_{\nu }^{2m}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.27 $$b_{\nu }^{2m+2}\left(k^2\right)=\frac{1}{2}(H_m+\nu (\nu +1)k^2),$$

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Symbols:

$b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter, $\nu$: real parameter and $H_p$: eigenvalues

Permalink:

http://dlmf.nist.gov/29.15.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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16: 31.7 Relations to Other Functions

… ►equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

17: 29.8 Integral Equations

§29.8 Integral Equations

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29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

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18: Bibliography V

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Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§29.3(ii), §29.3(vii), §29.5, §29.7(i), §29.7(i).

Encodings:

BibTeX

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19: 29.6 Fourier Series

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29.6.2 $$H=2a_{\nu }^{2m}\left(k^2\right)-\nu (\nu +1)k^2,$$

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Defines:

$H$ (locally)

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Referenced by:

§29.6(i)

Permalink:

http://dlmf.nist.gov/29.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(i), §29.6 and Ch.29

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29.6.17 $$H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2,$$

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Defines:

$H$ (locally)

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Permalink:

http://dlmf.nist.gov/29.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(ii), §29.6 and Ch.29

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29.6.32 $$H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2,$$

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Defines:

$H$ (locally)

Symbols:

$b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Permalink:

http://dlmf.nist.gov/29.6.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

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29.6.47 $$H=2b_{\nu }^{2m+2}\left(k^2\right)-\nu (\nu +1)k^2,$$

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Defines:

$H$ (locally)

Symbols:

$b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Permalink:

http://dlmf.nist.gov/29.6.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

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20: Bibliography F

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

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