Lamé equation

SearchAdvancedHelp

(0.002 seconds)

1—10 of 39 matching pages

1: 29.19 Physical Applications

… ►

§29.19(i) Lamé Functions

►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. … ►

§29.19(ii) Lamé Polynomials

…

2: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by …In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

3: 29.9 Stability

§29.9 Stability

►The Lamé equation (29.2.1) with specified values of $k,h,\nu$ is called stable if all of its solutions are bounded on $ℝ$; otherwise the equation is called unstable. …

4: 29.2 Differential Equations

… ►

§29.2(i) Lamé’s Equation

… ►

§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

5: 29.4 Graphics

… ►

§29.4(i) Eigenvalues of Lamé’s Equation: Line Graphs

… ►

§29.4(ii) Eigenvalues of Lamé’s Equation: Surfaces

…

6: 29.3 Definitions and Basic Properties

… ► ►

Table 29.3.1: Eigenvalues of Lamé’s equation.

► ►►

eigenvalue $h$	parity	period
…

►

… ►

29.3.2

ⓘ

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, $k$: real parameter and $\nu$: real parameter

Permalink:

http://dlmf.nist.gov/29.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.3(ii), §29.3 and Ch.29

… ►satisfies the continued-fraction equation … ►

Table 29.3.2: Lamé functions.

► ►►

boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
…

►

…

7: 29.7 Asymptotic Expansions

… ►

29.7.1 $$a_{\nu }^m\left(k^2\right)\sim p\kappa -{\tau }_0-{\tau }_1{\kappa }^{-1}-{\tau }_2{\kappa }^{-2}-\mathrm{\cdots },$$

ⓘ

Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $\sim$: Poincaré asymptotic expansion, $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\kappa$ and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►

29.7.5 $$b_{\nu }^{m+1}\left(k^2\right)-a_{\nu }^m\left(k^2\right)=O\left({\nu }^{m+\frac{3}{2}}{\left(\frac{1-k}{1+k}\right)}^{\nu }\right),$$ $\nu \to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $b_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $m$: nonnegative integer, $k$: real parameter and $\nu$: real parameter

Referenced by:

§29.7(i)

Permalink:

http://dlmf.nist.gov/29.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.

8: 29.1 Special Notation

… ►

$${(s_{\nu }^m(k^2))}^2=\frac{4}{\pi }{\int }_0^K{\left({\mathrm{𝐸𝑠}}_{\nu }^m(x,k^2)\right)}^{2}dx.$$

9: 29.5 Special Cases and Limiting Forms

… ►

29.5.1 $$a_{\nu }^m\left(0\right)=b_{\nu }^m\left(0\right)=m^2,$$

ⓘ

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/29.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

… ►

29.5.4 $$\underset{k\to 1-}{lim}{a}_{\nu }^m\left(k^2\right)=\underset{k\to 1-}{lim}{b}_{\nu }^{m+1}\left(k^2\right)=\nu (\nu +1)-{\mu }^2,$$

ⓘ

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, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

…

10: 29.17 Other Solutions

… ►

§29.17(i) Second Solution

…