Lamé functions

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1: 29.19 Physical Applications

§29.19 Physical Applications

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§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. …Brack et al. (2001) shows that Lamé functions occur at bifurcations in chaotic Hamiltonian systems. Bronski et al. (2001) uses Lamé functions in the theory of Bose–Einstein condensates. …

2: 29 Lamé Functions

Chapter 29 Lamé Functions

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3: 29.22 Software

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§29.22(i) Lamé Functions

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LA1: Eigenvalues for Lamé functions.

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LA2: Lamé functions.

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LA5: Coefficients ${\tau }_j$ of the asymptotic expansions for the eigenvalues of the Lamé functions; see §29.7(i).

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4: 29.17 Other Solutions

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§29.17(ii) Algebraic Lamé Functions

►Algebraic Lamé functions are solutions of (29.2.1) when $\nu$ is half an odd integer. … ►

§29.17(iii) Lamé–Wangerin Functions

►Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. …

5: 29.20 Methods of Computation

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§29.20(i) Lamé Functions

… ►The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). … ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to \mathrm{\infty }$. …

6: 29.5 Special Cases and Limiting Forms

§29.5 Special Cases and Limiting Forms

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29.5.2 $${\mathrm{𝐸𝑐}}_{\nu }^0(z,0)=2^{-\frac{1}{2}},$$

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Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $z$: complex variable and $\nu$: real parameter

Permalink:

http://dlmf.nist.gov/29.5.E2

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Annotations for §29.5 and Ch.29

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

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Annotations for §29.5 and Ch.29

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29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

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See also:

Annotations for §29.5 and Ch.29

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7: 29.21 Tables

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

8: 29.3 Definitions and Basic Properties

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§29.3(i) Eigenvalues

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Table 29.3.1: Eigenvalues of Lamé’s equation.

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eigenvalue $h$	parity	period
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§29.3(iv) Lamé Functions

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Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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9: 29.10 Lamé Functions with Imaginary Periods

§29.10 Lamé Functions with Imaginary Periods

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10: 29.7 Asymptotic Expansions

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§29.7(i) Eigenvalues

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§29.7(ii) Lamé Functions

… ►In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$ and ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$. …