Lamé polynomials

SearchAdvancedHelp

(0.002 seconds)

1—10 of 37 matching pages

1: 29.14 Orthogonality

§29.14 Orthogonality

►Lamé polynomials are orthogonal in two ways. … ►

29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.5 $${\mathrm{𝑐𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑐𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

►

29.14.6 $${\mathrm{𝑑𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑑𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

ⓘ

Symbols:

${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

…

2: 29.12 Definitions

§29.12 Definitions

►

§29.12(i) Elliptic-Function Form

… ►There are eight types of Lamé polynomials, defined as follows: … ►

Table 29.12.1: Lamé polynomials.

► ►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
…

►

…

3: 29.19 Physical Applications

§29.19 Physical Applications

… ►

§29.19(ii) Lamé Polynomials

…

4: 29.13 Graphics

… ►

§29.13(i) Eigenvalues for Lamé Polynomials

… ►

►►

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

►

§29.13(ii) Lamé Polynomials: Real Variable

… ►

§29.13(iii) Lamé Polynomials: Complex Variable

… ►

►

►

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

5: 29.15 Fourier Series and Chebyshev Series

… ►

Polynomial ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$

… ►

Polynomial ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$

…

6: 29.20 Methods of Computation

… ►These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. … ►

§29.20(ii) Lamé Polynomials

… ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … ►

§29.20(iii) Zeros

►Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …

7: 29.16 Asymptotic Expansions

§29.16 Asymptotic Expansions

►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. …

8: 29.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. ►The main functions treated in this chapter are the eigenvalues $a_{\nu }^{2m}\left(k^2\right)$, $a_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+1}\left(k^2\right)$, $b_{\nu }^{2m+2}\left(k^2\right)$, the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)$, ${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)$, and the Lamé polynomials ${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$, ${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$. …

9: 29.22 Software

… ►

§29.22(ii) Lamé Polynomials

… ►

LA3: Eigenvalues for Lamé polynomials.

►

LA4: Lamé polynomials.

…

10: 29.21 Tables

… ►

Ince (1940a) tabulates the eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^{m+1}\left(k^2\right)$ (with $a_{\nu }^{2m+1}$ and $b_{\nu }^{2m+1}$ interchanged) for $k^2=0.1,0.5,0.9$, $\nu =-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

►

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.