About the Project

Lamé polynomials

AdvancedHelp

(0.003 seconds)

1—10 of 37 matching pages

1: 29.14 Orthogonality
§29.14 Orthogonality
β–ΊLamé polynomials are orthogonal in two ways. … β–Ί
29.14.4 𝑠𝐸 2 ⁒ n + 1 m ⁑ ( s , k 2 ) ⁒ 𝑠𝐸 2 ⁒ n + 1 m ⁑ ( K ⁑ + i ⁒ t , k 2 ) ,
β–Ί
29.14.5 𝑐𝐸 2 ⁒ n + 1 m ⁑ ( s , k 2 ) ⁒ 𝑐𝐸 2 ⁒ n + 1 m ⁑ ( K ⁑ + i ⁒ t , k 2 ) ,
β–Ί
29.14.6 𝑑𝐸 2 ⁒ n + 1 m ⁑ ( s , k 2 ) ⁒ 𝑑𝐸 2 ⁒ n + 1 m ⁑ ( K ⁑ + i ⁒ t , k 2 ) ,
2: 29.12 Definitions
§29.12 Definitions
β–Ί
§29.12(i) Elliptic-Function Form
β–ΊThere are eight types of Lamé polynomials, defined as follows: … β–Ί
Table 29.12.1: Lamé polynomials.
β–Ί β–Ίβ–Ί
Ξ½
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 ⁑ i ⁒ K ⁑ )
β–Ί
3: 29.19 Physical Applications
§29.19 Physical Applications
β–Ί
§29.19(ii) Lamé Polynomials
4: 29.13 Graphics
β–Ί
§29.13(i) Eigenvalues for Lamé Polynomials
β–Ί
β–ΊSee accompanying textβ–Ί
Figure 29.13.4: a 4 m ⁑ ( k 2 ) , b 4 m ⁑ ( k 2 ) as functions of k 2 for m = 0 , 1 , 2 , 3 , 4 ( a ’s), m = 1 , 2 , 3 , 4 ( b ’s). Magnify
β–Ί
§29.13(ii) Lamé Polynomials: Real Variable
β–Ί
§29.13(iii) Lamé Polynomials: Complex Variable
β–Ί
β–Ί
See accompanying text
β–Ί
Figure 29.13.23: | 𝑒𝐸 4 1 ⁑ ( x + i ⁒ y , 0.9 ) | for 3 ⁒ K ⁑ x 3 ⁒ K ⁑ , 0 y 2 ⁒ K ⁑ . … Magnify 3D Help
5: 29.15 Fourier Series and Chebyshev Series
β–Ί
Polynomial 𝑒𝐸 2 ⁒ n m ⁑ ( z , k 2 )
β–Ί
Polynomial 𝑠𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 )
β–Ί
Polynomial 𝑐𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 )
β–Ί
Polynomial 𝑑𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 )
β–Ί
Polynomial 𝑠𝑐𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 )
6: 29.20 Methods of Computation
β–ΊThese 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. … β–Ί
§29.20(ii) Lamé Polynomials
β–ΊThe corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. … β–Ί
§29.20(iii) Zeros
β–ΊZeros 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
β–ΊHargrave 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 ⁑ z K ⁑ , 0 ⁑ z K ⁑ , when n ⁒ k and n ⁒ k assume large real values. …
8: 29.1 Special Notation
β–ΊAll derivatives are denoted by differentials, not by primes. β–ΊThe main functions treated in this chapter are the eigenvalues a Ξ½ 2 ⁒ m ⁑ ( k 2 ) , a Ξ½ 2 ⁒ m + 1 ⁑ ( k 2 ) , b Ξ½ 2 ⁒ m + 1 ⁑ ( k 2 ) , b Ξ½ 2 ⁒ m + 2 ⁑ ( k 2 ) , the Lamé functions 𝐸𝑐 Ξ½ 2 ⁒ m ⁑ ( z , k 2 ) , 𝐸𝑐 Ξ½ 2 ⁒ m + 1 ⁑ ( z , k 2 ) , 𝐸𝑠 Ξ½ 2 ⁒ m + 1 ⁑ ( z , k 2 ) , 𝐸𝑠 Ξ½ 2 ⁒ m + 2 ⁑ ( z , k 2 ) , and the Lamé polynomials 𝑒𝐸 2 ⁒ n m ⁑ ( z , k 2 ) , 𝑠𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 ) , 𝑐𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 ) , 𝑑𝐸 2 ⁒ n + 1 m ⁑ ( z , k 2 ) , 𝑠𝑐𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) , 𝑠𝑑𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) , 𝑐𝑑𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) , 𝑠𝑐𝑑𝐸 2 ⁒ n + 3 m ⁑ ( z , k 2 ) . …
9: 29.22 Software
β–Ί
§29.22(ii) Lamé Polynomials
β–Ί
  • LA3: Eigenvalues for Lamé polynomials.

  • β–Ί
  • LA4: Lamé polynomials.

  • 10: 29.21 Tables
    β–Ί
  • Ince (1940a) tabulates the eigenvalues a Ξ½ m ⁑ ( k 2 ) , b Ξ½ m + 1 ⁑ ( k 2 ) (with a Ξ½ 2 ⁒ m + 1 and b Ξ½ 2 ⁒ m + 1 interchanged) for k 2 = 0.1 , 0.5 , 0.9 , Ξ½ = 1 2 , 0 ⁒ ( 1 ) ⁒ 25 , and m = 0 , 1 , 2 , 3 . Precision is 4D.

  • β–Ί
  • 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.