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1: 29.19 Physical Applications
§29.19 Physical Applications
§29.19(i) Lamé Functions
Simply-periodic Lamé functions ( ν 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
3: 29.22 Software
§29.22(i) Lamé Functions
  • LA1: Eigenvalues for Lamé functions.

  • LA2: Lamé functions.

  • LA5: Coefficients τ j of the asymptotic expansions for the eigenvalues of the Lamé functions; see §29.7(i).

  • 4: 29.17 Other Solutions
    §29.17(ii) Algebraic Lamé Functions
    Algebraic Lamé functions are solutions of (29.2.1) when ν is half an odd integer. …
    §29.17(iii) Lamé–Wangerin Functions
    Lamé–Wangerin functions are solutions of (29.2.1) with the property that ( sn ( z , k ) ) 1 / 2 w ( z ) is bounded on the line segment from i K to 2 K + i K . …
    5: 29.20 Methods of Computation
    §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 . …
    6: 29.5 Special Cases and Limiting Forms
    §29.5 Special Cases and Limiting Forms
    29.5.2 Ec ν 0 ( z , 0 ) = 2 - 1 2 ,
    29.5.5 lim k 1 - Ec ν m ( z , k 2 ) Ec ν m ( 0 , k 2 ) = lim k 1 - Es ν m + 1 ( z , k 2 ) Es ν m + 1 ( 0 , k 2 ) = 1 ( cosh z ) μ F ( 1 2 μ - 1 2 ν , 1 2 μ + 1 2 ν + 1 2 1 2 ; tanh 2 z ) , m even,
    29.5.6 lim k 1 - Ec ν m ( z , k 2 ) d Ec ν m ( z , k 2 ) / d z | z = 0 = lim k 1 - Es ν m + 1 ( z , k 2 ) d Es ν m + 1 ( z , k 2 ) / d z | z = 0 = tanh z ( cosh z ) μ F ( 1 2 μ - 1 2 ν + 1 2 , 1 2 μ + 1 2 ν + 1 3 2 ; tanh 2 z ) , m odd,
    7: 29.21 Tables
  • 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
    §29.3(i) Eigenvalues
    Table 29.3.1: Eigenvalues of Lamé’s equation.
    eigenvalue h parity period
    §29.3(iv) Lamé Functions
    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 )
    9: 29.10 Lamé Functions with Imaginary Periods
    §29.10 Lamé Functions with Imaginary Periods
    10: 29.7 Asymptotic Expansions
    §29.7(i) Eigenvalues
    §29.7(ii) Lamé Functions
    In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions Ec ν m ( z , k 2 ) and Es ν m ( z , k 2 ) . …