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11: 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. … A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …
12: 2.8 Differential Equations with a Parameter
2.8.1 d 2 w / d z 2 = ( u 2 f ( z ) + g ( z ) ) w ,
in which u is a real or complex parameter, and asymptotic solutions are needed for large | u | that are uniform with respect to z in a point set D in or . For example, u can be the order of a Bessel function or degree of an orthogonal polynomial. …
2.8.3 d 2 W d ξ 2 = ( u 2 z ˙ 2 f ( z ) + ψ ( ξ ) ) W ,
2.8.8 d 2 W / d ξ 2 = ( u 2 ξ m + ψ ( ξ ) ) W ,
13: 30.9 Asymptotic Approximations and Expansions
§30.9(i) Prolate Spheroidal Wave Functions
30.9.1 λ n m ( γ 2 ) - γ 2 + γ q + β 0 + β 1 γ - 1 + β 2 γ - 2 + ,
The cases of large m , and of large m and large | γ | , are studied in Abramowitz (1949). …The behavior of λ n m ( γ 2 ) for complex γ 2 and large | λ n m ( γ 2 ) | is investigated in Hunter and Guerrieri (1982).
14: 28.34 Methods of Computation
  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • (b)

    Use of asymptotic expansions and approximations for large q (§§28.8(ii)28.8(iv)).

  • (c)

    Use of asymptotic expansions for large z or large q . See §§28.25 and 28.26.

  • 15: 18.15 Asymptotic Approximations
    For large β , fixed α , and 0 n / β c , Dunster (1999) gives asymptotic expansions of P n ( α , β ) ( z ) that are uniform in unbounded complex z -domains containing z = ± 1 . …This reference also supplies asymptotic expansions of P n ( α , β ) ( z ) for large n , fixed α , and 0 β / n c . … The asymptotic behavior of the classical OP’s as x ± with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1. … These approximations apply when the parameters are large, namely α and β (subject to restrictions) in the case of Jacobi polynomials, λ in the case of ultraspherical polynomials, and | α | + | x | in the case of Laguerre polynomials. …
    16: 30.11 Radial Spheroidal Wave Functions
    30.11.4 A n ± m ( γ 2 ) = 2 k m - n ( - 1 ) k a n , k ± m ( γ 2 ) ( 0 ) .
    §30.11(iii) Asymptotic Behavior
    30.11.6 S n m ( j ) ( z , γ ) = { ψ n ( j ) ( γ z ) + O ( z - 2 e | z | ) , j = 1 , 2 , ψ n ( j ) ( γ z ) ( 1 + O ( z - 1 ) ) , j = 3 , 4 .
    30.11.7 𝒲 { S n m ( 1 ) ( z , γ ) , S n m ( 2 ) ( z , γ ) } = 1 γ ( z 2 - 1 ) .
    17: 8.20 Asymptotic Expansions of E p ( z )
    §8.20(i) Large z
    §8.20(ii) Large p
    so that A k ( λ ) is a polynomial in λ of degree k - 1 when k 1 . …
    18: 10.20 Uniform Asymptotic Expansions for Large Order
    §10.20 Uniform Asymptotic Expansions for Large Order
    In the following formulas for the coefficients A k ( ζ ) , B k ( ζ ) , C k ( ζ ) , and D k ( ζ ) , u k , v k are the constants defined in §9.7(i), and U k ( p ) , V k ( p ) are the polynomials in p of degree 3 k defined in §10.41(ii). …
    §10.20(iii) Double Asymptotic Properties
    For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of z see §10.41(v).
    19: 30.16 Methods of Computation
    If | γ 2 | is large we can use the asymptotic expansions in §30.9. … For d sufficiently large, construct the d × d tridiagonal matrix A = [ A j , k ] with nonzero elements … If | γ 2 | is large, then we can use the asymptotic expansions referred to in §30.9 to approximate Ps n m ( x , γ 2 ) . …
    30.16.7 j = 1 d e j , d 2 ( n + m + 2 j - 2 p ) ! ( n - m + 2 j - 2 p ) ! 1 2 n + 4 j - 4 p + 1 = ( n + m ) ! ( n - m ) ! 1 2 n + 1 .
    30.16.8 a n , k m ( γ 2 ) = lim d e k + p , d ,
    20: 8.27 Approximations
  • DiDonato (1978) gives a simple approximation for the function F ( p , x ) = x - p e x 2 / 2 x e - t 2 / 2 t p d t (which is related to the incomplete gamma function by a change of variables) for real p and large positive x . This takes the form F ( p , x ) = 4 x / h ( p , x ) , approximately, where h ( p , x ) = 3 ( x 2 - p ) + ( x 2 - p ) 2 + 8 ( x 2 + p ) and is shown to produce an absolute error O ( x - 7 ) as x .

  • Verbeeck (1970) gives polynomial and rational approximations for E p ( x ) = ( e - x / x ) P ( z ) , approximately, where P ( z ) denotes a quotient of polynomials of equal degree in z = x - 1 .