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1: 2.10 Sums and Sequences
§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
2.10.25 f ( z ) = n = - f n z n , 0 < | z | < r .
2: Bibliography B
  • W. G. C. Boyd (1995) Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.
  • 3: 18.15 Asymptotic Approximations
    18.15.23 F 0 ( ζ ) = - 5 48 ζ 2 + ( x - 1 x ζ ) 1 2 ( 1 2 α 2 - 1 8 - 1 4 x x - 1 + 5 24 ( x x - 1 ) 2 ) , 0 x < .
    4: 29.20 Methods of Computation
    Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). … 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 . … A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. … The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …
    5: 2.1 Definitions and Elementary Properties
    2.1.13 f ( x ) = s = 0 n - 1 a s x - s + O ( x - n )
    6: 2.2 Transcendental Equations
    §2.2 Transcendental Equations
    Higher approximations are obtainable by successive resubstitutions. … In place of (2.2.1) assume that …where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …For other examples see de Bruijn (1961, Chapter 2).
    7: Bibliography V
  • A. van Wijngaarden (1953) On the coefficients of the modular invariant J ( τ ) . Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.
  • P. Verbeeck (1970) Rational approximations for exponential integrals E n ( x ) . Acad. Roy. Belg. Bull. Cl. Sci. (5) 56, pp. 1064–1072.
  • R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • H. Volkmer (2008) Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.
  • 8: 9.8 Modulus and Phase
    §9.8(iv) Asymptotic Expansions
    9.8.20 M 2 ( x ) 1 π ( - x ) 1 / 2 k = 0 1 3 5 ( 6 k - 1 ) k ! ( 96 ) k 1 x 3 k ,
    9.8.21 N 2 ( x ) ( - x ) 1 / 2 π k = 0 1 3 5 ( 6 k - 1 ) k ! ( 96 ) k 1 + 6 k 1 - 6 k 1 x 3 k ,
    Also, approximate values (25S) of the coefficients of the powers x - 15 , x - 18 , , x - 56 are available in Sherry (1959).
    9: 2.5 Mellin Transform Methods
    §2.5 Mellin Transform Methods
    This is allowable in view of the asymptotic formula …
    §2.5(ii) Extensions
    The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985). … See also Brüning (1984) for a different approach. …
    10: Bibliography O
  • A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.
  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
  • F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
  • F. W. J. Olver (1980a) Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.
  • F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.