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asymptotic behavior of coefficients

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1: 30.8 Expansions in Series of Ferrers Functions
30.8.6 a n , k - m ( γ 2 ) = ( n - m ) ! ( n + m + 2 k ) ! ( n + m ) ! ( n - m + 2 k ) ! a n , k m ( γ 2 ) .
2: Bibliography W
  • G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
  • 3: Bibliography J
  • W. B. Jones and W. Van Assche (1998) Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.
  • 4: 2.10 Sums and Sequences
    What is the asymptotic behavior of f n as n or n - ? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …
  • (c)

    The coefficients in the Laurent expansion

    2.10.27 g ( z ) = n = - g n z n , 0 < | z | < r ,

    have known asymptotic behavior as n ± .

  • 5: 1.8 Fourier Series
    The series (1.8.1) is called the Fourier series of f ( x ) , and a n , b n are the Fourier coefficients of f ( x ) . …
    Asymptotic Estimates of Coefficients
    If f ( x ) and g ( x ) are continuous, have the same period and same Fourier coefficients, then f ( x ) = g ( x ) for all x .
    Lebesgue Constants
    6: 28.4 Fourier Series
    §28.4(vi) Behavior for Small q
    7: 2.4 Contour Integrals
    §2.4(i) Watson’s Lemma
    with known asymptotic behavior as t + . …For examples see Olver (1997b, pp. 315–320). … For integral representations of the b 2 s and their asymptotic behavior as s see Boyd (1995). … For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …
    8: 8.12 Uniform Asymptotic Expansions for Large Parameter
    §8.12 Uniform Asymptotic Expansions for Large Parameter
    where g k , k = 0 , 1 , 2 , , are the coefficients that appear in the asymptotic expansion (5.11.3) of Γ ( z ) . …where d 0 , 0 = - 1 3 , … For the asymptotic behavior of c k ( η ) as k see Dunster et al. (1998) and Olde Daalhuis (1998c). …
    Inverse Function
    9: 30.9 Asymptotic Approximations and Expansions
    §30.9 Asymptotic Approximations and Expansions
    §30.9(i) Prolate Spheroidal Wave Functions
    The asymptotic behavior of λ n m ( γ 2 ) and a n , k m ( γ 2 ) as n in descending powers of 2 n + 1 is derived in Meixner (1944). …The asymptotic behavior of Ps n m ( x , γ 2 ) and Qs n m ( x , γ 2 ) as x ± 1 is given in Erdélyi et al. (1955, p. 151). …
    10: 3.6 Linear Difference Equations
    3.6.1 a n w n + 1 - b n w n + c n w n - 1 = d n ,
    The values of w N and w N + 1 needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). … A new problem arises, however, if, as n , the asymptotic behavior of w n is intermediate to those of two independent solutions f n and g n of the corresponding inhomogeneous equation (the complementary functions). … Thus the asymptotic behavior of the particular solution E n ( 1 ) is intermediate to those of the complementary functions J n ( 1 ) and Y n ( 1 ) ; moreover, the conditions for Olver’s algorithm are satisfied. … Here [ 0 , k ] , and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions. …