asymptotic behavior
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1: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
►§30.9(i) Prolate Spheroidal Wave Functions
… ►The asymptotic behavior of and as in descending powers of is derived in Meixner (1944). …The asymptotic behavior of and as is given in Erdélyi et al. (1955, p. 151). …2: 18.16 Zeros
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Asymptotic Behavior
… ►Asymptotic Behavior
… ►For an asymptotic expansion of as that applies uniformly for , see Olver (1959, §14(i)). …3: Bibliography J
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Asymptotic behavior of integrals.
SIAM Rev. 14 (2), pp. 286–317.
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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.
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4: 19.12 Asymptotic Approximations
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►With denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of and near the singularity at is given by the following convergent series:
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►For the asymptotic behavior of and as and see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007).
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5: 33.23 Methods of Computation
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►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21).
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►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions.
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6: 19.10 Relations to Other Functions
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►In each case when , the quantity multiplying supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0.
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7: 28.31 Equations of Whittaker–Hill and Ince
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Asymptotic Behavior
…8: 30.11 Radial Spheroidal Wave Functions
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§30.11(iii) Asymptotic Behavior
…9: 3.6 Linear Difference Equations
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►A new problem arises, however, if, as , the asymptotic behavior of is intermediate to those of two independent solutions and of the corresponding inhomogeneous equation (the complementary functions).
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►Thus the asymptotic behavior of the particular solution is intermediate to those of the complementary functions and ; moreover, the conditions for Olver’s algorithm are satisfied.
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►Here , and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions.
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