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double asymptotic properties

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1: 10.69 Uniform Asymptotic Expansions for Large Order
All fractional powers take their principal values. All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). …
2: 10.41 Asymptotic Expansions for Large Order
§10.41(iv) Double Asymptotic Properties
§10.41(v) Double Asymptotic Properties (Continued)
3: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20(iii) Double Asymptotic Properties
4: 10.74 Methods of Computation
Moreover, because of their double asymptotic properties10.41(v)) these expansions can also be used for large x or | z | , whether or not ν is large. …
5: 12.10 Uniform Asymptotic Expansions for Large Parameter
A 2 ( τ ) = 1 288 τ 2 ( 6160 τ 4 + 18480 τ 3 + 19404 τ 2 + 8028 τ + 945 ) .
In addition, it enjoys a double asymptotic property: it holds if either or both μ and t tend to infinity. …The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). …
6: Mathematical Introduction
In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). …
7: 13.21 Uniform Asymptotic Approximations for Large κ
For asymptotic expansions having double asymptotic properties see Skovgaard (1966). …
8: 2.1 Definitions and Elementary Properties
§2.1 Definitions and Elementary Properties
§2.1(iii) Asymptotic Expansions
If the set X in §2.1(iii) is a closed sector α ph x β , then by definition the asymptotic property (2.1.13) holds uniformly with respect to ph x [ α , β ] as | x | . The asymptotic property may also hold uniformly with respect to parameters. … As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. …
9: Bibliography N
  • G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.
  • G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.
  • NetNUMPAC (free Fortran library)
  • NMS (free collection of Fortran subroutines)
  • C. J. Noble (2004) Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.
  • 10: Bibliography F
  • B. R. Fabijonas (2004) Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.
  • FDLIBM (free C library)
  • C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.
  • A. S. Fokas and Y. C. Yortsos (1981) The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
  • R. C. Forrey (1997) Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.