# double asymptotic properties

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##### 2: 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). …
##### 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 $\nu$ is large. …
##### 5: 12.10 Uniform Asymptotic Expansions for Large Parameter
$\mathsf{A}_{2}(\tau)=\tfrac{1}{288}\tau^{2}(6160\tau^{4}+18480\tau^{3}+19404% \tau^{2}+8028\tau+945).$
In addition, it enjoys a double asymptotic property: it holds if either or both $\mu$ 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 $\kappa$
For asymptotic expansions having double asymptotic properties see Skovgaard (1966). …
##### 8: 2.1 Definitions and Elementary Properties
###### §2.1(iii) Asymptotic Expansions
If the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha\leq\operatorname{ph}x\leq\beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\operatorname{ph}x\in[\alpha,\beta]$ as $|x|\to\infty$. 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 S
• I. M. Sheffer (1939) Some properties of polynomial sets of type zero. Duke Math. J. 5, pp. 590–622.
• P. N. Shivakumar and J. Xue (1999) On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.
• I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.
• R. P. Stanley (1989) Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.
• B. I. Suleĭmanov (1987) The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity. Differ. Uravn. 23 (5), pp. 834–842 (Russian).