# double asymptotic properties

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## 1—10 of 22 matching pages

##### 1: 10.41 Asymptotic Expansions for Large Order

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###### §10.41(iv) Double Asymptotic Properties

… ►###### §10.41(v) Double Asymptotic Properties (Continued)

…##### 2: 10.69 Uniform Asymptotic Expansions for Large Order

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►All fractional powers take their principal values.
►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv).
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##### 3: 10.20 Uniform Asymptotic Expansions for Large Order

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###### §10.20(iii) Double Asymptotic Properties

…##### 4: 10.74 Methods of Computation

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►Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large $x$ or $|z|$, whether or not $\nu $ is large.
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##### 5: 12.10 Uniform Asymptotic Expansions for Large Parameter

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$${\mathsf{A}}_{2}(\tau )=\frac{1}{288}{\tau}^{2}(6160{\tau}^{4}+18480{\tau}^{3}+19404{\tau}^{2}+8028\tau +945).$$

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►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).
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##### 6: Mathematical Introduction

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►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)).
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##### 7: 13.21 Uniform Asymptotic Approximations for Large $\kappa $

##### 8: 2.1 Definitions and Elementary Properties

###### §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 \le \mathrm{ph}x\le \beta $, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty}$. 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

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The resurgence properties of the large order asymptotics of the Anger-Weber function I.
J. Class. Anal. 4 (1), pp. 1–39.
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The resurgence properties of the large order asymptotics of the Anger-Weber function II.
J. Class. Anal. 4 (2), pp. 121–147.
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Evaluation of negative energy Coulomb (Whittaker) functions.
Comput. Phys. Comm. 159 (1), pp. 55–62.
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##### 10: Bibliography F

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Algorithm 838: Airy functions.
ACM Trans. Math. Software 30 (4), pp. 491–501.
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An asymptotic expansion of the double gamma function.
J. Approx. Theory 111 (2), pp. 298–314.
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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions.
Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.
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Computing the hypergeometric function.
J. Comput. Phys. 137 (1), pp. 79–100.
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