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31: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
… ►When the zeros are asymptotically given by and , where is a large positive integer and … ►§12.11(iii) Asymptotic Expansions for Large Parameter
►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …32: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►§28.8(ii) Sips’ Expansions
… ►§28.8(iii) Goldstein’s Expansions
… ►Barrett’s Expansions
… ►33: Bibliography O
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Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. II.
Analysis and Applications (Singapore) 1 (1), pp. 121–128.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. III.
Analysis and Applications (Singapore) 8 (2), pp. 199–210.
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Some new asymptotic expansions for Bessel functions of large orders.
Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.
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Tables for Bessel Functions of Moderate or Large Orders.
National Physical Laboratory Mathematical Tables, Vol. 6.
Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
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34: 10.17 Asymptotic Expansions for Large Argument
§10.17 Asymptotic Expansions for Large Argument
… ►§10.17(ii) Asymptotic Expansions of Derivatives
… ►§10.17(iii) Error Bounds for Real Argument and Order
… ►§10.17(v) Exponentially-Improved Expansions
… ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).35: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
►§10.19(i) Asymptotic Forms
… ►§10.19(ii) Debye’s Expansions
… ►§10.19(iii) Transition Region
… ►See also §10.20(i).36: 13.20 Uniform Asymptotic Approximations for Large
§13.20 Uniform Asymptotic Approximations for Large
►§13.20(i) Large , Fixed
… ► … ►§13.20(v) Large , Other Expansions
… ►37: 10.20 Uniform Asymptotic Expansions for Large Order
§10.20 Uniform Asymptotic Expansions for Large Order
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10.20.5
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§10.20(iii) Double Asymptotic Properties
►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of see §10.41(v).38: 12.9 Asymptotic Expansions for Large Variable
§12.9 Asymptotic Expansions for Large Variable
… ►§12.9(ii) Bounds and Re-Expansions for the Remainder Terms
…39: 8.25 Methods of Computation
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►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation.
For large
the corresponding asymptotic expansions (generally divergent) are used instead.
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