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21: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
… ►Products
… ► ►-Derivative
… ►§10.40(iv) Exponentially-Improved Expansions
…22: 11.13 Methods of Computation
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►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation.
For large
and/or the asymptotic expansions given in §11.6 should be used instead.
…
►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that and can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity.
The solution needs to be integrated backwards for small , and either forwards or backwards for large
depending whether or not exceeds .
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23: 28.34 Methods of Computation
24: 29.16 Asymptotic Expansions
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►The approximations for Lamé polynomials hold uniformly on the rectangle , , when and assume large real values.
…
25: 33.10 Limiting Forms for Large or Large
§33.10 Limiting Forms for Large or Large
►§33.10(i) Large
… ►§33.10(ii) Large Positive
… ►§33.10(iii) Large Negative
…26: Ronald F. Boisvert
…
►He is also Editor at Large for the DLMF Project.
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27: 28.26 Asymptotic Approximations for Large
28: 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). …29: 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
… ►30: Bibliography O
…
►
Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
►
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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