asymptotic expansions for large variable
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21: Bibliography F
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Uniform asymptotic expansions for hypergeometric functions with large parameters IV.
Anal. Appl. (Singap.) 12 (6), pp. 667–710.
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The third Appell function for one large variable.
J. Approx. Theory 165, pp. 60–69.
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Uniform asymptotic expansions of certain classes of Meijer -functions for a large parameter.
SIAM J. Math. Anal. 4 (3), pp. 482–507.
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Uniform asymptotic expansions of a class of Meijer -functions for a large parameter.
SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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22: 13.19 Asymptotic Expansions for Large Argument
§13.19 Asymptotic Expansions for Large Argument
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13.19.3
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►Error bounds and exponentially-improved expansions are derivable by combining §§13.7(ii) and 13.7(iii) with (13.14.2) and (13.14.3).
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►For an asymptotic expansion of as that is valid in the sector and where the real parameters , are subject to the growth conditions , , see Wong (1973a).
23: 33.9 Expansions in Series of Bessel Functions
§33.9 Expansions in Series of Bessel Functions
►§33.9(i) Spherical Bessel Functions
… ►§33.9(ii) Bessel Functions and Modified Bessel Functions
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33.9.7
►For other asymptotic expansions of see Fröberg (1955, §8) and Humblet (1985).
24: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
►§11.6(i) Large , Fixed
… ►§11.6(ii) Large , Fixed
… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …25: 11.13 Methods of Computation
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§11.13(ii) Series Expansions
►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 . …26: 10.70 Zeros
§10.70 Zeros
►Asymptotic approximations for large zeros are as follows. …If is a large positive integer, then ►
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27: 13.7 Asymptotic Expansions for Large Argument
§13.7 Asymptotic Expansions for Large Argument
►§13.7(i) Poincaré-Type Expansions
… ►§13.7(ii) Error Bounds
… ►§13.7(iii) Exponentially-Improved Expansion
… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).28: 11.9 Lommel Functions
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►and
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§11.9(ii) Expansions in Series of Bessel Functions
… ►§11.9(iii) Asymptotic Expansion
►For fixed and , … ►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). …29: 2.11 Remainder Terms; Stokes Phenomenon
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