Debye expansions
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11: 6.12 Asymptotic Expansions
§6.12 Asymptotic Expansions
►§6.12(i) Exponential and Logarithmic Integrals
… ►For the function see §9.7(i). … ►§6.12(ii) Sine and Cosine Integrals
… ► …12: 28.16 Asymptotic Expansions for Large
13: 10.69 Uniform Asymptotic Expansions for Large Order
§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). … ►14: 12.9 Asymptotic Expansions for Large Variable
§12.9 Asymptotic Expansions for Large Variable
►§12.9(i) Poincaré-Type Expansions
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12.9.1
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§12.9(ii) Bounds and Re-Expansions for the Remainder Terms
►Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). …15: 8.25 Methods of Computation
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§8.25(i) Series Expansions
►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. … ►§8.25(iii) Asymptotic Expansions
►DiDonato and Morris (1986) describes an algorithm for computing and for , , and from the uniform expansions in §8.12. …16: 9.15 Mathematical Applications
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►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point.
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17: 12.6 Continued Fraction
18: Edward Neuman
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►Neuman has published several papers on approximations and expansions, special functions, and mathematical inequalities.
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19: 30.10 Series and Integrals
20: 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).