exponentially-improved expansions
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1: 6.12 Asymptotic Expansions
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►For these and other error bounds see Olver (1997b, pp. 109–112) with .
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►When the remainders are bounded in magnitude by times the first neglected terms.
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►For exponentially-improved asymptotic expansions, use (6.5.5), (6.5.6), and §6.12(i).
2: 8.20 Asymptotic Expansions of
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►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).
►For an exponentially-improved asymptotic expansion of see §2.11(iii).
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3: 7.12 Asymptotic Expansions
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►For these and other error bounds see Olver (1997b, pp. 109–112), with and replaced by ; compare (7.11.2).
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►They are bounded by times the first neglected terms when .
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►For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i).
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4: 2.11 Remainder Terms; Stokes Phenomenon
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§2.11(iii) Exponentially-Improved Expansions
… ►For this reason the expansion of in supplied by (2.11.8), (2.11.10), and (2.11.13) is said to be exponentially improved. … ►§2.11(v) Exponentially-Improved Expansions (continued)
… ►For another approach see Paris (2001a, b). …5: 8.22 Mathematical Applications
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►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon.
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6: 12.16 Mathematical Applications
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7: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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►For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when , and Wong and Zhao (1999a) when .
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►This reference includes exponentially-improved asymptotic expansions for when , together with a smooth interpretation of Stokes phenomena.
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8: 13.19 Asymptotic Expansions for Large Argument
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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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9: 13.7 Asymptotic Expansions for Large Argument
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