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exponentially-improved expansions

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1: 6.12 Asymptotic Expansions
For these and other error bounds see Olver (1997b, pp. 109–112) with α = 0 . … When 1 4 π | ph z | < 1 2 π the remainders are bounded in magnitude by csc ( 2 | ph z | ) times the first neglected terms. … For exponentially-improved asymptotic expansions, use (6.5.5), (6.5.6), and §6.12(i).
2: 8.20 Asymptotic Expansions of E p ( z )
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 E p ( z ) see §2.11(iii). …
3: 7.12 Asymptotic Expansions
For these and other error bounds see Olver (1997b, pp. 109–112), with α = 1 2 and z replaced by z 2 ; compare (7.11.2). … They are bounded by | csc ( 4 ph z ) | times the first neglected terms when 1 8 π | ph z | < 1 4 π . … For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i). …
4: 2.11 Remainder Terms; Stokes Phenomenon
§2.11(iii) Exponentially-Improved Expansions
For this reason the expansion of E p ( z ) in | ph z | π - δ 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
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. …
6: 12.16 Mathematical Applications
7: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
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 ρ > 0 , and Wong and Zhao (1999a) when - 1 < ρ < 0 . … This reference includes exponentially-improved asymptotic expansions for E a , b ( z ) when | z | , together with a smooth interpretation of Stokes phenomena. …
8: 13.19 Asymptotic Expansions for Large Argument
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). …
9: 13.7 Asymptotic Expansions for Large Argument
§13.7(iii) Exponentially-Improved Expansion
10: 8.11 Asymptotic Approximations and Expansions
where δ denotes an arbitrary small positive constant. … For an exponentially-improved asymptotic expansion2.11(iii)) see Olver (1991a). … For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c). …