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1: 12.16 Mathematical Applications
2: 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).
3: 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. …
4: 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). …
5: 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). …
6: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
  • F. W. J. Olver (1991a) Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22 (5), pp. 1460–1474.
  • F. W. J. Olver (1991b) Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.
  • F. W. J. Olver (1993a) Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.
  • 7: 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). …
    8: 8.11 Asymptotic Approximations and Expansions
    where δ denotes an arbitrary small positive constant. … For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). … For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c). … For sharp error bounds and an exponentially-improved extension, see Nemes (2016). …
    9: 12.9 Asymptotic Expansions for Large Variable
    §12.9(ii) Bounds and Re-Expansions for the Remainder Terms
    10: 7.20 Mathematical Applications
    The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). …