# exponentially-improved expansions

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## 1—10 of 26 matching pages

##### 1: 6.12 Asymptotic Expansions
For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha=0$. … When $\frac{1}{4}\pi\leq|\operatorname{ph}z|<\frac{1}{2}\pi$ the remainders are bounded in magnitude by $\csc\left(2|\operatorname{ph}z|\right)$ 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 $E_{p}\left(z\right)$
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 $E_{p}\left(z\right)$ see §2.11(iii). …
##### 3: 7.12 Asymptotic Expansions
For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha=\frac{1}{2}$ and $z$ replaced by $z^{2}$; compare (7.11.2). … They are bounded by $|\csc\left(4\operatorname{ph}z\right)|$ times the first neglected terms when $\frac{1}{8}\pi\leq|\operatorname{ph}z|<\frac{1}{4}\pi$. … 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-ImprovedExpansions
For this reason the expansion of $E_{p}\left(z\right)$ in $|\operatorname{ph}z|\leq\pi-\delta$ supplied by (2.11.8), (2.11.10), and (2.11.13) is said to be exponentially improved. …
###### §2.11(v) Exponentially-ImprovedExpansions (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. …
##### 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 $\rho>0$, and Wong and Zhao (1999a) when $-1<\rho<0$. … This reference includes exponentially-improved asymptotic expansions for $E_{a,b}\left(z\right)$ when $|z|\to\infty$, 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). …
##### 10: 8.11 Asymptotic Approximations and Expansions
where $\delta$ 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). …