# exponentially-improved expansions

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## 1—10 of 26 matching pages

##### 1: 6.12 Asymptotic Expansions

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►For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$.
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►When $$ the remainders are bounded in magnitude by $\mathrm{csc}\left(2|\mathrm{ph}z|\right)$ 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 ${E}_{p}\left(z\right)$

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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 ${E}_{p}\left(z\right)$ 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 $\alpha =\frac{1}{2}$ and $z$ replaced by ${z}^{2}$; compare (7.11.2).
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►They are bounded by $|\mathrm{csc}\left(4\mathrm{ph}z\right)|$ 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 ${E}_{p}\left(z\right)$ in $|\mathrm{ph}z|\le \pi -\delta $ 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 $\rho >0$, and Wong and Zhao (1999a) when $$.
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►This reference includes exponentially-improved asymptotic expansions for ${E}_{a,b}\left(z\right)$ when $|z|\to \mathrm{\infty}$, 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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