# improved

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## 1—10 of 56 matching pages

##### 1: 12.16 Mathematical Applications

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##### 2: DLMF Project News

*error generating summary*

##### 3: 17.18 Methods of Computation

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►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9.
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##### 4: 19 Elliptic Integrals

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##### 5: 27.17 Other Applications

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►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

##### 6: 6.12 Asymptotic Expansions

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►For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$.
►For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with $p=1$.
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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).

##### 7: 19.38 Approximations

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►The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi $ near $\pi /2$ with the improvements made in the 1970 reference.
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##### 8: 33.11 Asymptotic Expansions for Large $\rho $

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33.11.4
$${H}_{\mathrm{\ell}}^{\pm}(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta}_{\mathrm{\ell}}}(f(\eta ,\rho )\pm \mathrm{i}g(\eta ,\rho )),$$

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33.11.7
$$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$

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##### 9: 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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##### 10: Bibliography O

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Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one.
Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
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Improved error bounds for second-order differential equations with two turning points.
J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.
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Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral.
SIAM J. Math. Anal. 22 (5), pp. 1460–1474.
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Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms.
SIAM J. Math. Anal. 22 (5), pp. 1475–1489.
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Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function.
SIAM J. Math. Anal. 24 (3), pp. 756–767.
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