# applications to asymptotic expansions

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## 1—10 of 52 matching pages

##### 1: 2.11 Remainder Terms; Stokes Phenomenon

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►For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997).
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►shows that this direct estimate is correct to almost 3D.
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##### 2: 3.9 Acceleration of Convergence

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►For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).

##### 3: 12.18 Methods of Computation

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►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs.
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions.
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##### 4: Bibliography W

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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules.
Ann. Inst. Statist. Math. 45 (3), pp. 467–475.
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##### 5: Bibliography T

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions.
SIAM J. Math. Anal. 21 (1), pp. 241–261.
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##### 6: 18.40 Methods of Computation

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►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree.
►However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev.
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##### 7: 9.15 Mathematical Applications

###### §9.15 Mathematical Applications

►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. For descriptions of, and references to, the underlying theory see §§2.4(v) and 2.8(iii).##### 8: 2.6 Distributional Methods

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►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi).
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##### 9: 29.20 Methods of Computation

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►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8.
Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i).
Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6.
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►The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to \mathrm{\infty}$.
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►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree.
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##### 10: 10.74 Methods of Computation

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►Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection.
►For large positive real values of $\nu $ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used.
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►In the interval $$, ${J}_{\nu}\left(x\right)$ needs to be integrated in the forward direction and ${Y}_{\nu}\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)).
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►For applications of the continued-fraction expansions (10.10.1), (10.10.2), (10.33.1), and (10.33.2) to the computation of Bessel functions and modified Bessel functions see Gargantini and Henrici (1967), Amos (1974), Gautschi and Slavik (1978), Tretter and Walster (1980), Thompson and Barnett (1986), and Cuyt et al. (2008).
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►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of $2D$ graphs in $\mathbb{R}$ (e.
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