expansions in doubly-infinite partial fractions

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21: 10.74 Methods of Computation

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§10.74(i) Series Expansions

… ►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. … ► … ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … ►

Fourier–Bessel Expansion

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22: 10.69 Uniform Asymptotic Expansions for Large Order

§10.69 Uniform Asymptotic Expansions for Large Order

… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). … ►

23: Edward Neuman

… ► 1943 in Rydultowy, Poland) is Professor in the Department of Mathematics, Southern Illinois University at Carbondale. ►Neuman has published several papers on approximations and expansions, special functions, and mathematical inequalities. ►Neuman served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions.

24: 2.1 Definitions and Elementary Properties

… ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. ►Most operations on asymptotic expansions can be carried out in exactly the same manner as for convergent power series. … ►Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. … ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. … ►Care is needed in understanding and manipulating generalized asymptotic expansions. …

25: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►Then … ►In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null. … ►The final expansion then has the form … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …

26: 6.12 Asymptotic Expansions

§6.12 Asymptotic Expansions

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§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

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27: 10.41 Asymptotic Expansions for Large Order

… ►The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ , with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real $z$-axis. … ►For expansions in inverse factorial series see Dunster et al. (1993). … ►Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\mathrm{\ell }-1)}$. … ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …

28: 2.11 Remainder Terms; Stokes Phenomenon

… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … ►In this way we arrive at hyperasymptotic expansions. … ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. …

29: 15.19 Methods of Computation

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§15.19(i) Maclaurin Expansions

… ►Gauss quadrature approximations are discussed in Gautschi (2002b). … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). … ►

§15.19(v) Continued Fractions

►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

30: 30.10 Series and Integrals

… ►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). …For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).