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11: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

§30.9(ii) Oblate Spheroidal Wave Functions

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§30.9(iii) Other Approximations and Expansions

►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …

12: 2.11 Remainder Terms; Stokes Phenomenon

… ►If we permit the use of nonelementary functions as approximants, then even more powerful re-expansions become available. … ►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). … ►The first of these two references also provides an introduction to the powerful Borel transform theory. … ►The following example, based on Weniger (1996), illustrates their power. … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

13: Bibliography K

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The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.21(xiii).

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BibTeX

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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14: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ► $H\left(x\right)$ being the Heaviside step-function, see (1.16.13). … ►The example chosen is inversion from the ${\alpha }_n,{\beta }_n$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). … ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …