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21: 10.41 Asymptotic Expansions for Large Order

… ►The curve $E_{1}B{E}_2$ in the $z$-plane is the upper boundary of the domain $𝐊$ depicted in Figure 10.20.3 and rotated through an angle $-\frac{1}{2}\pi$. … ►For derivations of the results in this subsection, and also error bounds, see Olver (1997b, pp. 374–378). … ►In the case of (10.41.13) with positive real values of $z$ the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). … ►This is a consequence of the error bounds associated with these expansions. …

22: 9.9 Zeros

… ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). … ►Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of $\mathrm{Bi}$ and ${\mathrm{Bi}}^{\prime }$ in the upper half plane. …

23: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►uniformly for bounded values of $|z|$; also …uniformly for bounded positive values of $x$. For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4). … ►the upper or lower sign being taken according as $x\gtrless 2\mu$. … ►This reference also supplies error bounds and corresponding approximations when $x$, $\kappa$, and $\mu$ are replaced by $\mathrm{i}x$, $\mathrm{i}\kappa$, and $\mathrm{i}\mu$, respectively. …

24: Bibliography D

… ►

Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162 (10), pp. 1793–1804.

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External Links:

ISSN 0021-9045, Document, FullText, MathReview (Francisco Marcellán), ZentralBlatt

Referenced by:

(18.16.12), (18.16.13), §18.16(ii), §18.16(ii), §18.16(iv).

Encodings:

BibTeX

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities. Proc. Roy. Soc. London Ser. A 452, pp. 1351–1367.

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External Links:

ISSN 0962-8444, FullText, MathReview (Alexander Tovbis), ZentralBlatt

Referenced by:

§2.11(iii), §8.20(ii).

Encodings:

BibTeX

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

BibTeX

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25: 25.11 Hurwitz Zeta Function

… ►

25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

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Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $k$: nonnegative integer

Referenced by:

§25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.11.E36

Removal (effective with 1.1.4):

This equation has been removed because it is just (25.15.1) combined with (25.15.3).

Clarification (effective with 1.0.23):

The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

See also:

Annotations for §25.11(x), §25.11 and Ch.25

… ►uniformly with respect to bounded nonnegative values of $\alpha$. …