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21: Errata

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Immediately below (27.11.2), the bound ${\theta }_0$ for Dirichlet’s divisor problem (currently still unsolved) has been changed from $\frac{12}{37}$ Kolesnik (1969) to $\frac{131}{416}$ Huxley (2003).

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right)$$

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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The upper and lower bounds given have been replaced with stronger bounds.

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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22: 2.8 Differential Equations with a Parameter

… ►In addition, ${𝒱}_{{𝒬}_j}\left(A_1\right)$ and ${𝒱}_{{𝒬}_j}\left(A_n\right)$ must be bounded on ${𝚫}_j({\alpha }_j)$. ►For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►For results, including error bounds, see Olver (1977c). …

23: Bibliography L

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Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.

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External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.37.

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Bessel functions: Monotonicity and bounds. J. London Math. Soc. (2) 61 (1), pp. 197–215.

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External Links:

ISSN 0024-6107, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.14, §10.15.

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

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Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.

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ISSN 0036-1410, Document, MathReview (Walter Schempp), ZentralBlatt

Referenced by:

§2.6(iii).

Encodings:

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Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.

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External Links:

ISSN 0205-3217, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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24: 30.9 Asymptotic Approximations and Expansions

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$$2^{20}{\beta }_5=-527q^7-61529q^5-\mathrm{10\hspace{0.33em}43961}{q}^3-\mathrm{22\hspace{0.33em}41599}q+32m^2(5739q^5+\mathrm{1\hspace{0.33em}27550}{q}^3+\mathrm{2\hspace{0.33em}98951}q)-2048m^4(355q^3+1505q)+65536m^{6}q.$$

… ►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). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …

25: Bibliography D

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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).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

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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).

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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).

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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.

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26: Bibliography G

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Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.

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Referenced by:

§18.39(iii).

Encodings:

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Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144 (1-2), pp. 7–27.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.16(iv), §18.16(iv), §18.16(vi).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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Error bounds in equilibrium statistical mechanics. J. Math. Phys. 9, pp. 655–663.

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External Links:

ZentralBlatt

Referenced by:

§18.39(iii), §18.40(ii).

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New method for constructing wavefunctions for bound states and scattering. J. Chem. Phys. 51, pp. 14–25.

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External Links:

Document, MathReview (K. J. Le Couteur)

Referenced by:

§9.11(iv), §9.17(iii).

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27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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Bounded and Unbounded Linear Operators

… ►A linear operator $T$ on $V$ is bounded with norm $\Vert T\Vert$ if … ►If $T$ is a bounded linear operator on $V$ then its adjoint is the bounded linear operator $T^{\ast }$ such that, for $v,w\in V$, … ►If $T$ is a bounded operator then its spectrum is a closed bounded subset of $ℂ$. If $T$ is self-adjoint (bounded or unbounded) then $\sigma (T)$ is a closed subset of $ℝ$ and the residual spectrum is empty. …

28: Bibliography B

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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

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A new bound for the smallest $x$ with $\pi (x)>\mathrm{li}(x)$ . Math. Comp. 69 (231), pp. 1285–1296.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§27.12, §6.16(ii).

Encodings:

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Bounds and algorithms for the $K$-Bessel function of imaginary order. LMS J. Comput. Math. 16, pp. 78–108.

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External Links:

ISSN 1461-1570, Document, MathReview (Jürgen Müller), ZentralBlatt

Referenced by:

§10.45, §10.45.

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Error bounds for the method of steepest descents. Proc. Roy. Soc. London Ser. A 440, pp. 493–518.

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External Links:

ISSN 0962-8444, FullText, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§2.3(iii), §2.4(iii), §9.10(ii).

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Criterion for Existence of a Bound State In One Dimension. American Journal of Physics 68 (2), pp. 160–161.

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External Links:

Document

Referenced by:

§1.18(viii).

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29: 2.11 Remainder Terms; Stokes Phenomenon

… ►When a rigorous bound or reliable estimate for the remainder term is unavailable, it is unsafe to judge the accuracy of an asymptotic expansion merely from the numerical rate of decrease of the terms at the point of truncation. …First, it is impossible to bound the tail by majorizing its terms. … ►However, regardless whether we can bound the remainder, the accuracy achievable by direct numerical summation of a divergent asymptotic series is always limited. … ►For error bounds see Dunster (1996c). … ►For example, using double precision $d_{20}$ is found to agree with (2.11.31) to 13D. …

30: 1.8 Fourier Series

… ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►Suppose that $f(x)$ is continuous and of bounded variation on $[0,\mathrm{\infty })$. …