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11: 5.11 Asymptotic Expansions

… ►uniformly for bounded real values of $x$. ►

§5.11(ii) Error Bounds and Exponential Improvement

… ►For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b). … ►For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990). … ►For realistic error bounds in (5.11.14) see Frenzen (1987a, 1992). …

12: Bibliography D

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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).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

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ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

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

Referenced by:

5th item, §14.32.

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13: Bibliography P

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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.12.

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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ISSN 1065-2469, Document, MathReview, ZentralBlatt

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§10.37.

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

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6th item.

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Algorithm 680: Evaluation of the complex error function. ACM Trans. Math. Software 16 (1), pp. 47.

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Notes:

Single-precision Fortran, maximum accuracy 14S.

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ISSN 0098-3500, Document, GAMS (module 680; package TOMS), MathReview, ZentralBlatt

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1st item.

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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.

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Document, MathReview, ZentralBlatt

Referenced by:

§3.11(ii).

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14: 18.40 Methods of Computation

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►The bottom and top of the steps at the $x_i$ are lower and upper bounds to ${\int }_a^{x_i}d\mu (x)$ as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). … ►

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Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points $x_{i,N}$, $i=1,2,\mathrm{\dots },N$ for $N=40$. … Magnify

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15: Bibliography C

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The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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External Links:

MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§7.17(ii).

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Some series and bounds for incomplete elliptic integrals. J. Math. and Phys. 40, pp. 125–134.

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External Links:

MathReview (L. Robin), ZentralBlatt

Referenced by:

§19.15, (19.5.1), (19.5.2), (19.5.4), (19.5.4_1), (19.5.4_2), (19.5.4_3), §19.9(ii).

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Bound states in one and two spatial dimensions. J. Math. Phys. 44 (2), pp. 406–422.

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ISSN 0022-2488, Document, Link, MathReview (Pavel B. Kurasov)

Referenced by:

§1.18(viii).

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.

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FullText, MathReview, ZentralBlatt

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2nd item.

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16: Bibliography L

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

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

Referenced by:

§10.14, §10.15.

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

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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.

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ISSN 0010-4655, Document, Code, ZentralBlatt

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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).

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