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11: 27.12 Asymptotic Formulas: Primes

… ►The best available asymptotic error estimate (2009) appears in Korobov (1958) and Vinogradov (1958): there exists a positive constant $d$ such that … ►If $a$ is relatively prime to the modulus $m$, then there are infinitely many primes congruent to $a(modm)$. …

12: 2.3 Integrals of a Real Variable

… ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by … ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). … ►However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points. … ►For proofs of the results of this subsection, error bounds, and an example, see Olver (1974). For other estimates of the error term see Lyness (1971). …

13: 3.5 Quadrature

… ►When $f\in C^{\mathrm{\infty }}$, the Romberg method affords a means of obtaining high accuracy in many cases with a relatively simple adaptive algorithm. … ►Also, the error constant (3.5.20) is given by … ►Equation (3.5.36), without the error term, becomes … ►

Example

… ►where $\mathrm{erfc}z$ is the complementary error function, and from (7.12.1) it follows that …

14: Bibliography Z

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Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.

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External Links:

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

Referenced by:

4th item, §7.25(iii).

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Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

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15: 3.10 Continued Fractions

… ►For special functions see §5.10 (gamma function), §7.9 (error function), §8.9 (incomplete gamma functions), §8.17(v) (incomplete beta function), §8.19(vii) (generalized exponential integral), §§10.10 and 10.33 (quotients of Bessel functions), §13.6 (quotients of confluent hypergeometric functions), §13.19 (quotients of Whittaker functions), and §15.7 (quotients of hypergeometric functions). … ►To achieve a prescribed accuracy, either a priori knowledge is needed of the value of $n$, or $n$ is determined by trial and error. … ►The recurrences are continued until $(\nabla C_n)/C_n$ is within a prescribed relative precision. …

16: Bibliography S

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On Lindelöf’s error bound for Stirling’s series. J. Reine Angew. Math. 404, pp. 135–139.

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ISSN 0075-4102, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§5.11(ii).

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General relativity and quantum mechanics: towards a generalization of the Lambert $W$ function: a generalization of the Lambert $W$ function. Appl. Algebra Engrg. Comm. Comput. 17 (1), pp. 41–47.

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ISSN 0938-1279, Document, FullText, MathReview (E. Capelas de Oliveira), ZentralBlatt

Referenced by:

§4.13, §4.13.

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On the calculation of the inverse of the error function. Math. Comp. 22 (101), pp. 144–158.

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FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(ii).

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

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On the relative extrema of the Hermite orthogonal functions. J. Indian Math. Soc. (N.S.) 15, pp. 129–134.

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External Links:

MathReview (J. Todd), ZentralBlatt

Referenced by:

§18.14(i).

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BibTeX

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17: 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).

Encodings:

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Calcolo delle funzioni speciali $\mathrm{\Gamma }(x)$, $\log \mathrm{\Gamma }(x)$, $\beta (x,y)$, $\mathrm{erf}(x)$, $\mathrm{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

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Notes:

Translated title: Computation of the special functions $\mathrm{\Gamma }(x),\log \mathrm{\Gamma }(x),\beta (x,y),\mathrm{erf}(x),\mathrm{erfc}(x)$ to a high degree of precision

External Links:

MathReview, ZentralBlatt

Referenced by:

§5.24(ii).

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The Mathematical Theory of Black Holes. In General Relativity and Gravitation (Padova, 1983), pp. 5–26.

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External Links:

MathReview Entry

Referenced by:

§31.17(ii).

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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

Referenced by:

2nd item.

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18: 2.11 Remainder Terms; Stokes Phenomenon

… ►The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … ►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of $\mathrm{ph}z=\pi$, is increasingly significant as $\mathrm{ph}z$ passes from $\pi$ to $\frac{3}{2}\pi$, and becomes the dominant contribution after $\mathrm{ph}z$ passes $\frac{3}{2}\pi$. … ►Here $\mathrm{erfc}$ is the complementary error function (§7.2(i)), and … ►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 error bounds see Dunster (1996c). …