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11—20 of 24 matching pages

11: 10.75 Tables

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Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

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Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

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Olver (1960) tabulates $a^{\prime}_{n,m}$ , $\mathsf{j}_{n}\left(a^{\prime}_{n,m}\right)$ , $b^{\prime}_{n,m}$ , $\mathsf{y}_{n}\left(b^{\prime}_{n,m}\right)$ , $n=1(1)20$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

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Zhang and Jin (1996, p. 322) tabulates $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , $x=0(1)20$ , 7S.

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12: 9.9 Zeros

… ►They are denoted by

a_{k}

a^{\prime}_{k}

b_{k}

b^{\prime}_{k}

, respectively, arranged in ascending order of absolute value for

k=1,2,\ldots.

… ►For large

k

►

9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.7

\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

… ►For error bounds for the asymptotic expansions of

a_{k}

b_{k}

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

13: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. …They tend to thin out among the large integers, but this thinning out is not completely regular. … ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►

Table 27.2.2: Functions related to division.

► ►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
…
7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

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14: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►

§12.10(vi) Modifications of Expansions in Elementary Functions

… ► … ►

Modified Expansions

… ►

15: Bibliography W

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.

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Notes:: Translated from the 1989 Japanese original. With 1 IBM-PC floppy disk (5.25 inch) containing a large general purpose mathematical software collection written in Fortran, including some special function codes. Double precision.
External Links:: ISBN 0-444-82000-0, MathReview, ZentralBlatt
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.

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External Links:: ISSN 0036-1399, Document, MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §29.7(ii).
Encodings:: BibTeX

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J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:: Document, ZentralBlatt
Referenced by:: §33.5(ii).
Encodings:: BibTeX

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R. Wong (1973a) An asymptotic expansion of

{W}_{k,\,m}(z)

with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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External Links:: FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt
Referenced by:: §13.19.
Encodings:: BibTeX

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16: Bibliography L

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R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:: ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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J. L. López and P. J. Pagola (2010) The confluent hypergeometric functions

M(a,b;z)

and

U(a,b;z)

for large

b

and

z

. J. Comput. Appl. Math. 233 (6), pp. 1570–1576.

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External Links:: ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §13.8(ii), §13.8(ii).
Encodings:: BibTeX

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J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:: Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II
External Links:: ISSN 1073-2772, Pdf, MathReview, ZentralBlatt
Referenced by:: §13.21(i), §13.24(ii).
Encodings:: BibTeX

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J. L. López and N. M. Temme (2010b) Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §24.16(i), §24.16(i).
Encodings:: BibTeX

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17: 19.36 Methods of Computation

… ►Descending Gauss transformations of

\Pi\left(\phi,\alpha^{2},k\right)

(see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function

\operatorname{cel}\left(k_{c},p,a,b\right)

is computed by successive Bartky transformations (Bulirsch and Stoer (1968), Bulirsch (1969b)). The function

\operatorname{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). … ►Bulirsch (1969a, b) extend Bartky’s transformation to

\operatorname{el3}\left(x,k_{c},p\right)

by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

18: Bibliography K

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) 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:: BibTeX

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M. K. Kerimov (1980) 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:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) 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).
Encodings:: BibTeX

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U. J. Knottnerus (1960) Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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External Links:: MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

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T. H. Koornwinder (2009) 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).
Encodings:: BibTeX

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19: 3.4 Differentiation

… ►The

B_{k}^{n}

are the differentiated Lagrangian interpolation coefficients: … ►

B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

… ►

B_{-3}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),

►

B_{-2}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),

… ►With the choice

r=k

(which is crucial when

k

is large because of numerical cancellation) the integrand equals

e^{k}

at the dominant points

\theta=0,2\pi

, and in combination with the factor

k^{-k}

in front of the integral sign this gives a rough approximation to

1/k!

. …

20: 9.18 Tables

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Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

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