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21—30 of 203 matching pages

21: 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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22: 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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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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Zhang and Jin (1996, p. 323) tabulates the first $20$ real zeros of $\operatorname{ber}x$ , $\operatorname{ber}'x$ , $\operatorname{bei}x$ , $\operatorname{bei}'x$ , $\operatorname{ker}x$ , $\operatorname{ker}'x$ , $\operatorname{kei}x$ , $\operatorname{kei}'x$ , 8D.

23: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

24: Bibliography G

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GAP (website) The GAP Group, Centre for Interdisciplinary Research in Computational Algebra, University of St. Andrews, United Kingdom.

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Notes:: Groups, Algorithms, Programming: A system for computational discrete algebra with emphasis on computational group theory.
External Links:: GAP
Referenced by:: 2nd item.
Encodings:: BibTeX

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W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) 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).
Encodings:: BibTeX

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G. H. Golub and C. F. Van Loan (1996) Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD.

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External Links:: ISBN 0-8018-5413-X; 0-8018-5414-8, MathReview, ZentralBlatt
Referenced by:: §3.2(i), §3.2(i), §3.2(ii), §3.2(vii).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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25: Bibliography R

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H. E. Rauch and A. Lebowitz (1973) Elliptic Functions, Theta Functions, and Riemann Surfaces. The Williams & Wilkins Co., Baltimore, MD.

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External Links:: MathReview (H. H. Martens), ZentralBlatt
Referenced by:: §20.11(iv), §20.12(ii).
Encodings:: BibTeX

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J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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Referenced by:: 3rd item.
Encodings:: BibTeX

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W. P. Reinhardt (2018) Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:: arXiv:1812.02196
Referenced by:: §18.40(ii).
Encodings:: BibTeX

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J. Riordan (1958) An Introduction to Combinatorial Analysis. John Wiley & Sons Inc., New York.

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Notes:: Reprinted by Princeton University Press, Princeton, NJ, 1980 and reissued by Dover Publications Inc., New York, 2002.
External Links:: MathReview (L. Moser), ZentralBlatt
Referenced by:: §1.4(iii), §26.14(ii), §26.18, §26.3(ii), §26.3(iii), Ch.26.
Encodings:: BibTeX

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26: Brian R. Judd

… … ► Dieke Professor Emeritus in the Department of Physics and Astronomy at The Johns Hopkins University, Baltimore. ►Judd’s books include Operator Techniques in Atomic Spectroscopy, published by McGraw-Hill in 1963 and reprinted by Princeton University Press in 1998, Second Quantization and Atomic Spectroscopy, published by Johns Hopkins in 1967, Topics in Atomic and Nuclear Theory (with J. … ►Judd is an Honorary Fellow of Brasenose College, Oxford University, and a Fellow of the American Physical Society. …

27: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$ , with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\operatorname{Ei}\left(x\right)$ , with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$ , with accuracies up to 20S.

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28: 13.30 Tables

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Zhang and Jin (1996, pp. 411–423) tabulates $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ for $a=-5(.5)5$ , $b=0.5(.5)5$ , and $x=0.1,1,5,10,20,30$ , 8S (for $M\left(a,b,x\right)$ ) and 7S (for $U\left(a,b,x\right)$ ).

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29: 28.16 Asymptotic Expansions for Large $q$

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28.16.1

\lambda_{\nu}\left(h^{2}\right)\sim-2h^{2}+2sh-\dfrac{1}{8}(s^{2}+1)-\dfrac{1}% {2^{7}h}(s^{3}+3s)-\dfrac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\dfrac{1}{2^{17}h^% {3}}(33s^{5}+410s^{3}+405s)-\dfrac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}% +486)-\dfrac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

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Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $h$ : parameter and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.16 and Ch.28

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30: 7.24 Approximations

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Cody (1969) provides minimax rational approximations for $\operatorname{erf}x$ and $\operatorname{erfc}x$ . The maximum relative precision is about 20S.

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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