W. P. Reinhardt (2021a) Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

ⓘ

External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

►

W. P. Reinhardt (2021b) Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

ⓘ

External Links:: ISSN 1521-9615, Document, Link
Referenced by:: §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… ►

J. Rys, M. Dupuis, and H. F. King (1983) Computation of electron repulsion integrals using the Rys quadrature method. J. Comput. Chem. 4 (2), pp. 154–175.

ⓘ

External Links:: Link
Referenced by:: §18.32, §18.39(iii).
Encodings:: BibTeX

23: Bibliography N

… ►

D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

… ►

W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

ⓘ

External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

… ►

E. Neuman (2004) Inequalities involving Bessel functions of the first kind. JIPAM. J. Inequal. Pure Appl. Math. 5 (4), pp. Article 94, 4 pp. (electronic).

ⓘ

External Links:: ISSN 1443-5756, FullText, MathReview, ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

… ►

E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

ⓘ

Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

… ►

H. M. Nussenzveig (1992) Diffraction Effects in Semiclassical Scattering. Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.

ⓘ

Notes:: Also available electronically through Cambridge Books Online, ISBN 9780511599903.
External Links:: ISBN 9780521383189, Document
Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

…

24: 10.75 Tables

… ►

•

Bickley et al. (1952) tabulates $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ or $x^{n}Y_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ ( $.01$ or $.1$ ) $10(.1)25$ , 8D (for $J_{n}\left(x\right)$ ), 8S (for $Y_{n}\left(x\right)$ or $x^{n}Y_{n}\left(x\right)$ ); $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)25$ , 10D (for $J_{n}\left(x\right)$ ), 10S (for $Y_{n}\left(x\right)$ ).

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_{0}\left(x\right)$ to 15D, $J_{1}\left(x\right)$ , $J_{2}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ to 10D, $Y_{2}\left(x\right)$ to 8D, $x=0(.1)17.5$ ; $Y_{n}\left(x\right)-(2/\pi)J_{n}\left(x\right)\ln x$ , $n=0,1$ , $x=0(.1)2$ , 8D; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)20$ , 5D or 5S; $J_{n}\left(x\right)$ , $Y_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 10S; modulus and phase functions $\sqrt{x}M_{n}\left(x\right)$ , $\theta_{n}\left(x\right)-x$ , $n=0,1,2$ , $\ifrac{1}{x}=0(.01)0.1$ , 8D.

►

•

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.

►

•

Zhang and Jin (1996, pp. 185–195) tabulates $J_{n}\left(x\right)$ , $J_{n}'\left(x\right)$ , $Y_{n}\left(x\right)$ , $Y_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1$ , 5, 10, 25, 50, 100, 9S; $J_{n+\alpha}\left(x\right)$ , $J_{n+\alpha}'\left(x\right)$ , $Y_{n+\alpha}\left(x\right)$ , $Y_{n+\alpha}'\left(x\right)$ , $n=0(1)5,10,30,50,100$ , $\alpha=\tfrac{1}{4},\tfrac{1}{3},\tfrac{1}{2},\tfrac{2}{3},\tfrac{3}{4}$ , $x=1,5,10,50$ , 8S; real and imaginary parts of $J_{n+\alpha}\left(z\right)$ , $J_{n+\alpha}'\left(z\right)$ , $Y_{n+\alpha}\left(z\right)$ , $Y_{n+\alpha}'\left(z\right)$ , $n=0(1)15,20(10)50,100$ , $\alpha=0,\tfrac{1}{2}$ , $z=4+2i$ , $20+10i$ , 8S.

… ►

•

Olver (1960) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(\tfrac{1}{2})20\tfrac{1}{2}$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ ; see §10.21(viii), and more fully Olver (1954).

…

25: 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. …

26: 18.39 Applications in the Physical Sciences

… ►In the case of a single electron, charge

-e

and mass

m_{e}

, interacting with a fixed (infinite mass) nucleus of charge

+Ze

at the co-ordinate origin, with the choice of SI units,

V(r)=-Z{e}^{2}/(4\pi\epsilon_{0}r)

. … ►The non-relativistic Schrödinger equation describing a single, bound (negative energy) electron, in an

L^{2}

eigenstate of energy

E

is: … ►,

\hbar=m_{e}={e}^{2}=4\pi\epsilon_{0}=1

, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of

a . u .

to SI units is spelled out. … ►Interactions between electrons, in many electron atoms, breaks this degeneracy as a function of

l

, but

n

still dominates. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

27: Gloria Wiersma

… ►Then she began working with the staff of the Physics Laboratory Office of Electronic Commerce in Scientific and Engineering Data, developing and refining the Laboratory website until her retirement in 2007. …

28: Bibliography M

… ►

I. G. Macdonald (2000) Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

ⓘ

External Links:: ISSN 1286-4889, FullText, MathReview, ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

W. Magnus and S. Winkler (1966) Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

ⓘ

Notes:: Reprinted by Dover Publications, Inc., New York, 1979.
External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.
Encodings:: BibTeX

… ►

N. W. McLachlan (1961) Bessel Functions for Engineers. 2nd edition, Clarendon Press, Oxford.

ⓘ

Notes:: The first edition was published in 1934 by Clarendon Press. The second edition was published originally in 1955.
External Links:: ZentralBlatt
Referenced by:: §10.73(iii).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

…

29: Bibliography W

… ►

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

ⓘ

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

… ►

H. S. Wilf and D. Zeilberger (1992b) Rational function certification of multisum/integral/“

q

” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

ⓘ

External Links:: ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

… ►

J. Wimp (1981) Sequence Transformations and their Applications. Mathematics in Science and Engineering, Vol. 154, Academic Press Inc., New York.

ⓘ

External Links:: ISBN 0-12-757940-0, MathReview (Jan Zítko), ZentralBlatt
Referenced by:: §3.9(vi).
Encodings:: BibTeX

… ►

World Combinatorics Exchange (website)

ⓘ

Notes:: A service of the Electronic Journal of Combinatorics with links to databases, software, and other resources.
External Links:: World Combinatorics Exchange
Referenced by:: 8th item.
Encodings:: BibTeX

…

30: 6.20 Approximations

… ►

•

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

►

•

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

►

•

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.

…