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21—30 of 252 matching pages

21: Bibliography M

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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

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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.

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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

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B. M. McCoy (1992) Spin Systems, Statistical Mechanics and Painlevé Functions. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 377–391.

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Notes:: Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990
External Links:: MathReview, ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

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

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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

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

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

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22: 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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S. W. Weinberg (2013) Lectures on Quantum Mechanics. Cambridge University Press, Cambridge, UK.

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External Links:: ISBN 978-1-107-02872-2
Referenced by:: §18.39(ii).
Encodings:: BibTeX

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E. P. Wigner (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Notes:: Expanded and improved ed. Translated from the German by J. J. Griffin.
External Links:: ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt
Referenced by:: §34.12.
Encodings:: BibTeX

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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.

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

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

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

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23: Bibliography N

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

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

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J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §13.28(i).
Encodings:: BibTeX

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

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

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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.

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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

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24: 26.20 Physical Applications

… ►The latter reference also describes chemical applications of other combinatorial techniques. ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). For an application of statistical mechanics to combinatorics, see Bressoud (1999). ►Other applications to problems in engineering, crystallography, biology, and computer science can be found in Beckenbach (1981) and Graham et al. (1995).

25: 10.75 Tables

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•

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 $Y_{n}\left(x\right)$ ).

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•

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.

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•

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).

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26: 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. …

27: Bibliography G

… ►

W. Gautschi (1964a) Algorithm 222: Incomplete beta function ratios. Comm. ACM 7 (3), pp. 143–144.

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Notes:: Variable-precision Algol program. For a Certification see same journal 7 (1964), p. 244.
External Links:: ISSN 0001-0782, Document, Certification
Referenced by:: §8.28(iv).
Encodings:: BibTeX

►

W. Gautschi (1964b) Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

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Notes:: For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(iii).
Encodings:: BibTeX

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W. Gautschi (1969) Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:: Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iii).
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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28: Bibliography S

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K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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K. Schulten and R. G. Gordon (1975a) Exact recursive evaluation of

3j

- and

6j

-coefficients for quantum-mechanical coupling of angular momenta. J. Mathematical Phys. 16 (10), pp. 1961–1970.

ⓘ

External Links:: Document, MathReview (J.-L. Calais)
Referenced by:: §34.13, §34.3(iii).
Encodings:: BibTeX

►

K. Schulten and R. G. Gordon (1975b) Semiclassical approximations to

3j

- and

6j

-coefficients for quantum-mechanical coupling of angular momenta. J. Mathematical Phys. 16 (10), pp. 1971–1988.

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External Links:: Document, MathReview (J.-L. Calais)
Referenced by:: §34.8.
Encodings:: BibTeX

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T. C. Scott, R. Mann, and R. E. Martinez (2006) 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.

ⓘ

External Links:: ISSN 0938-1279, Document, FullText, MathReview (E. Capelas de Oliveira), ZentralBlatt
Referenced by:: §4.13, §4.13.
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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29: Bibliography

… ►

M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

ⓘ

External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

… ►

S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

… ►

J. V. Armitage (1989) The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System. In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.), London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §25.17.
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.

…