DLMF: Untitled Document

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B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

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Notes:: Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.
External Links:: ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt
Referenced by:: §19.29(ii), §19.36(i), §19.36(i), §19.39(iv).
Encodings:: BibTeX

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B. C. Carlson and J. L. Gustafson (1994) Asymptotic approximations for symmetric elliptic integrals. SIAM J. Math. Anal. 25 (2), pp. 288–303.

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External Links:: ISSN 0036-1410, Document, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.12, §19.27(ii), §19.27(iii), §19.27(iv), §19.27(v), §19.27(vi).
Encodings:: BibTeX

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B. C. Carlson and G. S. Rushbrooke (1950) On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §34.12.
Encodings:: BibTeX

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B. C. Carlson (1970) Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:: FullText, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §19.24(ii), §19.9(ii).
Encodings:: BibTeX

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H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:: Reprinted by Dover, New York, 1950.
External Links:: ZentralBlatt
Referenced by:: §6.16(i).
Encodings:: BibTeX

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I. G. Macdonald (1995) Symmetric Functions and Hall Polynomials. 2nd edition, The Clarendon Press, Oxford University Press, New York-Oxford.

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Notes:: With contributions by A. Zelevinsky. Paperback version published by Oxford University Press, July 1999.
External Links:: ISBN 0-19-853489-2; 0-19-850450-0, MathReview (John R. Stembridge), ZentralBlatt
Referenced by:: §18.37(iii), §26.19, §35.4(i), §35.4(ii).
Encodings:: BibTeX

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I. G. Macdonald (1998) Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-0770-6, MathReview (Angèle M. Hamel), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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I. D. Macdonald (1968) The Theory of Groups. Clarendon Press, Oxford.

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Notes:: Reprinted in 1988 by Krieger Publishing Co., Malabar, FL
External Links:: ISBN 0-19-853137-0;0-89464-287-1, MathReview (A. P. Street), ZentralBlatt
Referenced by:: §23.20(ii).
Encodings:: BibTeX

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N. W. Macfadyen and P. Winternitz (1971) Crossing symmetric expansions of physical scattering amplitudes: The

O(2,1)

group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.

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External Links:: Document, MathReview
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.

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External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §15.19(i), §15.19(i).
Encodings:: BibTeX

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… ►Bronski et al. (2001) uses Lamé functions in the theory of Bose–Einstein condensates. … ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …

… ►Suppose the potential energy of a gas of

n

point charges with positions

x_{1},x_{2},\dots,x_{n}

and free to move on the infinite line

-\infty<x<\infty

, is given by ►

5.20.1

W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln|x_{\ell% }-x_{j}|.

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $j$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $W$ : potential energy
Permalink:: http://dlmf.nist.gov/5.20.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

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5.20.2

P(x_{1},\dots,x_{n})=C\exp\left(-W/(kT)\right),

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Symbols:: $\exp\NVar{z}$ : exponential function, $n$ : nonnegative integer, $x$ : real variable, $W$ : potential energy, $T$ : temperature and $C$ : constant
Permalink:: http://dlmf.nist.gov/5.20.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

… ►

Elementary Particles

►Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. …

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL

(2,\mathbb{R})

, and spherical functions on certain nonsymmetric Gelfand pairs. … ►

§15.17(v) Monodromy Groups

…

… ►

Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of

J_{0}(z)-iJ_{1}(z)

and of Bessel functions

J_{m}(z)

of any real order

m

. Linear Algebra Appl. 194, pp. 35–70.

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External Links:: ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

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K. Ireland and M. Rosen (1990) A Classical Introduction to Modern Number Theory. 2nd edition, Springer-Verlag, New York.

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External Links:: ISBN 0-387-97329-X, MathReview (Glenn Stevens), ZentralBlatt
Referenced by:: §24.10(i), §24.10(ii), §24.10(iii), §24.17(iii).
Encodings:: BibTeX

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A. R. Its and V. Yu. Novokshënov (1986) The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-16483-9, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iv), §32.4(i), Profile Alexander A. Its.
Encodings:: BibTeX

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C. Itzykson and J. Drouffe (1989) Statistical Field Theory: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. Vol. 2, Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-37012-4; 0-521-40806-7, MathReview (C. A. Hurst), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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C. Itzykson and J. B. Zuber (1980) Quantum Field Theory. International Series in Pure and Applied Physics, McGraw-Hill International Book Co., New York.

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Notes:: Reprinted by Dover, 2006.
External Links:: ISBN 0-07-032071-3, MathReview (V. E. Korepin)
Referenced by:: §22.19(ii).
Encodings:: BibTeX

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D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:: MathReview (J. Browkin), ZentralBlatt
Referenced by:: §25.12(i).
Encodings:: BibTeX

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A. H. Zemanian (1987) Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

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Notes:: Reprint of 1965 McGraw-Hill Publication
External Links:: ISBN 0-486-65479-6, MathReview Entry
Referenced by:: §1.16(ix).
Encodings:: BibTeX

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A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.

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External Links:: ISSN 0021-9045, Link, MathReview (T. S. Chihara), ZentralBlatt
Referenced by:: §18.33(iii).
Encodings:: BibTeX

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D. G. Zill and B. C. Carlson (1970) Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.

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External Links:: FullText, Document, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §19.21(i), §19.21(iii), §19.22(iii), §19.25(i), §19.26(i).
Encodings:: BibTeX

…

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K. Yang and M. de Llano (1989) Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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External Links:: Document
Referenced by:: §1.18(viii).
Encodings:: BibTeX

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A. P. Yutsis, I. B. Levinson, and V. V. Vanagas (1962) Mathematical Apparatus of the Theory of Angular Momentum. Israel Program for Scientific Translations for National Science Foundation and the National Aeronautics and Space Administration, Jerusalem.

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Notes:: Translated from the Russian by A. Sen and R. N. Sen
External Links:: MathReview (R. F. Stora), ZentralBlatt
Referenced by:: §34.11.
Encodings:: BibTeX

§26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). ►Other areas of combinatorial analysis include graph theory, coding theory, and combinatorial designs. These have applications in operations research, probability theory, and statistics. …

§23.21 Physical Applications

… ►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form

(1-x^{2})(1-k^{2}x^{2})

. The Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

g_{3}+g_{2}x-4x^{3}

. … ►

•

Quantum field theory. See Witten (1987).

… ►

•

String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

axially symmetric potential theory

21—30 of 204 matching pages

21: Bibliography C

22: Bibliography M

23: 29.19 Physical Applications

24: 5.20 Physical Applications

Elementary Particles

25: 15.17 Mathematical Applications

§15.17(v) Monodromy Groups

26: Bibliography I

27: Bibliography Z

28: Bibliography Y

29: 26.19 Mathematical Applications

§26.19 Mathematical Applications

30: 23.21 Physical Applications

§23.21 Physical Applications