complex physical systems

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1: 8.23 Statistical Applications

§8.23 Statistical Applications

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2: 8.24 Physical Applications

… ►The function

\gamma\left(a,x\right)

appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …

3: Bibliography M

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I. G. Macdonald (1982) Some conjectures for root systems. SIAM J. Math. Anal. 13 (6), pp. 988–1007.

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External Links:: ISSN 0036-1410, Document, MathReview (S. Milne), ZentralBlatt
Referenced by:: §17.13, §17.14.
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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R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

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N. Michel (2007) Precise Coulomb wave functions for a wide range of complex

\ell

\eta

and

z

. Computer Physics Communications 176 (3), pp. 232–249.

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Notes:: C++ code. Maximum accuracy 10D
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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C. Micu and E. Papp (2005) Applying

q

-Laguerre polynomials to the derivation of

q

-deformed energies of oscillator and Coulomb systems. Romanian Reports in Physics 57 (1), pp. 25–34.

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Referenced by:: §17.17.
Encodings:: BibTeX

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4: 14.31 Other Applications

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§14.31(ii) Conical Functions

►The conical functions

\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)

appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). … ►

§14.31(iii) Miscellaneous

►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

5: 12.17 Physical Applications

§12.17 Physical Applications

►The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation … ►

12.17.2

\nabla^{2}=\frac{{\partial}^{2}}{{\partial x}^{2}}+\frac{{\partial}^{2}}{{% \partial y}^{2}}+\frac{{\partial}^{2}}{{\partial z}^{2}}

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable, $y$ : real variable and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/12.17.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §12.17 and Ch.12

… ► … ►Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. …

6: 31.17 Physical Applications

§31.17 Physical Applications

… ►Then … ►

§31.17(ii) Other Applications

►Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). … ►More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …

7: Bibliography T

… ►

N. M. Temme (1996b) Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley & Sons Inc., New York.

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External Links:: ISBN 0-471-11313-1, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.13(iv), §10.23(iii), Ch.12, §13.14(ii), §13.19, §13.2(i), §13.2(iii), §13.2(vii), §13.6(i), §13.6(ii), §13.6(iii), §13.6(iv), §13.7(i), Ch.13, §14.31(iii), Ch.14, §15.2(ii), §15.4(i), Ch.15, §24.17(i), §24.9, §3.5(i), §5.11(iii), §5.12, §5.19(i), §5.2(i), §5.2(ii), §5.5(i), §5.5(iii), §5.7(ii), (5.9.2_5), §5.9(i), §5.9(ii), Ch.5, §6.11, §6.16(i), §6.7(iii), Ch.6, §7.2(i), §7.2(ii), §7.2(iii), §7.20(ii), Ch.7, §8.11(i), §8.17(ii), §8.17(iii), §8.17(iv), §8.17(v), §8.18(ii), §8.18(ii), §8.18(ii), §8.19(i), §8.25(iv), §8.6(i), §8.6(iii), Ch.8, Ch.9, Profile Nico M. Temme.
Encodings:: BibTeX

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I. J. Thompson and A. R. Barnett (1985) COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:: Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item, §33.23(vi), 3rd item.
Encodings:: BibTeX

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I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:: Document, Code, ZentralBlatt
Referenced by:: §33.23(vi), I. J. Thompson and A. R. Barnett (1985).
Encodings:: BibTeX

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W. J. Thompson (1994) Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:: With 1 Macintosh floppy disk (3.5 inch; DD). Programs for the calculation of $3j,6j,9j$ symbols are included.
External Links:: ISBN 0-471-55264-X, MathReview (J. F. Cornwell), ZentralBlatt
Referenced by:: §34.3(vii).
Encodings:: BibTeX

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J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

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External Links:: MathReview (H. Bückner), ZentralBlatt
Referenced by:: §6.18(i).
Encodings:: BibTeX

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8: Bibliography

… ►

J. A. Adam (2002) The mathematical physics of rainbows and glories. Phys. Rep. 356 (4-5), pp. 229–365 (English).

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External Links:: Document, FullText, ZentralBlatt
Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

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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.

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

… ►

N. W. Ashcroft and N. D. Mermin (1976) Solid State Physics. Holt, Rinehart and Winston, New York.

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External Links:: ISBN 0-03-083993-9
Referenced by:: §1.18(vii), §4.44, §8.22(ii).
Encodings:: BibTeX

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M. Audin (1999) Spinning Tops: A Course on Integrable Systems. Cambridge Studies in Advanced Mathematics, Vol. 51, Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-56129-9; 0-521-77919-7, MathReview (Emma Previato), ZentralBlatt
Referenced by:: §22.19(iv).
Encodings:: BibTeX

… ►

M. Aymar, C. H. Greene, and E. Luc-Koenig (1996) Multichannel Rydberg spectroscopy of complex atoms. Reviews of Modern Physics 68, pp. 1015–1123.

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External Links:: Document
Referenced by:: §33.22(ii).
Encodings:: BibTeX

9: 18.38 Mathematical Applications

… ►

Integrable Systems

►The Toda equation provides an important model of a completely integrable system. …While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. …Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. …

10: 23.21 Physical Applications

§23.21 Physical Applications

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§23.21(iii) Ellipsoidal Coordinates

… ►

23.21.1

\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,

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Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $z$ : complex and $\rho$ : roots
Permalink:: http://dlmf.nist.gov/23.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

… ►Physical applications of modular functions include: … ►

•

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