Hamiltonian%20structure

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11: Bibliography P

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Optical properties of layer antiferromagnets with ${\mathrm{K}}_2{\mathrm{NiF}}_4$ structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

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External Links:

Document

Referenced by:

§19.35(ii).

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The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. Philos. Mag. (7) 37, pp. 311–317.

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

§36.2(ii).

Encodings:

BibTeX

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

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12: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►

§33.22(iv) Klein–Gordon and Dirac Equations

… ►The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. …

13: 3.8 Nonlinear Equations

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3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

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Permalink:

http://dlmf.nist.gov/3.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. … ►

3.8.16 $$\frac{dx}{d{a}_{19}}=-\frac{{20}^{19}}{19!}=(-4.30\mathrm{\dots })\times {10}^7.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/3.8.E16

Encodings:

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See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. …

14: 26.20 Physical Applications

… ►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). …

15: 26.22 Software

… ►For algorithms for counting and analyzing combinatorial structures see Knuth (1993), Nijenhuis and Wilf (1975), and Stanton and White (1986).

16: Alexander I. Bobenko

… ► Eitner), published by Springer in 2000, and Discrete Differential Geometry: Integrable Structure (with Y. …

17: 36.13 Kelvin’s Ship-Wave Pattern

… ►The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency $\omega$ as a function of wavevector $𝐤$: …

18: 8 Incomplete Gamma and Related
Functions

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19: 28 Mathieu Functions and Hill’s Equation

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20: Bibliography R

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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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Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics. Annual Review of Physical Chemistry 33, pp. 223–255.

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Referenced by:

§1.18(viii).

Encodings:

BibTeX

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