as eigenfunctions of a q-difference operator

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31: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For

Z_{1}Z_{2}=-1

and

m=m_{e}

, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius,

a_{0}=\hbar/(m_{e}c\alpha)

, and to a multiple of the Rydberg constant, … ►The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). 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. … ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions,

F_{\ell}\left(\eta,\rho\right)

and

G_{\ell}\left(\eta,\rho\right)

, or

f\left(\epsilon,\ell;r\right)

and

h\left(\epsilon,\ell;r\right)

, to determine the scattering

S

-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). … ►

•

Eigenstates using complex-rotated coordinates $r\to re^{\mathrm{i}\theta}$ , so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

…

32: 13.27 Mathematical Applications

… ►Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …

33: 34.9 Graphical Method

… ►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation. …

34: 26.22 Software

… ►Also included are websites operated by university departments and consortia, research institutions, and peer-reviewed journals. … ►A more complete list of available software for computing these functions is found in the Software Index. … ►

•

GAP (website). A system for computational discrete algebra.

… ►

•

Magma (website). A computational algebra system.

…

35: 1.1 Special Notation

… ► ►►►►►►

$x, y$	real variables.
…
$\phi$	a testing function.
…
$\det(\mathbf{A})$	determinant of the square matrix $\mathbf{A}$
$\operatorname{tr}(\mathbf{A})$	trace of the square matrix $\mathbf{A}$
…
$\mathcal{L}$	linear operator defined on a manifold $\mathcal{M}$
…

►In the physics, applied maths, and engineering literature a common alternative to

\overline{a}

a^{*}

a

being a complex number or a matrix; the Hermitian conjugate of

\mathbf{A}

is usually being denoted

\mathbf{A}^{{\dagger}}

36: About MathML

… ►The DLMF uses MathML Core, a subset designed specifically for browsers. … ►Of course you are encouraged to use a modern, up-to-date browser. As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. … ►For other browsers, you may see a ? or a box like

indicating missing symbols, and thus insufficient fonts. …

37: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

§1.3(iv) Matrices as Linear Operators

►

Linear Operators in Finite Dimensional Vector Spaces

►Square matices can be seen as linear operators because

\mathbf{A}(\alpha\mathbf{a}+\beta\mathbf{b})=\alpha\mathbf{A}\mathbf{a}+\beta% \mathbf{A}\mathbf{b}

for all

\alpha,\beta\in\mathbb{C}

and

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

, the space of all

n

-dimensional vectors. … ►Real symmetric (

\mathbf{A}=\mathbf{A}^{\mathrm{T}}

) and Hermitian (

\mathbf{A}={\mathbf{A}}^{{\rm H}}

) matrices are self-adjoint operators on

\mathbf{E}_{n}

. …

38: Guide to Searching the DLMF

… ►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: ►

term:

a textual word, a number, or a math symbol.

… ►

Boolean operator:

and and or

►

proximity operator:

adj, prec/n, and near/n, where n is any positive natural number.

… ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. …

39: William P. Reinhardt

… ►He is both an Associate Editor for Chemistry and a Senior Associate Editor for the DLMF. … ►Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. …Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions. … ►This is closely connected with his interests in classical dynamical “chaos,” an area where he coauthored a book, Chaos in atomic physics with Reinhold Blümel. … ►He has been a National Lecturer for Sigma Xi and Phi Beta Kappa, as well as a Sloan, Dreyfus, and Guggenheim Fellow, and Fulbright Senior Scholar (Australia). …

40: 16.8 Differential Equations

… ►is a value

z_{0}

z

at which all the coefficients

f_{j}(z)

j=0,1,\dots,n-1

, are analytic. … ►the function

w={{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)

satisfies the differential equation …Equation (16.8.4) has a regular singularity at

z=0

, and an irregular singularity at

z=\infty

, whereas (16.8.5) has regular singularities at

z=0

1

, and

\infty

. … ►where

*

indicates that the entry

1-a_{j}+a_{j}

is omitted. … ►When

p=q+1

and some of the

a_{j}

differ by an integer a limiting process can again be applied. …