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11: 11.12 Physical Applications

… ►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …

12: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). …

13: 19.38 Approximations

… ►They are valid over parts of the complex

k

and

\phi

planes. …

14: Bonita V. Saunders

… ►As the principal developer of graphics for the DLMF, she has collaborated with other NIST mathematicians, computer scientists, and student interns to produce informative graphs and dynamic interactive visualizations of elementary and higher mathematical functions over both simply and multiply connected domains. …

15: Philip J. Davis

… ►In 1957, Davis took over as Chief, Numerical Analysis Section when John Todd and his wife Olga Taussky-Todd, feeling a strong pull toward teaching and research, left to pursue full-time positions at the California Institute of Technology. … ►NBS mathematician Irene Stegun took over management of the A&S project which was already well on its way, and led the work to publication in 1964. … ►Davis’s comments about our uninspired graphs sparked the research and design of techniques for creating interactive 3D visualizations of function surfaces, which grew in sophistication as our knowledge and the technology for developing 3D graphics on the web advanced over the years. …

16: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

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

. … ►

1.3.20

\mathbf{u}=\sum_{i=1}^{n}c_{i}\mathbf{a}_{i},

c_{i}=\left\langle\mathbf{u},\mathbf{a}_{i}\right\rangle

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $n$ : nonnegative integer
Referenced by:: Erratum (V1.2.0) Section 1.3
Permalink:: http://dlmf.nist.gov/1.3.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §1.3(iv), §1.3(iv), §1.3 and Ch.1

…

17: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With

e

denoting here the elementary charge, the Coulomb potential between two point particles with charges

Z_{1}e,Z_{2}e

and masses

m_{1},m_{2}

separated by a distance

s

V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s

, where

Z_{j}

are atomic numbers,

\varepsilon_{0}

is the electric constant,

\alpha

is the fine structure constant, and

\hbar

is the reduced Planck’s constant. The reduced mass is

m=m_{1}m_{2}/(m_{1}+m_{2})

, and at energy of relative motion

E

with relative orbital angular momentum

\ell\hbar

, the Schrödinger equation for the radial wave function

w(s)

is given by … ►

{\sf k}=(2mE/\hbar^{2})^{1/2},

… ►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, ►

R_{\infty}=m_{e}c{\alpha}^{2}/(2\hbar)

. …

18: 35.2 Laplace Transform

… ►where the integration variable

\mathbf{X}

ranges over the space

{\boldsymbol{\Omega}}

. … ►where the integral is taken over all

\mathbf{Z}=\mathbf{U}+\mathrm{i}\mathbf{V}

such that

\mathbf{U}>\mathbf{X}_{0}

and

\mathbf{V}

ranges over

\boldsymbol{\mathcal{S}}

. …

19: 27.18 Methods of Computation: Primes

… ►The ECPP (Elliptic Curve Primality Proving) algorithm handles primes with over 20,000 digits. …

20: 18.39 Applications in the Physical Sciences

… ►where

x

is a spatial coordinate,

m

the mass of the particle with potential energy

V(x)

\hbar=h/(2\pi)

is the reduced Planck’s constant, and

(a,b)

a finite or infinite interval. ►Here the term

-\frac{\hbar^{2}}{2m}\tfrac{{\partial}^{2}}{{\partial x}^{2}}

represents the quantum kinetic energy of a single particle of mass

m

, and

V(x)

its potential energy. … ►and

\hbar=k=m=1

, has eigenfunctions … ►The eigenfunctions of

\mathrm{L}^{2}

are the spherical harmonics

Y_{{l},{m_{l}}}\left(\theta,\phi\right)

with eigenvalues

\hbar^{2}l(l+1)

, each with degeneracy

2l+1

m_{l}=-l,-l+1,\dots,l

. … ►,

\hbar=m_{e}={e}^{2}=4\pi\epsilon_{0}=1

, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of

a . u .

to SI units is spelled out. …