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21: 27.1 Special Notation

$d, k, m, n$	positive integers (unless otherwise indicated).
…
$\sum_{d\mathbin{\|}n}$ , $\prod_{d\mathbin{\|}n}$	sum, product taken over divisors of $n$ .
$\sum_{\left(m,n\right)=1}$	sum taken over $m$ , $1\leq m\leq n$ and $m$ relatively prime to $n$ .
…
$\sum_{p}$ , $\prod_{p}$	sum, product extended over all primes.
…

22: Need Help?

… ►

Finding Things

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How do I search within DLMF? See Guide to Searching the DLMF.
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See also the Index or Notations sections.
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Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

23: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

… ►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. …

27: Bernard Deconinck

… ►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. …

28: 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

…

29: 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)

. …

30: 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}