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31: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The adjoint of a matrix $𝐀$ is the matrix ${𝐀}^{\ast }$ such that $⟨𝐀𝐚,𝐛⟩=⟨𝐚,{𝐀}^{\ast }𝐛⟩$ for all $𝐚,𝐛\in {𝐄}_n$. … ►

1.3.20 $$𝐮=\sum _{i=1}^n{c}_i{𝐚}_i,$$ $c_i=⟨𝐮,{𝐚}_i⟩$.

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Symbols:

$⟨𝐮,𝐯⟩$: 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

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32: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►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. 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 $\mathrm{\ell }\mathrm{\hslash }$, the Schrödinger equation for the radial wave function $w(s)$ is given by … ►

$𝗄={(2mE/{\mathrm{\hslash }}^2)}^{1/2},$

… ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. …

33: 35.2 Laplace Transform

… ►where the integration variable $𝐗$ ranges over the space $𝛀$. … ►where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. …

34: 27.18 Methods of Computation: Primes

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

35: Bibliography D

… ►

Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

External Links:

Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

Encodings:

BibTeX

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36: 18.39 Applications in the Physical Sciences

… ►where $x$ is a spatial coordinate, $m$ the mass of the particle with potential energy $V(x)$, $\mathrm{\hslash }=h/(2\pi )$ is the reduced Planck’s constant, and $(a,b)$ a finite or infinite interval. ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►and $\mathrm{\hslash }=k=m=1$, has eigenfunctions … ►The eigenfunctions of ${\mathrm{L}}^2$ are the spherical harmonics $Y_{l,m_l}(\theta ,\varphi )$ with eigenvalues ${\mathrm{\hslash }}^{2}l(l+1)$, each with degeneracy $2l+1$ as $m_l=-l,-l+1,\mathrm{\dots },l$. … ►, $\mathrm{\hslash }=m_e=e^2=4\pi {ϵ}_0=1$, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of $a.u.$ to SI units is spelled out. …

37: 27.6 Divisor Sums

… ►Sums of number-theoretic functions extended over divisors are of special interest. … ►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …

38: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …

39: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …

40: 1.6 Vectors and Vector-Valued Functions

… ►Much vector algebra involves summation over suffices of products of vector components. In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over). … ►

§1.6(v) Surfaces and Integrals over Surfaces

… ►The integral of a continuous function $f(x,y,z)$ over a surface $S$ is …