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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: Bibliography O

… ►

Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:

Online fulltext contains postpublication annotations.

External Links:

FullText

Referenced by:

§27.18.

Encodings:

BibTeX

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37: 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. …

38: 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. …

39: Bibliography S

… ►

FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

10th item.

Encodings:

BibTeX

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Automatic computing methods for special functions. III. The sine, cosine, exponential integrals, and related functions. J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 291–311.

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Notes:

Includes Fortran code, maximum accuracy 18S.

External Links:

MathReview (Martin E. Price), ZentralBlatt

Referenced by:

§6.21(ii).

Encodings:

BibTeX

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Department of Computer Science, Stony Brook University, New York.

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Notes:

A collection of over 70 combinatorial algorithms.

External Links:

Stony Brook Algorithm Repository

Referenced by:

7th item.

Encodings:

BibTeX

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40: 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 …