reduced%20residue%20system

SearchAdvancedHelp

(0.002 seconds)

21—30 of 231 matching pages

21: 1.11 Zeros of Polynomials

… ►Set $z=w-\frac{1}{3}a$ to reduce $f(z)=z^3+a{z}^2+bz+c$ to $g(w)=w^3+pw+q$, with $p=(3b-a^2)/3$, $q=(2a^3-9ab+27c)/27$. … ►Set $z=w-\frac{1}{4}a$ to reduce $f(z)=z^4+a{z}^3+b{z}^2+cz+d$ to … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

22: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.11 $$L^2{Y}_{l,m}={\mathrm{\hslash }}^{2}l(l+1)Y_{l,m},$$ $l=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and $L$: angular momentum operator

Referenced by:

§18.39(ii)

Permalink:

http://dlmf.nist.gov/14.30.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

… ►where $\mathrm{\hslash }$ is the reduced Planck’s constant. …

23: Bibliography H

… ►

Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

ⓘ

External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

… ►

Numerical Tools for the Study of Finite Gap Solutions of Integrable Systems. Ph.D. Thesis, Technischen Universität Berlin.

ⓘ

Notes:

(All relevant material, plus corrections, are included in Deconinck et al. (2004).)

External Links:

ZentralBlatt

Referenced by:

§21.5(i).

Encodings:

BibTeX

… ►

Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

ⓘ

External Links:

ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

…

24: 18.39 Applications in the Physical Sciences

… ►

Introduction and One-Dimensional (1D) Systems

… ►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. … ►As in classical dynamics this sum is the total energy of the one particle system. … ►

1D Quantum Systems with Analytically Known Stationary States

… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

25: 15.13 Zeros

… ►If $a$, $b$, $c$, $c-a$, or $c-b\in \{0,-1,-2,\mathrm{\dots }\}$, then $F(a,b;c;z)$ is not defined, or reduces to a polynomial, or reduces to ${(1-z)}^{c-a-b}$ times a polynomial. …

26: 27.16 Cryptography

… ►To code a piece $x$, raise $x$ to the power $r$ and reduce $x^r$ modulo $n$ to obtain an integer $y$ (the coded form of $x$) between $1$ and $n$. … ►In other words, to recover $x$ from $y$ we simply raise $y$ to the power $s$ and reduce modulo $n$. …

27: 30.12 Generalized and Coulomb Spheroidal Functions

… ►which reduces to (30.2.1) if $\alpha =0$. … ►which also reduces to (30.2.1) when $\alpha =0$. …

28: 14.29 Generalizations

… ►As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when ${\mu }_1={\mu }_2=\mu$. …

29: 29.11 Lamé Wave Equation

… ►In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

30: William P. Reinhardt

… ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.