equivalent expressions

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1: 31.3 Basic Solutions

… ►

§31.3(iii) Equivalent Expressions

… ►There are 192 automorphisms in all, so there are

192/8=24

equivalent expressions for each of the 8. …

2: Guide to Searching the DLMF

… ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. …

3: 18.39 Applications in the Physical Sciences

… ►This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional

L^{2}

functions of (18.39.53), provided that such information involves potentials, or projections onto

L^{2}

functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). …

4: 15.5 Derivatives and Contiguous Functions

… ►By repeated applications of (15.5.11)–(15.5.18) any function

F\left(a+k,b+\ell;c+m;z\right)

, in which

k,\ell,m

are integers, can be expressed as a linear combination of

F\left(a,b;c;z\right)

and any one of its contiguous functions, with coefficients that are rational functions of

a, b, c

, and

z

. ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …

5: 34.4 Definition: $\mathit{6j}$ Symbol

… ►Except in degenerate cases the combination of the triangle inequalities for the four

\mathit{3j}

symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths

j_{1},j_{2},j_{3},l_{1},l_{2},l_{3}

; see Figure 34.4.1. … ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►Equivalently, … ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

6: 27.2 Functions

… ►It can be expressed as a sum over all primes

p\leq x

: … ►An equivalent form states that the

n

th prime

p_{n}

(when the primes are listed in increasing order) is asymptotic to

n\ln n

n\to\infty

: … ►

27.2.8

a^{\phi\left(n\right)}\equiv 1\pmod{n},

ⓘ

Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\equiv$ : modular equivalence, $n$ : positive integer and $a$ : primitive root
A&S Ref:: 24.3.2 II.B
Referenced by:: §27.16, §27.8
Permalink:: http://dlmf.nist.gov/27.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §27.2(i), §27.2 and Ch.27

►and if

\phi\left(n\right)

is the smallest positive integer

f

such that

a^{f}\equiv 1\pmod{n}

, then

a

is a primitive root mod

n

. …It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …

7: 33.11 Asymptotic Expansions for Large $\rho$

… ►

33.11.1

{H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},

ⓘ

Symbols:: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $a$ : coefficient and $b$ : coefficient
A&S Ref:: 14.5.9
Referenced by:: Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)
Permalink:: http://dlmf.nist.gov/33.11.E1
Encodings:: pMML, png, TeX
Errata (effective with 1.0.22):: Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.
Reported 2019-01-29 by Gergő Nemes
Errata (effective with 1.0.19):: Originally the factor in the denominator on the right-hand side was written incorrectly as $(\mp 2\mathrm{i}\rho)^{k}$ . This has been corrected to $(\pm 2\mathrm{i}\rho)^{k}$ .
Reported 2018-05-15 by Ian Thompson
See also:: Annotations for §33.11 and Ch.33

… ►An equivalent formulation is given by …

8: 2.1 Definitions and Elementary Properties

… ►Condition (2.1.13) is equivalent to … ►Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. …

9: 23.20 Mathematical Applications

… ►or equivalently, on replacing

x

x/z

and

y

y/z

(projective coordinates), into the form … ►In terms of

(x,y)

the addition law can be expressed

(x,y)+o=(x,y)

(x,y)+(x,-y)=o

; otherwise

(x_{1},y_{1})+(x_{2},y_{2})=(x_{3},y_{3})

, where …

10: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). … ►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity

\rho/({F_{\ell}}^{2}\left(\eta,\rho\right)+{G_{\ell}}^{2}\left(\eta,\rho\right))

(Mott and Massey (1956, pp. 63–65)). …