equivalent expressions

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11—18 of 18 matching pages

11: 34.2 Definition: $\mathit{3j}$ Symbol

… ►When both conditions are satisfied the

\mathit{3j}

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

\mathit{3j}

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

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

12: 1.13 Differential Equations

… ►Two solutions

w_{1}(z)

and

w_{2}(z)

are called a fundamental pair if any other solution

w(z)

is expressible as … ►The following three statements are equivalent:

w_{1}(z)

and

w_{2}(z)

comprise a fundamental pair in

D

;

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}

does not vanish in

D

;

w_{1}(z)

and

w_{2}(z)

are linearly independent, that is, the only constants

A

and

B

such that … ►If

w_{0}(z)

is any one solution, and

w_{1}(z)

w_{2}(z)

are a fundamental pair of solutions of the corresponding homogeneous equation (1.13.1), then every solution of (1.13.8) can be expressed as …

13: 28.29 Definitions and Basic Properties

… ►Equivalently, … ►The case

c=0

is equivalent to … ►For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of

Q(x)

; see Magnus and Winkler (1966, §2.3, pp. 28–36). …

14: 1.17 Integral and Series Representations of the Dirac Delta

… ►Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9)

\&

(1.14.11), (1.14.10)

\&

(1.14.12), respectively. … ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

15: 19.21 Connection Formulas

… ►The complete case of

R_{J}

can be expressed in terms of

R_{F}

and

R_{D}

: …

16: Errata

… ►

Equation (33.11.1)

33.11.1

{H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim{\mathrm{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}}

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 by Gergő Nemes on 2019-01-29

… ►

Subsection 26.7(iv)

In the final line of this subsection, $\operatorname{Wm}\left(n\right)$ was replaced by $\operatorname{Wp}\left(n\right)$ twice, and the wording was changed from “or, equivalently, $N={\mathrm{e}}^{\operatorname{Wm}\left(n\right)}$ ” to “or, specifically, $N={\mathrm{e}}^{\operatorname{Wp}\left(n\right)}$ ”.

Reported by Gergő Nemes on 2018-04-09

… ►

Notation

The overloaded operator $\equiv$ is now more clearly separated (and linked) to two distinct cases: equivalence by definition (in §§1.4(ii), 1.4(v), 2.7(i), 2.10(iv), 3.1(i), 3.1(iv), 4.18, 9.18(ii), 9.18(vi), 9.18(vi), 18.2(iv), 20.2(iii), 20.7(vi), 23.20(ii), 25.10(i), 26.15, 31.17(i)); and modular equivalence (in §§24.10(i), 24.10(ii), 24.10(iii), 24.10(iv), 24.15(iii), 24.19(ii), 26.14(i), 26.21, 27.2(i), 27.8, 27.9, 27.11, 27.12, 27.14(v), 27.14(vi), 27.15, 27.16, 27.19).

… ►

Subsection 2.1(iii)

A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

… ►

Equation (13.9.16)

Originally was expressed in term of asymptotic symbol $\sim$ . As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$ .

…

17: 19.33 Triaxial Ellipsoids

… ►or equivalently, … ►Expressions in terms of Legendre’s integrals, numerical tables, and further references are given by Cronemeyer (1991). …

18: 18.2 General Orthogonal Polynomials

… ►The recurrence relations (18.2.10) can be equivalently written as … ►See Ismail (2009, §3.4) for another expression of the discriminant in the case of a general OP. … ►The monic OP’s

p_{n}(x)

with respect to the measure

\,\mathrm{d}\mu(x)

can be expressed in terms of the moments by …The recurrence coefficients

\alpha_{n}

and

\beta_{n}

in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by … ►The polynomials

p_{n}^{(1)}(x)

may be also be directly expressed in terms of the

p_{n}{(x)}

of (18.2.11_5): …