alternative forms

(0.001 seconds)

21—26 of 26 matching pages

21: Errata

… тЦ║The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … тЦ║

Equations (18.2.5), (18.2.6)

18.2.5

h_{n}=\int_{a}^{b}\left(p_{n}(x)\right)^{2}w(x)\,\mathrm{d}x

\text{ or }h_{n}=\sum_{x\in X}\left(p_{n}(x)\right)^{2}w_{x}

\text{ or }h_{n}=\int_{a}^{b}\left(p_{n}(x)\right)^{2}\,\mathrm{d}\mu(x)

18.2.6

\tilde{h}_{n}=\int_{a}^{b}x\left(p_{n}(x)\right)^{2}w(x)\,\mathrm{d}x

\text{ or }\tilde{h}_{n}=\sum_{x\in X}x\left(p_{n}(x)\right)^{2}w_{x}

\text{ or }\tilde{h}_{n}=\int_{a}^{b}x\left(p_{n}(x)\right)^{2}\,\mathrm{d}\mu% (x)

The third alternatives, involving $\,\mathrm{d}\mu(x)$ , were included.

… тЦ║

Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

… тЦ║

Subsections 10.6(i), 10.29(i)

Sentences were added just below (10.6.5) and (10.29.3) regarding results on modified quotients of the form $\ifrac{z\mathscr{C}_{\nu\pm 1}\left(z\right)}{\mathscr{C}_{\nu}\left(z\right)}$ and $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ , respectively (suggested by Art Ballato on 2021-04-29).

… тЦ║

Paragraph Steed’s Algorithm (in §3.10(iii))

A sentence was added to inform the reader of alternatives to Steed’s algorithm, namely the Lentz algorithm (see e.g., Lentz (1976)) and the modified Lentz algorithm (see e.g., Thompson and Barnett (1986)).

…

22: 1.10 Functions of a Complex Variable

… тЦ║A cut neighborhood is formed by deleting a ray emanating from the center. … тЦ║Alternatively, take

z_{0}

to be any point in

D

and set

F(z_{0})={\mathrm{e}}^{\alpha\ln\left(1-z_{0}\right)}{\mathrm{e}}^{\beta\ln% \left(1+z_{0}\right)}

where the logarithms assume their principal values. … тЦ║It should be noted that different branches of

(w-w_{0})^{1/\mu}

used in forming

(w-w_{0})^{n/\mu}

in (1.10.16) give rise to different solutions of (1.10.12). … тЦ║Let

F(x,z)

have a converging power series expansion of the form …

23: 19.34 Mutual Inductance of Coaxial Circles

… тЦ║A simpler form of the result is …

24: 34.2 Definition: $\mathit{3j}$ Symbol

… тЦ║They must form the sides of a triangle (possibly degenerate). … тЦ║

34.2.6

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},

тУШ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

… тЦ║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).

25: 5.9 Integral Representations

… тЦ║

5.9.2

\frac{1}{\Gamma\left(z\right)}=\frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z% }\,\mathrm{d}t,

тУШ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable
A&S Ref:: 6.1.4 (in a slightly different form.)
Referenced by:: §5.21, §5.9(i)
Permalink:: http://dlmf.nist.gov/5.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

… тЦ║Two alternative versions of Binet’s formula are …

26: 18.27 $q$ -Hahn Class

… тЦ║Together they form the $q$ -Askey scheme. … тЦ║All these systems of OP’s have orthogonality properties of the form …Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … тЦ║Alternative definitions and notations are …

alternative forms

21—26 of 26 matching pages

21: Errata

22: 1.10 Functions of a Complex Variable

23: 19.34 Mutual Inductance of Coaxial Circles

24: 34.2 Definition: 3 тБв j Symbol

25: 5.9 Integral Representations

26: 18.27 q -Hahn Class

24: 34.2 Definition: $\mathit{3j}$ Symbol

26: 18.27 $q$ -Hahn Class