Levi-Civita symbol for vectors

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21: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

Linear Operators in Finite Dimensional Vector Spaces

►Square matices can be seen as linear operators because

\mathbf{A}(\alpha\mathbf{a}+\beta\mathbf{b})=\alpha\mathbf{A}\mathbf{a}+\beta% \mathbf{A}\mathbf{b}

for all

\alpha,\beta\in\mathbb{C}

and

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

, the space of all

n

-dimensional vectors. … ►The adjoint of a matrix

\mathbf{A}

is the matrix

{\mathbf{A}}^{*}

such that

\left\langle\mathbf{A}\mathbf{a},\mathbf{b}\right\rangle=\left\langle\mathbf{a% },{\mathbf{A}}^{*}\mathbf{b}\right\rangle

for all

\mathbf{a},\mathbf{b}\in\mathbf{E}_{n}

. … ►Assuming

\{\mathbf{a}_{i}\}

is an orthonormal basis in

\mathbf{E}_{n}

, any vector

\mathbf{u}

may be expanded as ►

1.3.20

\mathbf{u}=\sum_{i=1}^{n}c_{i}\mathbf{a}_{i},

c_{i}=\left\langle\mathbf{u},\mathbf{a}_{i}\right\rangle

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : 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

…

22: 3.2 Linear Algebra

… ►

Iterative Refinement

… ►Because of rounding errors, the residual vector

\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}

is nonzero as a rule. … ►The $p$ -norm of a vector

\mathbf{x}=[x_{1},\dots,x_{n}]^{\rm T}

is given by … ►The sensitivity of the solution vector

\mathbf{x}

in (3.2.1) to small perturbations in the matrix

\mathbf{A}

and the vector

\mathbf{b}

is measured by the condition number … ►

§3.2(vi) Lanczos Tridiagonalization of a Symmetric Matrix

…

23: 21.3 Symmetry and Quasi-Periodicity

… ►

21.3.4

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)=e^{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}\theta% \genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Referenced by:: Erratum (V1.0.5) for Equation (21.3.4)
Permalink:: http://dlmf.nist.gov/21.3.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: Originally the vector $\mathbf{m}_{2}$ on the right-hand-side was given incorrectly as $\mathbf{m}_{1}$ .
Reported 2012-08-27 by Klaas Vantournhout
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

… ►

21.3.5

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}\middle|% \boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\boldsymbol{{\alpha}}\cdot\mathbf{% m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}{2}\mathbf{m}_{2}\cdot% \boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}\cdot\mathbf{z}\right)}% \theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

… ►

21.3.6

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-1)^{4\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\beta}}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

…

24: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols, different asymptotic forms are obtained depending on which parameters are large. … ►and the symbol

o\left(1\right)

denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). … ►Uniform approximations in terms of Airy functions for the

\mathit{3j}

and

\mathit{6j}

symbols are given in Schulten and Gordon (1975b). For approximations for the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

… … ► 1958 in Ried im Innkreis, Austria) is Professor of Mathematics (successor to Bruno Buchberger), Director of the Research Institute for Symbolic Computation (RISC), and Director of the Doctoral Program on Computational Mathematics at the Johannes Kepler University, Linz, Austria. … ►He is on the editorial boards for the Journal of Symbolic Computation and The Ramanujan Journal, and is Managing Editor of Annals of Combinatorics. He is also Editor-in-Chief of the Springer book series Texts and Monographs in Symbolic Computation. …

26: 15.11 Riemann’s Differential Equation

… ►The complete set of solutions of (15.11.1) is denoted by Riemann’s $P$ -symbol: ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►

15.11.4

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►Symbolically: ►

15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

…

27: Bibliography D

… ►

H. F. Davis and A. D. Snider (1987) Introduction to Vector Analysis. 5th edition, Allyn and Bacon Inc., Boston, MA.

ⓘ

Notes:: Seventh edition printed by WCB/McGraw-Hill, 1995.
External Links:: ISBN 0-697-06356-9; 0-697-16099-8, MathReview (R. H. Bowman), ZentralBlatt
Referenced by:: §1.5(ii).
Encodings:: BibTeX

… ►

Derive (commercial interactive system) Texas Instruments, Inc..

ⓘ

Notes:: Symbolic and extended-precision numerical computing system.
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

R. McD. Dodds and G. Wiechers (1972) Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

ⓘ

Notes:: Single-precision Fortran
External Links:: Document, Code
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

B. I. Dunlap and B. R. Judd (1975) Novel identities for simple

n

j

symbols. J. Mathematical Phys. 16, pp. 318–319.

ⓘ

External Links:: Document, MathReview (M. J. Westwater)
Referenced by:: §34.5(vi).
Encodings:: BibTeX

…

28: 16.24 Physical Applications

… ►

§16.24(iii) $\mathit{3j}$ , $\mathit{6j}$ , and $\mathit{9j}$ Symbols

►The

\mathit{3j}

symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner

\mathit{6j}

symbols. …Lastly, special cases of the

\mathit{9j}

symbols are

{{}_{5}F_{4}}

functions with unit argument. …

29: 34.7 Basic Properties: $\mathit{9j}$ Symbol

§34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

§34.7(ii) Symmetry

… ►

§34.7(iv) Orthogonality

… ►

§34.7(vi) Sums

… ►It constitutes an addition theorem for the

\mathit{9j}

symbol. …

30: 27.1 Special Notation

… ► ►►►►

$d, k, m, n$	positive integers (unless otherwise indicated).
…
$(n\|P)$	Jacobi symbol; see §27.9.
$(n\|p)$	Legendre symbol; see §27.9.