finite

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2: 36.15 Methods of Computation

… ►

§36.15(iv) Integration along Finite Contour

►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of

\Phi

, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

… ►Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. …

4: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

… ►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

5: 29.20 Methods of Computation

… ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices

\mathbf{M}

given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

6: 5.16 Sums

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

7: 29.19 Physical Applications

… ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …

8: 27.10 Periodic Number-Theoretic Functions

… ►Every function periodic (mod

k

) can be expressed as a finite Fourier series of the form … ►is a periodic function of

n\pmod{k}

and has the finite Fourier-series expansion … ►The finite Fourier expansion of a primitive Dirichlet character

\chi\pmod{k}

has the form …

9: 26.1 Special Notation

… ► ►►►

$x$	real variable.
…
$\left\|A\right\|$	number of elements of a finite set $A$ .
…

…

10: 15.15 Sums

… ►For compendia of finite sums and infinite series involving hypergeometric functions see Prudnikov et al. (1990, §§5.3 and 6.7) and Hansen (1975). …

finite

1—10 of 111 matching pages

1: 16.9 Zeros

2: 36.15 Methods of Computation

§36.15(iv) Integration along Finite Contour

3: Daniel W. Lozier

4: 34.6 Definition: $\mathit{9j}$ Symbol

5: 29.20 Methods of Computation

6: 5.16 Sums

7: 29.19 Physical Applications

8: 27.10 Periodic Number-Theoretic Functions

9: 26.1 Special Notation

10: 15.15 Sums

finite

1—10 of 111 matching pages

1: 16.9 Zeros

2: 36.15 Methods of Computation

§36.15(iv) Integration along Finite Contour

3: Daniel W. Lozier

4: 34.6 Definition: 9 ⁢ j Symbol

5: 29.20 Methods of Computation

6: 5.16 Sums

7: 29.19 Physical Applications

8: 27.10 Periodic Number-Theoretic Functions

9: 26.1 Special Notation

10: 15.15 Sums

4: 34.6 Definition: $\mathit{9j}$ Symbol