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12: Bibliography D

… ►

S. D. Daymond (1955) The principal frequencies of vibrating systems with elliptic boundaries. Quart. J. Mech. Appl. Math. 8 (3), pp. 361–372.

ⓘ

External Links:: Document, MathReview (M. A. Hyman), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

K. Dilcher (1988) Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. Mem. Amer. Math. Soc. 73 (386), pp. iv+94.

ⓘ

External Links:: ISSN 0065-9266, MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i), §24.12(iii), §24.16(ii).
Encodings:: BibTeX

… ►

B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

ⓘ

Notes:: For a numerical error in one of the zeros see Amos (1985).
External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

G. V. Dunne and K. Rao (2000) Lamé instantons. J. High Energy Phys. 2000 (1), pp. Paper 19, 8.

ⓘ

Notes:: (electronic only, article id. JHEP01(2000)019, 8 pp.).
External Links:: ISSN 1029-8479, Document, MathReview, ZentralBlatt
Referenced by:: §22.11.
Encodings:: BibTeX

… ►

L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

ⓘ

External Links:: Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §18.17(iii).
Encodings:: BibTeX

…

13: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►

34.5.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{% \frac{1}{2}}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}+\frac{1}{2}\end{Bmatrix}=(-1)^{J}\left(% \frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_% {3}+1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ \frac{1}{2}&j_{3}-\frac{1}{2}&j_{2}-\frac{1}{2}\end{Bmatrix}=(-1)^{J}\left(% \frac{(j_{2}+j_{3}-j_{1})(j_{1}+j_{2}+j_{3}+1)}{2j_{2}(2j_{2}+1)2j_{3}(2j_{3}+% 1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.4

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1% )}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac{1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

►

34.5.5

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}\end{Bmatrix}=(-1)^{J}\left(\frac{2(J+1)(J-2j_{1})(J-2j_{2})(J-% 2j_{3}+1)}{2j_{2}(2j_{2}+1)(2j_{2}+2)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{\frac% {1}{2}},

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.5(i), §34.5 and Ch.34

…

14: Bibliography S

… ►

H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

ⓘ

External Links:: FullText, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §3.5(viii).
Encodings:: BibTeX

… ►

J. Segura and A. Gil (1999) Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

ⓘ

Notes:: Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)
External Links:: ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt
Referenced by:: §14.30(i), 3rd item.
Encodings:: BibTeX

… ►

L. Shen (1981) The elliptical microstrip antenna with circular polarization. IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.

ⓘ

External Links:: FullText
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

K. Srinivasa Rao, V. Rajeswari, and C. B. Chiu (1989) A new Fortran program for the

9

j

angular momentum coefficient. Comput. Phys. Comm. 56 (2), pp. 231–248.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §34.13.
Encodings:: BibTeX

… ►

S. K. Suslov (2003) An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

ⓘ

External Links:: ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt
Referenced by:: §17.3(i).
Encodings:: BibTeX

…

15: Bibliography C

… ►

R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

B. C. Carlson (1978) Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

ⓘ

External Links:: Document, MathReview (Y. L. Luke), ZentralBlatt
Referenced by:: §19.26(i).
Encodings:: BibTeX

… ►

B. C. Carlson (1990) Landen Transformations of Integrals. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Appl. Math., Vol. 124, pp. 75–94.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §19.22(iii).
Encodings:: BibTeX

… ►

T. S. Chihara and M. E. H. Ismail (1993) Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt
Referenced by:: §18.28(iv).
Encodings:: BibTeX

… ►

J. S. Christiansen and M. E. H. Ismail (2006) A moment problem and a family of integral evaluations. Trans. Amer. Math. Soc. 358 (9), pp. 4071–4097.

ⓘ

External Links:: ISSN 0002-9947, Document, MathReview (Roelof Koekoek), ZentralBlatt
Referenced by:: §18.28(iv).
Encodings:: BibTeX

…

16: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients (

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols) are essential in the fields of nuclear, atomic, and molecular physics. …

\mathit{3j},\mathit{6j}

, and

\mathit{9j}

symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

17: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…

… ►Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

19: 23.6 Relations to Other Functions

… ►In this subsection

2\omega_{1}

2\omega_{3}

are any pair of generators of the lattice

\mathbb{L}

, and the lattice roots

e_{1}

e_{2}

e_{3}

are given by (23.3.9). … ►Again, in Equations (23.6.16)–(23.6.26),

2\omega_{1},2\omega_{3}

are any pair of generators of the lattice

\mathbb{L}

and

e_{1},e_{2},e_{3}

are given by (23.3.9). … ►Also,

\mathbb{L}_{\mspace{1.0mu}1}

\mathbb{L}_{\mspace{1.0mu}2}

\mathbb{L}_{\mspace{1.0mu}3}

are the lattices with generators

(4K,2\mathrm{i}{K^{\prime}})

(2K-2\mathrm{i}{K^{\prime}},2K+2\mathrm{i}{K^{\prime}})

(2K,4\mathrm{i}{K^{\prime}})

, respectively. … ►Let

z

be on the perimeter of the rectangle with vertices

0,2\omega_{1},2\omega_{1}+2\omega_{3},2\omega_{3}

. … ►Let

z

be a point of

\mathbb{C}

different from

e_{1},e_{2},e_{3}

, and define

w

by …

20: 18.17 Integrals

… ►For the beta function

\mathrm{B}\left(a,b\right)

see §5.12, and for the confluent hypergeometric function

{{}_{1}F_{1}}

see (16.2.1) and Chapter 13. … ►For the confluent hypergeometric function

{{}_{1}F_{1}}

see (16.2.1) and Chapter 13. … ►For the hypergeometric function

{{}_{2}F_{1}}

see §§15.1 and 15.2(i). … ►For the generalized hypergeometric function

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

see (16.2.1). … ►For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2000, pp. 788–806), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).

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11: DLMF Project News

12: Bibliography D

13: 34.5 Basic Properties: $\mathit{6j}$ Symbol

14: Bibliography S

15: Bibliography C

16: 34.12 Physical Applications

§34.12 Physical Applications

17: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

18: 4.26 Integrals

19: 23.6 Relations to Other Functions

20: 18.17 Integrals

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11—20 of 742 matching pages

11: DLMF Project News

12: Bibliography D

13: 34.5 Basic Properties: 6 ⁢ j Symbol

14: Bibliography S

15: Bibliography C

16: 34.12 Physical Applications

§34.12 Physical Applications

17: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

18: 4.26 Integrals

19: 23.6 Relations to Other Functions

20: 18.17 Integrals

13: 34.5 Basic Properties: $\mathit{6j}$ Symbol