three j symbols

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2: 16.24 Physical Applications

… ►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner

\mathit{6j}

symbols. …

3: 34.6 Definition: $\mathit{9j}$ Symbol

… ►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. …

4: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►In the following three equations it is assumed that the triangle conditions are satisfied by each

\mathit{3j}

symbol. …

5: 16.4 Argument Unity

… ►See Raynal (1979) for a statement in terms of

\mathit{3j}

symbols (Chapter 34). … ►These series contain

\mathit{6j}

symbols as special cases when the parameters are integers; compare §34.4. … ►One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … ►Relations between three solutions of three-term recurrence relations are given by Masson (1991). …

6: Bibliography C

… ►

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

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

… ►

B. C. Carlson (1999) Toward symbolic integration of elliptic integrals. J. Symbolic Comput. 28 (6), pp. 739–753.

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Notes:: Code written by J. FitzSimons to generate reduction tables for elliptic integrals symbolically in the Derive computer algebra system is available.
External Links:: ISSN 0747-7171, Document, Code, MathReview (Axel Riese), ZentralBlatt
Referenced by:: §19.29(i), §19.29(ii), §19.29(ii), §19.29(ii), §19.29(ii).
Encodings:: BibTeX

►

B. C. Carlson (2002) Three improvements in reduction and computation of elliptic integrals. J. Res. Nat. Inst. Standards Tech. 107 (5), pp. 413–418.

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External Links:: FullText
Referenced by:: §19.22(ii), §19.29(ii), §19.36(i), §19.8(i).
Encodings:: BibTeX

… ►

J. N. L. Connor and D. Farrelly (1981) Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.

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External Links:: ISSN 0009-2614, Document, MathReview (J. S. Joel)
Referenced by:: §36.14(iii).
Encodings:: BibTeX

… ►

R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert

W

function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.
Encodings:: BibTeX

…

7: Bibliography G

… ►

A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.

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External Links:: ISSN 0022-2488, Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

►

A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

►

A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

►

A. Gervois and H. Navelet (1986a) Some integrals involving three modified Bessel functions. I. J. Math. Phys. 27 (3), pp. 682–687.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

►

A. Gervois and H. Navelet (1986b) Some integrals involving three modified Bessel functions. II. J. Math. Phys. 27 (3), pp. 688–695.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

…

8: 19.15 Advantages of Symmetry

… ►Symmetry in

x, y, z

R_{F}\left(x,y,z\right)

R_{G}\left(x,y,z\right)

, and

R_{J}\left(x,y,z,p\right)

replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). …(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. … ►These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …

9: 18.37 Classical OP’s in Two or More Variables

… ►The following three conditions, taken together, determine

R^{(\alpha)}_{m,n}\left(z\right)

uniquely: ►

18.37.3

R^{(\alpha)}_{m,n}\left(z\right)=\sum_{j=0}^{\min(m,n)}c_{j}z^{m-j}{\overline{% z}}^{n-j},

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

►where

c_{j}

are real or complex constants, with

c_{0}\neq 0

; ►

18.37.4

\iint\limits_{x^{2}+y^{2}<1}R^{(\alpha)}_{m,n}\left(x+\mathrm{i}y\right)(x-iy)% ^{m-j}(x+iy)^{n-j}\*(1-x^{2}-y^{2})^{\alpha}\,\mathrm{d}x\,\mathrm{d}y=0,

j=1,2,\dots,\min(m,n)

;

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $\mathrm{i}$ : imaginary unit, $y$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.37.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

… ►

18.37.6

R^{(\alpha)}_{m,n}\left(z\right)=\sum_{j=0}^{\min(m,n)}\frac{(-1)^{j}{\left(% \alpha+1\right)_{m+n-j}}{\left(-m\right)_{j}}{\left(-n\right)_{j}}}{{\left(% \alpha+1\right)_{m}}{\left(\alpha+1\right)_{n}}j!}\*z^{m-j}\*{\overline{z}}^{n% -j}.

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\overline{\NVar{z}}$ : complex conjugate, $R^{(\NVar{\alpha})}_{\NVar{m},\NVar{n}}\left(\NVar{z}\right)$ : disk polynomial, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.37.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

…

10: Bibliography V

… ►

Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation

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External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, A. L. Van Buren and J. E. Boisvert (2002), A. L. Van Buren and J. E. Boisvert (2004), A. L. Van Buren and J. E. Boisvert (2007).
Encodings:: BibTeX

… ►

J. van de Lune, H. J. J. te Riele, and D. T. Winter (1986) On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (174), pp. 667–681.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (Jürgen G. Hinz), ZentralBlatt
Referenced by:: §25.10(ii), §25.18(ii).
Encodings:: BibTeX

… ►

A. van Wijngaarden (1953) On the coefficients of the modular invariant

J(\tau)

. Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.

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External Links:: MathReview (J. Lehner), ZentralBlatt
Referenced by:: §23.17(ii).
Encodings:: BibTeX

… ►

R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:: ISSN 0022-247X, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

… ►

I. M. Vinogradov (1937) Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).

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External Links:: ZentralBlatt
Referenced by:: §27.13(ii).
Encodings:: BibTeX

…