DLMF: Untitled Document

34.4.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m_{r}m^{\prime}_{s}}(-1)^{l_{1}+m^{\prime% }_{1}+l_{2}+m^{\prime}_{2}+l_{3}+m^{\prime}_{3}}\*\begin{pmatrix}j_{1}&j_{2}&j% _{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix},

â“˜

Defines:: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol
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., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer
Referenced by:: §34.10, §34.4, §34.9
Permalink:: http://dlmf.nist.gov/34.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

â–ºwhere the summation is taken over all admissible values of the

m

’s and

m^{\prime}

’s for each of the four

\mathit{3j}

symbols; compare (34.2.2) and (34.2.3). … â–º

34.4.2

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\*\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\*\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},

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Symbols:: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product
Referenced by:: Erratum (V1.0.7) for Equation (34.4.2)
Permalink:: http://dlmf.nist.gov/34.4.E2
Encodings:: pMML, png, TeX
Errata (effective with 1.0.7):: Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation.
Reported 2012-12-31 by Yu Lin
See also:: Annotations for §34.4 and Ch.34

â–ºwhere the summation is over all nonnegative integers

s

such that the arguments in the factorials are nonnegative. …

… â–º

§3.4(i) Equally-Spaced Nodes

… â–º … â–ºIf

f

can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) … â–ºFor partial derivatives we use the notation

u_{t,s}=u(x_{0}+th,y_{0}+sh)

. …

… â–º

E. L. Ince (1940b) Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

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External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.
Encodings:: BibTeX

… â–º

A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:: ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.36(ii).
Encodings:: BibTeX

â–º

A. Iserles, S. P. Nørsett, and S. Olver (2006) Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:: Proceedings of ENuMath, Santiago de Compostela (2005)
External Links:: ISBN 3-540-34287-7, MathReview, ZentralBlatt
Referenced by:: §3.5(vii).
Encodings:: BibTeX

… â–º

A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:: ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

… â–º

A. IviÄ‡ (1985) The Riemann Zeta-Function. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:: Reprinted by Dover, 2003.
External Links:: ISBN 0-471-80634-X, MathReview (S. W. Graham), ZentralBlatt
Referenced by:: (25.2.4), (25.2.5), §25.2(i), §25.2(ii), §25.2, Ch.25.
Encodings:: BibTeX

…

… â–º

Y. Takei (1995) On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. SÅ«rikaisekikenkyÅ«sho KÅkyÅ«roku (931), pp. 70–99.

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Notes:: Painlevé functions and asymptotic analysis (Japanese) (Kyoto, 1995)
External Links:: FullText, MathReview (Andrei A. Kapaev)
Referenced by:: §32.12(i).
Encodings:: BibTeX

… â–º

J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… â–º

J. W. Tanner and S. S. Wagstaff (1987) New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.

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External Links:: ISSN 0025-5718, FullText, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… â–º

S. A. Teukolsky (1972) Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.

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External Links:: Document
Referenced by:: §31.17(ii).
Encodings:: BibTeX

… â–º

J. S. Thompson (1996) High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

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External Links:: FullText
Referenced by:: §25.21(vii).
Encodings:: BibTeX

…

… â–º

National Physical Laboratory (1961) Modern Computing Methods. 2nd edition, Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London.

â“˜

External Links:: MathReview (A. S. Householder), ZentralBlatt
Referenced by:: §3.8(iv).
Encodings:: BibTeX

… â–º

National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

â“˜

External Links:: MathReview, ZentralBlatt
Referenced by:: §28.1, §28.26(i), 6th item, Notations S, Notations S.
Encodings:: BibTeX

… â–º

G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.16), (11.11.17), §11.11(iii).
Encodings:: BibTeX

… â–º

E. Neuman (1969a) Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

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Notes:: Includes Algol program.
External Links:: MathReview
Referenced by:: §19.39(iii).
Encodings:: BibTeX

… â–º

A. Nijenhuis and H. S. Wilf (1975) Combinatorial Algorithms. Academic Press, New York.

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External Links:: ISBN 0-12-519250-9, MathReview (Alon Itai), ZentralBlatt
Referenced by:: §26.22.
Encodings:: BibTeX

…

… â–º

10.32.5

K_{0}\left(z\right)=-\frac{1}{\pi}\int_{0}^{\pi}e^{\pm z\cos\theta}\left(% \gamma+\ln\left(2z(\sin\theta)^{2}\right)\right)\,\mathrm{d}\theta.

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Symbols:: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function and $z$ : complex variable
A&S Ref:: 9.6.17
Permalink:: http://dlmf.nist.gov/10.32.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §10.32(i), §10.32 and Ch.10

… â–º

Basset’s Integral

…

… â–º

H. M. Edwards (1974) Riemann’s Zeta Function. Academic Press, New York-London.

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Notes:: Pure and Applied Mathematics, Vol. 58. Reprinted by Dover Publications Inc., Mineola, NY, 2001.
External Links:: MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: Ch.25.
Encodings:: BibTeX

… â–º

Á. Elbert and A. Laforgia (2000) Further results on McMahon’s asymptotic approximations. J. Phys. A 33 (36), pp. 6333–6341.

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External Links:: ISSN 0305-4470, Document, MathReview (R. G. AÄrapetyan), ZentralBlatt
Referenced by:: §10.21(vi).
Encodings:: BibTeX

… â–º

W. J. Ellison (1971) Waring’s problem. Amer. Math. Monthly 78 (1), pp. 10–36.

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External Links:: Document, FullText, MathReview (J. W. S. Cassels), ZentralBlatt
Referenced by:: §27.13(iii), §27.13(iii).
Encodings:: BibTeX

… â–º

D. Erricolo (2006) Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Software 32 (4), pp. 622–634.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §28.36(iii).
Encodings:: BibTeX

… â–º

H. Exton (1983) The asymptotic behaviour of the inhomogeneous Airy function

{\rm Hi}(z)

. Math. Chronicle 12, pp. 99–104.

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External Links:: ISSN 0581-1155, MathReview (Ram Kishore Saxena), ZentralBlatt
Referenced by:: (9.12.24), §9.12(vii).
Encodings:: BibTeX

… â–º

L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

… â–º

H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:: Reprinted by Dover, New York, 1950.
External Links:: ZentralBlatt
Referenced by:: §6.16(i).
Encodings:: BibTeX

… â–º

M. A. Chaudhry and S. M. Zubair (1994) Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §8.16.
Encodings:: BibTeX

… â–º

R. M. Corless, D. J. Jeffrey, and H. Rasmussen (1992) Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.

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External Links:: ISSN 0021-9991, Document, MathReview, ZentralBlatt
Referenced by:: 4th item, 1st item.
Encodings:: BibTeX

… â–º

A. Cruz, J. Esparza, and J. Sesma (1991) Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(ix).
Encodings:: BibTeX

…

… â–º

§19.9(i) Complete Integrals

… â–ºRamanujan’s approximation and its leading error term yield the following approximation to

L(a,b)/(\pi(a+b))

: …Even for the extremely eccentric ellipse with

a=99

and

b=1

, this is correct within 0. …Barnard et al. (2000) shows that nine of the thirteen approximations, including Ramanujan’s, are from below and four are from above. … â–º

§19.9(ii) Incomplete Integrals

…

… â–º

B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.

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External Links:: ISBN 0-387-96794-X, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §15.7.
Encodings:: BibTeX

â–º

B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.

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External Links:: ISBN 0-387-97503-9, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

… â–º

M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.

â“˜

External Links:: Document
Referenced by:: §36.14(i).
Encodings:: BibTeX

… â–º

M. V. Berry (1989) Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

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External Links:: ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt
Referenced by:: §2.11(iv).
Encodings:: BibTeX

… â–º

I. Bloch, M. H. Hull, A. A. Broyles, W. G. Bouricius, B. E. Freeman, and G. Breit (1950) Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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External Links:: Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §33.7.
Encodings:: BibTeX

…

Fej%C3%A9r%E2%80%99s

31—40 of 610 matching pages

31: 34.4 Definition: $\mathit{6j}$ Symbol

32: 3.4 Differentiation

§3.4(i) Equally-Spaced Nodes

33: Bibliography I

34: Bibliography T

35: Bibliography N

36: 10.32 Integral Representations

Basset’s Integral

37: Bibliography E

38: Bibliography C

39: 19.9 Inequalities

§19.9(i) Complete Integrals

§19.9(ii) Incomplete Integrals

40: Bibliography B

Fej%C3%A9r%E2%80%99s

31—40 of 610 matching pages

31: 34.4 Definition: 6 â¢ j Symbol

32: 3.4 Differentiation

§3.4(i) Equally-Spaced Nodes

33: Bibliography I

34: Bibliography T

35: Bibliography N

36: 10.32 Integral Representations

Basset’s Integral

37: Bibliography E

38: Bibliography C

39: 19.9 Inequalities

§19.9(i) Complete Integrals

§19.9(ii) Incomplete Integrals

40: Bibliography B

31: 34.4 Definition: $\mathit{6j}$ Symbol