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11: 19.37 Tables

… ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

k^{2}=0(.01)1

to 10D by Fettis and Caslin (1964). ►Tabulated for

\phi=0(1^{\circ})90^{\circ}

k^{2}=0(.01)1

to 7S by Beli͡akov et al. (1962). … ►Tabulated for

\phi=0(5^{\circ})90^{\circ}

k=0(.01)1

to 10D by Fettis and Caslin (1964). … ►Tabulated (with different notation) for

\phi=0(15^{\circ})90^{\circ}

\alpha^{2}=0(.1)1

\operatorname{arcsin}k=0(15^{\circ})90^{\circ}

to 5D by Abramowitz and Stegun (1964, Chapter 17), and for

\phi=0(15^{\circ})90^{\circ}

\alpha^{2}=0(.1)1

\operatorname{arcsin}k=0(15^{\circ})90^{\circ}

to 7D by Zhang and Jin (1996, pp. 676–677). ►Tabulated for

\phi=5^{\circ}(5^{\circ})80^{\circ}(2.5^{\circ})90^{\circ}

\alpha^{2}=-1(.1)-0.1,0.1(.1)1

k^{2}=0(.05)0.9(.02)1

to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …

12: 34.4 Definition: $\mathit{6j}$ Symbol

… ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►

34.4.3

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $!$ : 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:: §18.38(iii)
Permalink:: http://dlmf.nist.gov/34.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §34.4 and Ch.34

►where

{{}_{4}F_{3}}

is defined as in §16.2. ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

13: Bibliography L

… ►

D. F. Lawden (1989) Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.
Encodings:: BibTeX

… ►

D. H. Lehmer (1940) On the maxima and minima of Bernoulli polynomials. Amer. Math. Monthly 47 (8), pp. 533–538.

ⓘ

External Links:: FullText, MathReview (W. E. Milne), ZentralBlatt
Referenced by:: §24.12(i), §24.9.
Encodings:: BibTeX

… ►

D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

ⓘ

Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

X. Li and R. Wong (2001) On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 17 (1), pp. 59–90.

ⓘ

External Links:: ISSN 0176-4276, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

… ►

N. A. Lukaševič (1968) Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).

ⓘ

Notes:: In Russian. English translation: Differential Equations 4(8), pp. 732–735
External Links:: MathReview (C. E. Langenhop), ZentralBlatt
Referenced by:: §32.10(v), §32.8(v).
Encodings:: BibTeX

…

14: Bibliography M

… ►

P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function

{\rm Ai}(x)

. J. Comput. Phys. 99 (2), pp. 337–340.

ⓘ

Notes:: There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$ .
External Links:: ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

ⓘ

External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

… ►

P. Midy (1975) An improved calculation of the general elliptic integral of the second kind in the neighbourhood of

x=0

. Numer. Math. 25 (1), pp. 99–101.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

S. C. Milne (1985c) A new symmetry related to

\mathit{SU}(n)

for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

ⓘ

External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

… ►

S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

ⓘ

External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

…

15: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99). … ►Equation (34.5.23) can be regarded as an alternative definition of the

\mathit{6j}

symbol. …

16: Bibliography G

… ►

W. Gautschi (1964b) Algorithm 236: Bessel functions of the first kind. Comm. ACM 7 (8), pp. 479–480.

ⓘ

Notes:: For certification see same journal 8(1965), pp. 105–106, for remark see ACM Trans. Math. Software, 1(1975), pp. 282–284. Includes Algol program, maximum accuracy: 11D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(iii).
Encodings:: BibTeX

►

W. Gautschi (1965) Algorithm 259: Legendre functions for arguments larger than one. Comm. ACM 8 (8), pp. 488–492.

ⓘ

Notes:: Variable-precision Algol. For remark see ACM Trans. Math. Software 3(2) (1977), p. 204–205
External Links:: ISSN 0001-0782, Document
Referenced by:: §14.34(ii).
Encodings:: BibTeX

… ►

W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

ⓘ

External Links:: Document, MathReview (E. Frank), ZentralBlatt
Referenced by:: §10.74(iv), §14.32, §3.10(iii), §3.6(iii).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2012) An improved algorithm and a Fortran 90 module for computing the conical function

P^{m}_{-1/2+i\tau}(x)

. Comput. Phys. Commun. 183 (3), pp. 794–799.

ⓘ

External Links:: ISSN 0010-4655, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: 5th item, §14.32.
Encodings:: BibTeX

… ►

K. Goldberg, F. T. Leighton, M. Newman, and S. L. Zuckerman (1976) Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

ⓘ

External Links:: MathReview (M. S. Cheema), ZentralBlatt
Referenced by:: §26.21.
Encodings:: BibTeX

…

18: Bibliography S

… ►

C. W. Schelin (1983) Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.

ⓘ

External Links:: ISSN 0002-9890, FullText, MathReview (J. Albrycht), ZentralBlatt
Referenced by:: §4.45(i).
Encodings:: BibTeX

… ►

B. I. Schneider, J. Segura, A. Gil, X. Guan, and K. Bartschat (2010) A new Fortran 90 program to compute regular and irregular associated Legendre functions. Comput. Phys. Comm. 181 (12), pp. 2091–2097.

ⓘ

External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: 8th item, §14.34(ii).
Encodings:: BibTeX

… ►

D. C. Shaw (1985) Perturbational results for diffraction of water-waves by nearly-vertical barriers. IMA J. Appl. Math. 34 (1), pp. 99–117.

ⓘ

External Links:: ISSN 0272-4960, Document, MathReview, ZentralBlatt
Referenced by:: §11.12.
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

… ►

D. M. Smith (2001) Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions. ACM Trans. Math. Software 27 (4), pp. 377–387.

ⓘ

External Links:: ISSN 0098-3500, Document, GAMS (module 814; package TOMS), ZentralBlatt
Referenced by:: 2nd item, 6th item, 3rd item, 1st item, 6th item, 5th item.
Encodings:: BibTeX

…

19: 19.9 Inequalities

… ►Further inequalities for

K\left(k\right)

and

E\left(k\right)

can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►Even for the extremely eccentric ellipse with

a=99

and

b=1

, this is correct within 0. … ►Sharper inequalities for

F\left(\phi,k\right)

are: … ►Inequalities for both

F\left(\phi,k\right)

and

E\left(\phi,k\right)

involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). … ►Other inequalities for

F\left(\phi,k\right)

can be obtained from inequalities for

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

given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

20: 19.21 Connection Formulas

… ►Legendre’s relation (19.7.1) can be written … ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►The complete case of

R_{J}

can be expressed in terms of

R_{F}

and

R_{D}

: … ►Because

R_{G}

is completely symmetric,

x, y, z

can be permuted on the right-hand side of (19.21.10) so that

(x-z)(y-z)\leq 0

if the variables are real, thereby avoiding cancellations when

R_{G}

is calculated from

R_{F}

and

R_{D}

(see §19.36(i)). … ►Connection formulas for

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

are given in Carlson (1977b, pp. 99, 101, and 123–124). …