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21: 23.4 Graphics

… ►(The figures in this subsection may be compared with the figures in §22.3(i).) … ►(The figures in this subsection may be compared with the figures in §22.3(iii).) …

A. Laforgia (1979) On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §10.21(vi).
Encodings:: BibTeX

… ►

E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:: ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt
Referenced by:: §30.12, §31.17(ii).
Encodings:: BibTeX

… ►

D. H. Lehmer (1941) Guide to Tables in the Theory of Numbers. Bulletin of the National Research Council, No. 105, National Research Council, Washington, D.C..

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Notes:: Reprinted in 1961.
External Links:: MathReview (R. D. Carmichael), ZentralBlatt
Referenced by:: §27.20, §27.20, §27.21, §27.21.
Encodings:: BibTeX

… ►

J. C. Light and T. Carrington Jr. (2000) Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

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External Links:: ISBN 9780470141731, Document, Link
Referenced by:: §18.38(i).
Encodings:: BibTeX

… ►

D. W. Lozier and F. W. J. Olver (1994) Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

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Notes:: An updated version exists online.
External Links:: FullText, MathReview (John P. Coleman), ZentralBlatt
Referenced by:: Ch.3.
Encodings:: BibTeX

…

23: 22.19 Physical Applications

… ►With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is … ►Classical motion in one dimension is described by Newton’s equation … ►Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. … ►The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …Elementary discussions of this topic appear in Lawden (1989, §5.7), Greenhill (1959, pp. 101–103), and Whittaker (1964, Chapter VI). …

24: DLMF Project News

error generating summary

25: Bibliography N

… ►

A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:: ISSN 0031-9015, Document, MathReview, ZentralBlatt
Referenced by:: §18.38(ii).
Encodings:: BibTeX

… ►

V. Yu. Novokshënov (1990) The Boutroux ansatz for the second Painlevé equation in the complex domain. Izv. Akad. Nauk SSSR Ser. Mat. 54 (6), pp. 1229–1251 (Russian).

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Notes:: In Russian. English translation: Math. USSR-Izv. 37(1991), no. 3, pp. 587–609.
External Links:: ISSN 0373-2436, Document, MathReview (Alexander I. Bobenko), ZentralBlatt
Referenced by:: §32.12(ii).
Encodings:: BibTeX

… ►

H. M. Nussenzveig (1992) Diffraction Effects in Semiclassical Scattering. Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.

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Notes:: Also available electronically through Cambridge Books Online, ISBN 9780511599903.
External Links:: ISBN 9780521383189, Document
Referenced by:: §9.16, §9.16.
Encodings:: BibTeX

… ►

J. F. Nye (2006) Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.

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External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §36.7(iv).
Encodings:: BibTeX

►

J. F. Nye (2007) Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:: ISSN 1364-5021, Document, ZentralBlatt
Referenced by:: §36.7(iv).
Encodings:: BibTeX

26: Bibliography V

… ►

J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

… ►

B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:: Document
Referenced by:: §14.31(i).
Encodings:: BibTeX

… ►

R. Vein and P. Dale (1999) Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York.

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External Links:: ISBN 0-387-98558-1, MathReview (Jorma Kaarlo Merikoski), ZentralBlatt
Referenced by:: §1.3(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

… ►

H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

…

27: Bibliography U

… ►

Unpublished Mathematical Tables (1944) Mathematics of Computation Unpublished Mathematical Tables Collection.

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Notes:: Archives of American Mathematics, Center for American History, The University of Texas at Austin. Covers 1944–1994. An online guide is arranged chronologically by date of issuance within Mathematics of Computation
External Links:: Guide
Referenced by:: §27.21.
Encodings:: BibTeX

…

28: Bibliography O

… ►

A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

… ►

M. N. Olevskiĭ (1950) Triorthogonal systems in spaces of constant curvature in which the equation

\Delta_{2}u+\lambda u=0

allows a complete separation of variables. Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).

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Notes:: In Russian.
External Links:: MathReview (H. Busemann), ZentralBlatt
Referenced by:: §31.17(i).
Encodings:: BibTeX

►

T. Oliveira e Silva (2006) Computing

\pi(x)

: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:: Online fulltext contains postpublication annotations.
External Links:: FullText
Referenced by:: §27.18.
Encodings:: BibTeX

… ►

F. W. J. Olver (1970) A paradox in asymptotics. SIAM J. Math. Anal. 1 (4), pp. 533–534.

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External Links:: Document, MathReview (L. Hoischen), ZentralBlatt
Referenced by:: §2.10(iv).
Encodings:: BibTeX

… ►

J. M. Ortega and W. C. Rheinboldt (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:: Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania
External Links:: MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))
Referenced by:: §3.8(vii).
Encodings:: BibTeX

…

29: 19.9 Inequalities

… ►Throughout this subsection

0<k<1

, except in (19.9.4). …The left-hand inequalities in (19.9.2) and (19.9.3) are equivalent, but the right-hand inequality of (19.9.3) is sharper than that of (19.9.2) when

0<k^{2}\leq 0.922

. …The lower bound in (19.9.4) is sharper than

2/\pi

when

0\leq k^{2}\leq 0.9960

. … ►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). … ►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). …

30: 19.5 Maclaurin and Related Expansions

… ►

19.5.1

K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.4))
Referenced by:: §19.36(iv), §19.5
Permalink:: http://dlmf.nist.gov/19.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

… ►

19.5.2

E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({-\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Source:: Carlson (1961b, (2.5))
Referenced by:: §19.36(iv)
Permalink:: http://dlmf.nist.gov/19.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

►

19.5.3

D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{% }_{2}F_{1}}\left({\tfrac{3}{2},\tfrac{1}{2}\atop 2};k^{2}\right),

ⓘ

Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $D\left(\NVar{k}\right)$ : complete elliptic integral of Legendre’s type, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.5 and Ch.19

… ►Coefficients of terms up to

\lambda^{49}

are given in Lee (1990), along with tables of fractional errors in

K\left(k\right)

and

E\left(k\right)

0.1\leq k^{2}\leq 0.9999

, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). …