DLMF: Untitled Document

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A. A. Kapaev (1991) Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187, pp. 139–170 (Russian).

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Notes:: English translation: J. Math. Sci. 73(1995), no. 4, pp. 500–517
External Links:: ISSN 0373-2703, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt
Referenced by:: §32.12(ii).
Encodings:: BibTeX

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M. K. Kerimov and S. L. Skorokhodov (1984b) Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 24 (8), pp. 1150–1163.

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 24(4), pp. 115–123
External Links:: ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §10.42, §10.74(vi), 3rd item.
Encodings:: BibTeX

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M. K. Kerimov and S. L. Skorokhodov (1985a) Calculation of the complex zeros of a Bessel function of the second kind and its derivatives. Zh. Vychisl. Mat. i Mat. Fiz. 25 (10), pp. 1457–1473, 1581 (Russian).

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Notes:: In Russian. English translation: USSR Comput. Math. Math. Phys., 25(1985), no. 5, pp. 117–128.
External Links:: ISSN 0044-4669, Document, MathReview (Jiří Hřebíček), ZentralBlatt
Referenced by:: §10.21(i), §10.21(ix), §10.74(vi), 3rd item.
Encodings:: BibTeX

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B. J. King, R. V. Baier, and S. Hanish (1970) A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 7012 Naval Res. Lab. Washingtion, D.C..

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Referenced by:: §30.18(iii).
Encodings:: BibTeX

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B. J. King and A. L. Van Buren (1970) A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives. NRL Report No. 7161 Naval Res. Lab. Washingtion, D.C..

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Referenced by:: §30.18(iii).
Encodings:: BibTeX

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… ►A descent of a permutation is a pair of adjacent elements for which the first is larger than the second. … ►In this subsection

S\left(n,k\right)

is again the Stirling number of the second kind (§26.8), and

B_{m}

is the

m

th Bernoulli number (§24.2(i)). … ►

26.14.7

\genfrac{<}{>}{0.0pt}{}{n}{k}=\sum_{j=0}^{n-k}(-1)^{n-k-j}j!\genfrac{(}{)}{0.0% pt}{}{n-j}{k}S\left(n,j\right),

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(iii), §26.14 and Ch.26

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26.14.12

S\left(n,m\right)=\frac{1}{m!}\sum_{k=0}^{n-1}\genfrac{<}{>}{0.0pt}{}{n}{k}% \genfrac{(}{)}{0.0pt}{}{k}{n-m},

n\geq m

,

n\geq 1

.

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Symbols:: $\genfrac{<}{>}{0.0pt}{}{\NVar{n}}{\NVar{k}}$ : Eulerian number, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §26.14(iii), §26.14 and Ch.26

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See accompanying text — Figure 10.3.10: ${H^{(1)}_{0}}\left(x+iy\right)$ , $-10\leq x\leq 5$ , $-2.8\leq y\leq 4$ . … Magnify 3D Help

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B. C. Carlson (1964) Normal elliptic integrals of the first and second kinds. Duke Math. J. 31 (3), pp. 405–419.

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External Links:: Document, MathReview (S. Hitotumatu), ZentralBlatt
Referenced by:: §19.22(iii), §19.23, §19.33(i), §19.33(i).
Encodings:: BibTeX

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B. C. Carlson (1987) A table of elliptic integrals of the second kind. Math. Comp. 49 (180), pp. 595–606, S13–S17.

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External Links:: ISSN 0025-5718, Document, Supplement, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §19.29(i).
Encodings:: BibTeX

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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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H. S. Cohl, J. Park, and H. Volkmer (2021) Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.

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External Links:: Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (14.13.2), §14.3(iii), §14.3(iii).
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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P. Painlevé (1906) Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

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External Links:: ZentralBlatt
Referenced by:: §32.2(iv).
Encodings:: BibTeX

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F. A. Paxton and J. E. Rollin (1959) Tables of the Incomplete Elliptic Integrals of the First and Third Kind. Technical report Curtiss-Wright Corp., Research Division, Quehanna, PA.

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Notes:: Reviewed in Math. Comp. 14(1960)209-210.
Referenced by:: §19.37(iii).
Encodings:: BibTeX

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L. Piela (2014) Ideas of Quantum Chemistry. second edition, Elsevier, Amsterdam-New York.

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External Links:: ISBN 978-0-444-59436-5
Referenced by:: §18.39(ii).
Encodings:: BibTeX

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R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:: Double-precision Fortran, maximum accuracy 20S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986b) Integrals and Series: Special Functions, Vol. 2. Gordon & Breach Science Publishers, New York.

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Notes:: A second edition was published in 1988. Table errata: Math. Comp. v. 65 (1996), no. 215, pp. 1382–1383
External Links:: ISBN 2-88124-097-6, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.23(iv), §10.43(v), §10.43(vi), §10.44(iv), §10.60(iv), §10.71(iii), §12.12, §12.13(ii), §18.17(ix), §18.18(ix), §25.7, §25.8, §5.13, §6.14(iii), §6.15, §7.14(iii), §7.15, §8.14, §8.15.
Encodings:: BibTeX

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… ►For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). … ►Further improvements in accuracy can be realized by making a second application of the Euler transformation; see Olver (1997b, pp. 540–543). … ►For large

|z|

, with

|\operatorname{ph}z|\leq\frac{3}{2}\pi-\delta

(

<\frac{3}{2}\pi

), the Whittaker function of the second kind has the asymptotic expansion (§13.19) …With

z=1.0

,

\kappa=2.3

,

\mu=0.5

, the values of

a_{n}

to 8D are supplied in the second column of Table 2.11.1. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

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G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

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K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

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W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

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G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 2nd item, 1st item.
Encodings:: BibTeX

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W. E. Bleick and P. C. C. Wang (1974) Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (2), pp. 575–580.

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Notes:: Erratum: same journal v. 48 (1975), p. 518
External Links:: FullText, MathReview (H. A. Lauwerier), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

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A. Gil and J. Segura (2003) Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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External Links:: ISSN 0036-1429, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi), §3.8(v).
Encodings:: BibTeX

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K. Gottfried and T. Yan (2004) Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

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External Links:: ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt
Referenced by:: §18.39(i).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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J. Raynal (1979) On the definition and properties of generalized

6

-

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

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K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §19.36(i), §19.36(ii).
Encodings:: BibTeX

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F. Riesz and B. Sz.-Nagy (1990) Functional analysis. Dover Books on Advanced Mathematics, Dover Publications, Inc., New York.

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Notes:: Translated from the second French edition by Leo F. Boron, Reprint of the 1955 original
External Links:: ISBN 0-486-66289-6, MathReview Entry
Referenced by:: §1.18(i), §1.4(v), §1.4(v).
Encodings:: BibTeX

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Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

{K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}

{E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}

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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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Equations (14.5.3), (14.5.4)

The constraints in (14.5.3), (14.5.4) on $\nu+\mu$ have been corrected to exclude all negative integers since the Ferrers function of the second kind is not defined for these values.

Reported by Hans Volkmer on 2021-06-02

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Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

A note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind $U$ on the right-hand side of (7.18.10) was inserted.

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Chapters 8, 20, 36

Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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of%20the%20second%20kind

11—20 of 20 matching pages

11: Bibliography K

12: 26.14 Permutations: Order Notation

13: 10.3 Graphics

14: Bibliography C

15: Bibliography P

16: 2.11 Remainder Terms; Stokes Phenomenon

17: Bibliography B

18: Bibliography G

19: Bibliography R

20: Errata