DLMF: Untitled Document

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K. W. J. Kadell (1994) A proof of the

q

-Macdonald-Morris conjecture for

BC_{n}

. Mem. Amer. Math. Soc. 108 (516), pp. vi+80.

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External Links:: ISSN 0065-9266, MathReview (Eric M. Opdam), ZentralBlatt
Referenced by:: §17.14.
Encodings:: BibTeX

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E. H. Kaufman and T. D. Lenker (1986) Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.

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External Links:: ISSN 0002-9890, FullText, MathReview (J. G. Herriot), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §24.15(iv).
Encodings:: BibTeX

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A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

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T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:: Document, MathReview (M. Mikolas), ZentralBlatt
Referenced by:: (18.14.8), §18.14(i), §18.17(ii), §18.18(ii).
Encodings:: BibTeX

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… ►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of

Q\left(a,x\right)

; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

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Subsection 17.9(iii)

The title of the paragraph which was previously “Gasper’s $q$ -Analog of Clausen’s Formula” has been changed to “Gasper’s $q$ -Analog of Clausen’s Formula (16.12.2)”.

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Paragraph Inversion Formula (in §35.2)

The wording was changed to make the integration variable more apparent.

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Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$ -argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$ , where $1\leq j,k\leq n$ . Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$ -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$ th-order differential equations. A reference to Ince (1926, §5.2) was added.

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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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References

Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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§36.5(ii) Cuspoids

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36.5.4

80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1=0,

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Symbols:: $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7

X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),

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Symbols:: $\operatorname{sign}\NVar{x}$ : sign of, $z$ : real parameter, $X$ : scaled coordinate and $Y$ : scaled coordinate
Permalink:: http://dlmf.nist.gov/36.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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§36.5(iii) Umbilics

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32.8.3

w(z;3)=\frac{3z^{2}}{z^{3}+4}-\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80},

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Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

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32.8.4

w(z;4)=-\frac{1}{z}+\frac{6z^{2}(z^{3}+10)}{z^{6}+20z^{3}-80}-\frac{9z^{5}(z^{% 3}+40)}{z^{9}+60z^{6}+11200}.

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Symbols:: $z$ : real
Permalink:: http://dlmf.nist.gov/32.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.8(ii), §32.8 and Ch.32

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Q_{3}(z)=z^{6}+20z^{3}-80,

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E. Jahnke and F. Emde (1945) Tables of Functions with Formulae and Curves. 4th edition, Dover Publications, New York.

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Notes:: Contains corrections and enlargements of earlier editions. Originally published by B. G. Teubner, Leipzig, in 1933. In German and English.
External Links:: MathReview, ZentralBlatt
Referenced by:: 2nd item, §20.15, §5.1, Notations P, E. Jahnke, F. Emde, and F. Lösch (1966).
Encodings:: BibTeX

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M. Jimbo, T. Miwa, Y. Môri, and M. Sato (1980) Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:: ISSN 0167-2789, Document, MathReview, ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

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X.-S. Jin and R. Wong (1999) Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:: ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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F. Johansson (2012) Efficient implementation of the Hardy-Ramanujan-Rademacher formula. LMS J. Comput. Math. 15, pp. 341–359.

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External Links:: ISSN 1461-1570, Document, Link, MathReview (Tim Huber), ZentralBlatt
Referenced by:: §27.20.
Encodings:: BibTeX

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B. R. Judd (1976) Modifications of Coulombic interactions by polarizable atoms. Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.

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External Links:: ISSN 0305-0041, Document, MathReview (Cataldo Agostinelli)
Referenced by:: §34.12.
Encodings:: BibTeX

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Table 26.6.1: Delannoy numbers

D(m,n)

.

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$m$	$n$
$m$	…
10	1	21	221	1561	8361	36365	1 34245	4 33905	12 56465	33 17445	80 97453

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Table 26.6.3: Narayana numbers

N(n,k)

.

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$n$	$k$
$n$	…
5	0	1	10	20	10	1
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Two-Point Formula

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Three-Point Formula

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Four-Point Formula

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Five-Point Formula

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Six-Point Formula

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V. I. Pagurova (1965) An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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Notes:: English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166
External Links:: Document, MathReview (E. Lukacs), ZentralBlatt
Referenced by:: §8.11(iv).
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. Poquérusse and S. Alexiou (1999) Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

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External Links:: Document
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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J. L. Powell (1947) Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

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External Links:: Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §33.4.
Encodings:: BibTeX

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T. Prellberg and A. L. Owczarek (1995) Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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External Links:: Document, ZentralBlatt
Referenced by:: §8.24(ii).
Encodings:: BibTeX

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Abel%E2%80%93Plana%20formula

11—20 of 317 matching pages

11: Bibliography K

12: 8.23 Statistical Applications

13: Errata

14: 36.5 Stokes Sets

§36.5(ii) Cuspoids

§36.5(iii) Umbilics

15: 32.8 Rational Solutions

16: 26.13 Permutations: Cycle Notation

17: Bibliography J

18: 26.6 Other Lattice Path Numbers

19: 3.4 Differentiation

Two-Point Formula

Three-Point Formula

Four-Point Formula

Five-Point Formula

Six-Point Formula

20: Bibliography P