Abel%E2%80%93Plana formula

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$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
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12: 8.23 Statistical Applications

… ►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). …

13: Bibliography J

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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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14: Errata

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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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Usability

Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

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Subsection 14.18(iii)

This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

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15: 36.5 Stokes Sets

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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(iii) Umbilics

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16: 22.7 Landen Transformations

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17: 3.4 Differentiation

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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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18: Bibliography P

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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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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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19: 12.14 The Function $W\left(a,x\right)$

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§12.14(iv) Connection Formula

… ►Other expansions, involving

\cos\left(\tfrac{1}{4}x^{2}\right)

and

\sin\left(\tfrac{1}{4}x^{2}\right)

, can be obtained from (12.4.3) to (12.4.6) by replacing

a

-ia

and

z

xe^{\ifrac{\pi i}{4}}

; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). …

20: 10.31 Power Series

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