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11: Bibliography N

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A. Natarajan and N. Mohankumar (1993) On the numerical evaluation of the generalised Fermi-Dirac integrals. Comput. Phys. Comm. 76 (1), pp. 48–50.

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External Links:: ISSN 0010-4655, Document, ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

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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

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… ►They were selected as recognized leaders in the research communities interested in the mathematics and applications of special functions and orthogonal polynomials; in the presentation of mathematics reference information online and in handbooks; and in the presentation of mathematics on the web. …

13: Bibliography D

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G. M. D’Ariano, C. Macchiavello, and M. G. A. Paris (1994) Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (5), pp. 4298–4302.

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External Links:: Document
Referenced by:: §13.28(iii).
Encodings:: BibTeX

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K. Dilcher, L. Skula, and I. Sh. Slavutskiǐ (1991) Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:: A frequently updated version, with links to reviews, exists online
External Links:: Online version, MathReview, ZentralBlatt
Referenced by:: Ch.24.
Encodings:: BibTeX

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B. I. Dunlap and B. R. Judd (1975) Novel identities for simple

n

j

symbols. J. Mathematical Phys. 16, pp. 318–319.

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External Links:: Document, MathReview (M. J. Westwater)
Referenced by:: §34.5(vi).
Encodings:: BibTeX

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P. L. Duren (1991) The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

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External Links:: ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

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14: Bibliography W

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J. V. Wehausen and E. V. Laitone (1960) Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:: Online edition, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

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S. W. Weinberg (2013) Lectures on Quantum Mechanics. Cambridge University Press, Cambridge, UK.

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External Links:: ISBN 978-1-107-02872-2
Referenced by:: §18.39(ii).
Encodings:: BibTeX

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H. S. Wilf and D. Zeilberger (1992a) An algorithmic proof theory for hypergeometric (ordinary and “

q

”) multisum/integral identities. Invent. Math. 108, pp. 575–633.

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External Links:: ISSN 0020-9910, Document, MathReview (David R. Masson), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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H. S. Wilf and D. Zeilberger (1992b) Rational function certification of multisum/integral/“

q

” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

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External Links:: ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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15: Bibliography L

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J. Lepowsky and S. Milne (1978) Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:: ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

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J. Lepowsky and R. L. Wilson (1982) A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:: ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt
Referenced by:: §17.16.
Encodings:: BibTeX

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L. Lorch (1992) On Bessel functions of equal order and argument. Rend. Sem. Mat. Univ. Politec. Torino 50 (2), pp. 209–216 (1993).

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External Links:: ISSN 0373-1243, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

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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

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16: Errata

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Subsection 25.10(ii)

In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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Equation (3.3.34)

In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

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Subsection 5.2(iii)

Three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection.

Suggested by Tom Koornwinder.

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Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

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References

Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

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17: Bibliography M

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I. G. Macdonald (1990) Hypergeometric Functions.

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Notes:: Unpublished Lecture Notes, University of London. An online version exists.
External Links:: FullText
Referenced by:: §35.7(ii), §35.8(iii).
Encodings:: BibTeX

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Maple (commercial interactive system) Maplesoft.

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Notes:: General purpose mathematical computing environment. Includes more than 100 special functions. Includes symbolic and arbitrary precision numerical computation.
Referenced by:: 1st item, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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J. Miller and V. S. Adamchik (1998) Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: (25.11.21), (25.11.22), (25.11.23), §25.11, §25.11(vi), §25.18(i), (25.6.15).
Encodings:: BibTeX

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S. C. Milne (1985b) An elementary proof of the Macdonald identities for

A_{l}^{(1)}

. Adv. in Math. 57 (1), pp. 34–70.

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External Links:: ISSN 0001-8708, Document, MathReview (Y. Lehrer-Ilamed), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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G. W. Morgenthaler and H. Reismann (1963) Zeros of first derivatives of Bessel functions of the first kind,

J_{n}^{\prime}\,(x)

21\leq n\leq 51

0\leq x\leq 100

. J. Res. Nat. Bur. Standards Sect. B 67B (3), pp. 181–183.

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External Links:: ISSN 0160-1741, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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18: 25.10 Zeros

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25.10.1

Z(t)\equiv\exp\left(i\vartheta(t)\right)\zeta\left(\tfrac{1}{2}+it\right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\equiv$ : equals by definition, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $Z(t)$ : zeros function and $\vartheta(t)$ : function
Keywords:: definition
Source:: Titchmarsh (1986b, (4.17.2), p. 89)
Permalink:: http://dlmf.nist.gov/25.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(i), §25.10 and Ch.25

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25.10.2

\vartheta(t)\equiv\operatorname{ph}\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}it% \right)-\tfrac{1}{2}t\ln\pi

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\equiv$ : equals by definition, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\vartheta(t)$ : function and $\chi(s)$ : function
Keywords:: definition
Proof sketch:: Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta(t)\equiv-\frac{1}{2}\operatorname{ph}\chi(\frac{1}{2}+\mathrm{i}t)$ , by using (25.9.2), (1.9.18), (1.9.20), $\Gamma\left(\overline{z}\right)=\overline{\Gamma\left(z\right)}$ , and (4.8.10).
Permalink:: http://dlmf.nist.gov/25.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.10(i), §25.10 and Ch.25

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

19: Bibliography C

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CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.

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Notes:: Computer Algebra and Orthogonal Polynomials: a Web service using Maple for online generation of formulas and graphs of orthogonal polynomials belonging to the Askey scheme. For further information see Swarttouw (1997) and Koepf (1999).
External Links:: CAOP
Referenced by:: 1st item.
Encodings:: BibTeX

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F. Chapeau-Blondeau and A. Monir (2002) Numerical evaluation of the Lambert

W

function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.

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External Links:: ISSN 1053-587X, Document, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §4.44, §4.45(iii).
Encodings:: BibTeX

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T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:: MathReview (I. I. Hirschman, Jr.), ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

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20: Bibliography V

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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

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