DLMF: Untitled Document

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W. Magnus, F. Oberhettinger, and R. P. Soni (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edition, Springer-Verlag, New York-Berlin.

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Notes:: Table errata: Math. Comp. v. 23 (1969), p. 471, v. 24 (1970), pp. 240 and 505, v. 25 (1971), no. 113, p. 201, v. 29 (1975), no. 130, p. 672, v. 30 (1976), no. 135, pp. 677–678, v. 32 (1978), no. 141, pp. 319–320, v. 36 (1981), no. 153, pp. 315–317, v. 41 (1983), no. 164, pp. 775–776, and v. 81 (2012), no. 280, p. 2251
External Links:: MathReview (Mary L. Boas), ZentralBlatt
Referenced by:: §10.15, §10.22(vi), §10.32(iv), §10.38, §10.43(vi), §10.60(iv), §10.9(iv), §11.5(iii), Ch.11, §12.12, §12.5(iv), Ch.12, §13.10(vi), §13.23(iv), §14.1, §14.11, §14.12(ii), §14.18(iv), §14.25, §14.3(iii), §14.5(v), §14.7(iv), Ch.14, (25.5.21), §25.5, (25.6.1), (25.6.2), §25.6(i), §8.1, Notations B, Notations I, Notations P, Notations P, Notations Q, Notations Q.
Encodings:: BibTeX

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C. S. Meijer (1932) Über die asymptotische Entwicklung von

{\displaystyle\int_{0}^{-\infty-i(\arg w-\mu)}e^{\nu z-w\sinh z}dz}

,

{\displaystyle(-\frac{\pi}{2}<\mu<\frac{\pi}{2})}

für große Werte von

|w|

und

|\nu|

. I, II. Proc. Akad. Wet. Amsterdam 35, pp. 1170–1180, 1291–1303 (German).

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External Links:: ISSN 0370-0348, ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

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S. C. Milne (1985a) A

q

-analog of the

{}_{5}F_{4}(1)

summation theorem for hypergeometric series well-poised in

\mathit{SU}(n)

. Adv. in Math. 57 (1), pp. 14–33.

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External Links:: ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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S. C. Milne (1988) A

q

-analog of the Gauss summation theorem for hypergeometric series in

U(n)

. Adv. in Math. 72 (1), pp. 59–131.

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External Links:: ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

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S. C. Milne (1997) Balanced

\sideset{{}_{3}}{{}_{2}}{\mathop{\Theta}}

summation theorems for

U(n)

basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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

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J. Achenbach (1986) Numerik: Implementierung von Zylinderfunktionen. Elektrotechnik, Friedr. Vieweg & Sohn, Braunschweig-Wiesbaden.

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External Links:: ISBN 3-528-04462-4, MathReview, ZentralBlatt
Referenced by:: 5th item, 5th item.
Encodings:: BibTeX

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W. A. Al-Salam (1990) Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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External Links:: Link, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §18.3.
Encodings:: BibTeX

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H. Alzer and S. Qiu (2004) Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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External Links:: ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt
Referenced by:: §19.9(i).
Encodings:: BibTeX

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T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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External Links:: MathReview (N. G. de Bruijn), ZentralBlatt
Referenced by:: (25.11.41), (25.11.42), §25.11(xii).
Encodings:: BibTeX

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T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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External Links:: MathReview (Giovanni Coppola), ZentralBlatt
Referenced by:: (25.16.2), (25.16.4), §25.16(i), §25.16.
Encodings:: BibTeX

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S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:: ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt
Referenced by:: §35.7(ii), §35.8(iv), §35.8(v).
Encodings:: BibTeX

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P. G. L. Dirichlet (1837) Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, pp. 45–81 (German).

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Notes:: Reprinted in G. Lejeune Dirichlet’s Werke, Band I, pp. 313–342, Reimer, Berlin, 1889 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.
External Links:: ISBN 0-8284-0225-6, Link, MathReview
Referenced by:: §25.15(i).
Encodings:: BibTeX

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P. G. L. Dirichlet (1849) Über die Bestimmung der mittleren Werthe in der Zahlentheorie. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1849, pp. 69–83 (German).

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Notes:: Reprinted in G. Lejeune Dirichlet’s Werke, Band II, pp. 49–66, Reimer, Berlin, 1897 and with corrections in G. Lejeune Dirichlet’s Werke, AMS Chelsea Publishing Series, Providence, RI, 1969.
External Links:: ISBN 0-8284-0225-6, Link, MathReview
Referenced by:: §27.11.
Encodings:: BibTeX

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J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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External Links:: ZentralBlatt
Referenced by:: §15.4(ii).
Encodings:: BibTeX

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•

Abramowitz and Stegun (1964) tabulates: $\zeta\left(n\right)$ , $n=2,3,4,\dots$ , 20D (p. 811); $\operatorname{Li}_{2}\left(1-x\right)$ , $x=0(.01)0.5$ , 9D (p. 1005); $f(\theta)$ , $\theta=15^{\circ}(1^{\circ})30^{\circ}(2^{\circ})90^{\circ}(5^{\circ})180^{\circ}$ , $f(\theta)+\theta\ln\theta$ , $\theta=0(1^{\circ})15^{\circ}$ , 6D (p. 1006). Here $f(\theta)$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ , $x=-5(.05)25$ , to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$ . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$ , and §22.17 lists tables for some Dirichlet $L$ -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

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H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:: Reprint of the 1897 original, with a foreword by Igor Krichever.
External Links:: ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt
Referenced by:: §20.9(ii).
Encodings:: BibTeX

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M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.

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External Links:: Document
Referenced by:: §36.14(i).
Encodings:: BibTeX

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R. Bulirsch and J. Stoer (1968) II. Darstellung von Funktionen in Rechenautomaten. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.),

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External Links:: MathReview (E. Kreyszig)
Referenced by:: §19.2(iii), §19.36(ii).
Encodings:: BibTeX

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C. G. J. Jacobi (1829) Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumptibus fratrum Bornträger.

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Notes:: Reprinted in Werke, Vol. I, pp. 49–239, Herausgegeben von C. W. Borchardt, Berlin, Verlag von G. Reimer, 1881.
External Links:: Werke TOC
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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H. von Koch (1901) Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).

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External Links:: ISSN 0025-5831; 1432-1807/e, Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.12.
Encodings:: BibTeX

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D. Hilbert (1909) Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl

n^{\mathrm{ter}}

Potenzen (Waringsches Problem). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, pp. 17–36 (German).

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Notes:: Also in Math. Annalen (1909), v. 67, pp. 281–300.
External Links:: Document, MathReview Entry, ZentralBlatt
Referenced by:: §27.13(iii).
Encodings:: BibTeX

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G. W. Hill (1981) Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions:

I_{1}(x)/I_{0}(x)

,

I_{1.5}(x)/I_{0.5}(x)

and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

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Notes:: Single-precision Fortran, minimum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 571; package TOMS)
Referenced by:: 5th item.
Encodings:: BibTeX

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§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences

x\equiv a_{1}\pmod{m_{1}},\dots,x\equiv a_{k}\pmod{m_{k}}

, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod

m

), where

m

is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod

m_{1}

), (mod

m_{2}

), (mod

m_{3}

), and (mod

m_{4}

), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

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§25.21(vi) Clausen’s Integral

…

von Staudt–Clausen theorem

11—20 of 130 matching pages

11: Bibliography M

12: Bibliography

13: Bibliography D

14: 25.19 Tables

15: Bibliography B

16: Bibliography J

17: Bibliography V

18: Bibliography H

19: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

20: 25.21 Software

§25.21(vi) Clausen’s Integral