DLMF: Untitled Document

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D. F. Lawden (1989) Elliptic Functions and Applications. Applied Mathematical Sciences, Vol. 80, Springer-Verlag, New York.

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External Links:: ISBN 0-387-96965-9, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.2(ii), §19.36(iii), §20.13, §20.15, §20.2(i), §20.2(ii), §20.2(iii), §20.2(iv), §20.4(i), §20.5(i), §20.5(ii), §20.7(i), §20.7(ii), §20.7(iv), §20.7(vi), §20.7(vii), §20.7(vii), §20.7(viii), Ch.20, §22.10(i), §22.10(ii), §22.12, §22.13(i), §22.14(i), §22.14(ii), §22.14(iii), §22.14(iii), §22.15(i), §22.15(ii), §22.15(ii), §22.16(ii), §22.16(iii), §22.17(i), §22.18(i), §22.18(iii), §22.19(ii), §22.19(i), §22.19(i), §22.19(ii), §22.19(ii), §22.19(iv), §22.19(v), §22.2, §22.20(vii), §22.21, §22.4(i), §22.4(iii), §22.5(i), §22.5(ii), §22.6(ii), §22.6(iii), §22.6(iv), §22.7(i), §22.7(ii), §22.8(i), §22.8(ii), Ch.22, §23.10(i), §23.10(i), §23.10(iii), §23.10(iv), §23.12, §23.2(ii), §23.2(iii), §23.2(iii), §23.21(i), §23.3(i), §23.3(ii), §23.6(iv), §23.6(i), §23.6(i), §23.6(ii), §23.6(ii), §23.6(iii), §23.7, §23.8(i), §23.8(ii), §23.8(iii), Ch.23.
Encodings:: BibTeX

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D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

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A. M. Legendre (1808) Essai sur la Théorie des Nombres. 2nd edition, Courcier, Paris.

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External Links:: ZentralBlatt
Referenced by:: §27.2(i).
Encodings:: BibTeX

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M. Lerch (1903) Zur Theorie der Gaußschen Summen. Math. Ann. 57 (4), pp. 554–567 (German).

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External Links:: ISSN 0025-5831, Document, MathReview Entry, ZentralBlatt
Referenced by:: §20.11(i).
Encodings:: BibTeX

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R. L. Liboff (2003) Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. third edition, Springer, New York.

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External Links:: ISBN 0387955518
Referenced by:: §18.39(iii).
Encodings:: BibTeX

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Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

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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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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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M. N. Barber and B. W. Ninham (1970) Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.

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External Links:: ZentralBlatt
Referenced by:: §16.24(i).
Encodings:: BibTeX

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M. V. Berry (1969) Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

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Referenced by:: §36.14(i), §9.16.
Encodings:: BibTeX

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D. Bleichenbacher (1996) Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.

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External Links:: FullText
Referenced by:: 2nd item.
Encodings:: BibTeX

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I. Bloch, M. H. Hull, A. A. Broyles, W. G. Bouricius, B. E. Freeman, and G. Breit (1950) Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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External Links:: Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §33.7.
Encodings:: BibTeX

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I. D. Macdonald (1968) The Theory of Groups. Clarendon Press, Oxford.

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Notes:: Reprinted in 1988 by Krieger Publishing Co., Malabar, FL
External Links:: ISBN 0-19-853137-0;0-89464-287-1, MathReview (A. P. Street), ZentralBlatt
Referenced by:: §23.20(ii).
Encodings:: BibTeX

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L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

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S. C. Milne (1996) New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function. Proc. Nat. Acad. Sci. U.S.A. 93 (26), pp. 15004–15008.

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External Links:: ISSN 0027-8424, MathReview (Ken Ono), ZentralBlatt
Referenced by:: §27.13(iv).
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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L. J. Mordell (1917) On the representation of numbers as a sum of

2r

squares. Quarterly Journal of Math. 48, pp. 93–104.

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External Links:: ZentralBlatt
Referenced by:: §27.13(iv).
Encodings:: BibTeX

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… ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

,

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

,

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

,

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

,

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. …

… ►In other cases the general theory of (12.2.2) is available. … ►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

by

-ia

and

z

by

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

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

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B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

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B_{-3}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),

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B_{-2}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),

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B_{2}^{6}=-\tfrac{1}{60}(9+9t-30t^{2}-20t^{3}+5t^{4}+3t^{5}),

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B_{2}^{7}=-\tfrac{1}{240}(72+36t-267t^{2}-80t^{3}+90t^{4}+12t^{5}-7t^{6}),

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L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

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External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

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H. S. Carslaw (1930) Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:: Reprinted by Dover, New York, 1950.
External Links:: ZentralBlatt
Referenced by:: §6.16(i).
Encodings:: BibTeX

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S. Chandrasekhar (1984) The Mathematical Theory of Black Holes. In General Relativity and Gravitation (Padova, 1983), pp. 5–26.

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External Links:: MathReview Entry
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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E. W. Cheney (1982) Introduction to Approximation Theory. 2nd edition, Chelsea Publishing Co., New York.

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Notes:: Reprinted by AMS Chelsea Publishing, Providence, RI, 1998
External Links:: ISBN 0-8218-1374-9, MathReview, ZentralBlatt
Referenced by:: §18.38(i).
Encodings:: BibTeX

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D. Colton and R. Kress (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-62838-X, MathReview, ZentralBlatt
Referenced by:: §10.73(i).
Encodings:: BibTeX

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Fuchs%E2%80%93Frobenius%20theory

21—30 of 228 matching pages

21: Bibliography L

22: 9.18 Tables

23: 36.5 Stokes Sets

24: Bibliography B

25: Bibliography M

26: 10.31 Power Series

27: 1.11 Zeros of Polynomials

28: 12.14 The Function $W\left(a,x\right)$

29: 3.4 Differentiation

30: Bibliography C