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21: 36 Integrals with Coalescing Saddles

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22: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

23: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

24: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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25: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.

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External Links:

ISSN 0010-3616, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.11(iii), §8.11(iii), Erratum (V1.1.10) for Section 8.11(iii).

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

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Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.

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External Links:

ISSN 0020-9910, Document, MathReview (Edward A. Bender), ZentralBlatt

Referenced by:

§26.12(i), §26.12(ii).

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26: Bibliography G

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

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Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26 (6), pp. 789–803.

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External Links:

ISSN 0747-7171, Document, MathReview (Gabino González Díez), ZentralBlatt

Referenced by:

1st item.

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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27: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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Asymptotic formula for the number of plane partitions of positive integers. C. R. Acad. Bulgare Sci. 59 (4), pp. 361–366.

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External Links:

ISSN 1310-1331, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§26.12(iv).

Encodings:

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28: Bibliography B

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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).

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

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Plane Algebraic Curves. Birkhäuser Verlag, Basel.

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Notes:

Translated from the German by John Stillwell

External Links:

ISBN 3-7643-1769-8, MathReview, ZentralBlatt

Referenced by:

§21.7(i).

Encodings:

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29: 5.21 Methods of Computation

… ►An effective way of computing $\mathrm{\Gamma }\left(z\right)$ in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …

30: 27.15 Chinese Remainder Theorem

… ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …