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31: 36 Integrals with Coalescing Saddles

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32: 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. …

33: 18.40 Methods of Computation

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§18.40(i) Computation of Polynomials

►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. … … ►Results of low (

~{}2

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

20

. …

34: 33.24 Tables

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•

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ 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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35: Bibliography

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M. J. Ablowitz and H. Segur (1977) 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.
Encodings:: BibTeX

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A. Adelberg (1992) 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:: BibTeX

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S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:: Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.
External Links:: Code, Document
Referenced by:: 1st item.
Encodings:: BibTeX

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D. E. Amos (1989) 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:: BibTeX

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T. M. Apostol and H. S. Zuckerman (1951) On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.

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External Links:: ISSN 0002-9939, Fulltext, MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §27.17.
Encodings:: BibTeX

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36: Bibliography S

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K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:: Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.
External Links:: ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt
Referenced by:: §3.3(vi), §3.4(i), §3.5(i).
Encodings:: BibTeX

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W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:: Abstract
Referenced by:: §18.16(v).
Encodings:: BibTeX

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37: Bibliography O

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A. B. Olde Daalhuis and N. M. Temme (1994) Uniform Airy-type expansions of integrals. SIAM J. Math. Anal. 25 (2), pp. 304–321.

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External Links:: ISSN 0036-1410, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1998c) On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

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A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:: Document, MathReview (David L. Elliott), ZentralBlatt
Referenced by:: §3.11(ii).
Encodings:: BibTeX

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F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:: ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt
Referenced by:: §13.20(iii), §13.20(iv).
Encodings:: BibTeX

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38: Bibliography G

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W. Gautschi (1994) 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).
Encodings:: BibTeX

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W. Gautschi (2004) Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation, Oxford University Press, New York.

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Notes:: OPQ, a package of Matlab routines for generating classical and Sobolev orthogonal polynomials, accompanies this book.
External Links:: OPQ, ISBN 0-19-850672-4, MathReview Entry, ZentralBlatt
Referenced by:: (18.2.27), (18.2.28), (18.2.29), (18.2.30), §18.2(ix), §18.40(ii), §18.40(ii), §18.40(i), 3rd item, §3.5(v), Erratum (V1.0.28) for Table 18.3.1.
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2003a) Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion. J. Comput. Appl. Math. 153 (1-2), pp. 225–234.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.45, §10.74(viii).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2014) 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).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) 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:: BibTeX

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39: Bibliography C

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R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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A. Ciarkowski (1989) Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point. Proc. Roy. Soc. London Ser. A 426, pp. 273–286.

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External Links:: ISSN 0080-4630, FullText, MathReview, ZentralBlatt
Referenced by:: §2.4(vi).
Encodings:: BibTeX

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J. P. Coleman (1987) Polynomial approximations in the complex plane. J. Comput. Appl. Math. 18 (2), pp. 193–211.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

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M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

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40: 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

\pmod{m}

, which is correct to 20 digits. …