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21: How to Cite

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NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

… ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …

22: 9 Airy and Related Functions

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23: 2.2 Transcendental Equations

… ►In place of (2.2.1) assume that ►

2.2.7 $$f(x)\sim x+f_0+f_1{x}^{-1}+f_2{x}^{-2}+\mathrm{\cdots },$$ $x\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: function and $f_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

… ►where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). Conditions for the validity of the reversion process in $ℂ$ are derived in Olver (1997b, pp. 14–16). Applications to real and complex zeros of Airy functions are given in Fabijonas and Olver (1999). …

24: 19 Elliptic Integrals

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25: 12 Parabolic Cylinder Functions

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26: Philip J. Davis

… ►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. … ►In 1961, Davis hired Frank W. J. Olver, a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. …Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. …Decades later, Olver became Editor-in-Chief and Mathematics Editor of the NIST Digital Library of Mathematical Functions (DLMF), a complete revision of A&S that was publicly released in 2010. … ►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined. …

27: Preface

… ►Olver (University of Maryland, College Park, and NIST), Daniel W. …In addition, Olver was author or co-author of five chapters. … ►Olver, D. …Olver, W. … ►Frank W. J. Olver …

28: Bibliography L

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A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.

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External Links:

ISSN 0305-4470, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.

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External Links:

ISSN 0022-0396, Document, MathReview (John Albert), ZentralBlatt

Referenced by:

§22.19(iii).

Encodings:

BibTeX

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Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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External Links:

ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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Airy and Bessel Functions by Parallel Integration of ODEs. In Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, R. F. Sincovec, D. E. Keyes, M. R. Leuze, L. R. Petzold, and D. A. Reed (Eds.), Philadelphia, PA, pp. 530–538.

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External Links:

ISBN 0-89871-315-3

Referenced by:

§10.74(ii), §3.7(iii), §9.17(ii).

Encodings:

BibTeX

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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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29: Bibliography

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Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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Notes:

Corrections appeared in later printings up to the 10th Printing, December, 1972. Reproductions by other publishers, in whole or in part, have been available since 1965.

External Links:

FullText, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§10.1, §10.21(ix), §10.47(i), §10.68(iv), §10.70, 4th item, 2nd item, 4th item, 5th item, 1st item, 1st item, 4th item, 1st item, 1st item, Ch.10, §11.10(vi), 1st item, 1st item, Ch.11, 1st item, Ch.12, 3rd item, Ch.13, 1st item, Ch.14, Ch.15, §18.14(i), §18.3, §18.41(i), §18.41(ii), §18.5(iv), §18.8, §19.1, §19.14(i), §19.14(ii), §19.25(v), §19.36(iii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.38, Ch.19, §20.15, Ch.20, §22.1, §22.15(ii), Ch.22, §23.1, §23.20(i), §23.23, §23.23, §23.9, §23.9, §23.9, Ch.23, §24.2(iv), §24.20, Ch.24, 1st item, §26.1, §26.1, §26.2, §26.21, §26.3(i), §26.4(i), §26.8(i), §27.2(ii), §27.21, §28.1, Ch.28, §30.1, Ch.30, 1st item, Ch.33, §4.44, §4.46, §5.11(i), §5.22(ii), §5.22(iii), §5.7(i), Ch.5, 1st item, 1st item, 1st item, Ch.6, 1st item, 2nd item, 1st item, Ch.7, (8.17.24), §8.22(ii), 1st item, Ch.8, 1st item, 1st item, 1st item, §9.8(i), Ch.9, Profile Frank W. J. Olver, About the Project, Preface, Possible Errors in DLMF, Notations E, Notations E, Notations F, Notations F, Notations H, Notations H, Notations J, Notations K, Notations P, Notations P, Notations S, Notations Y, H. E. Fettis and J. C. Caslin (1973).

Encodings:

BibTeX

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Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. 50 (4), pp. 419–428.

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External Links:

ISSN 0029-599X, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§8.25(ii).

Encodings:

BibTeX

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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument. ACM Trans. Math. Software 16 (2), pp. 178–182.

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Notes:

Double- and single-precision Fortran, maximum accuracy 18S or 14S.

External Links:

ISSN 0098-3500, Document, GAMS (module 683; package TOMS), MathReview, ZentralBlatt

Referenced by:

1st item, 1st item.

Encodings:

BibTeX

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Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§29.15(ii), 2nd item.

Encodings:

BibTeX

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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt

Referenced by:

§18.15(vi), §18.15(vi).

Encodings:

BibTeX

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30: 10 Bessel Functions

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