incomplete Airy functions

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1: 9.14 Incomplete Airy Functions

§9.14 Incomplete Airy Functions

►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

2: Bibliography L

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L. Levey and L. B. Felsen (1969) On incomplete Airy functions and their application to diffraction problems. Radio Sci. 4 (10), pp. 959–969.

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External Links:: ISSN 0048-6604, Document, MathReview (T. Erber)
Referenced by:: §9.14.
Encodings:: BibTeX

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3: Bibliography C

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E. D. Constantinides and R. J. Marhefka (1993) Efficient and accurate computation of the incomplete Airy functions. Radio Science 28 (4), pp. 441–457.

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External Links:: ISSN 0048-6604, Document
Referenced by:: §9.14.
Encodings:: BibTeX

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4: 9.10 Integrals

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9.10.15

\int_{0}^{\infty}e^{-pt}\operatorname{Ai}\left(-t\right)\,\mathrm{d}t={\frac{1% }{3}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{% \Gamma\left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^% {3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)},

\Re p>0

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part and $p$ : parameter
Keywords:: Laplace transform
Source:: Gibbs (1973, Problem 72–21)
Permalink:: http://dlmf.nist.gov/9.10.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §9.10(v), §9.10 and Ch.9

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9.10.16

\int_{0}^{\infty}e^{-pt}\operatorname{Bi}\left(-t\right)\,\mathrm{d}t={\frac{1% }{\sqrt{3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}% \right)}{\Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(\tfrac{1}{3},% \tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)},

\Re p>0

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part and $p$ : parameter
Keywords:: Laplace transform
Source:: Gibbs (1973, Problem 72–21)
Permalink:: http://dlmf.nist.gov/9.10.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §9.10(v), §9.10 and Ch.9

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5: Bibliography G

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W. Gautschi (1964a) Algorithm 222: Incomplete beta function ratios. Comm. ACM 7 (3), pp. 143–144.

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Notes:: Variable-precision Algol program. For a Certification see same journal 7 (1964), p. 244.
External Links:: ISSN 0001-0782, Document, Certification
Referenced by:: §8.28(iv).
Encodings:: BibTeX

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W. Gautschi (1979a) Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0098-3500, Document, GAMS (module 542; package TOMS), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:: MathReview (H. O. Pollak), ZentralBlatt
Referenced by:: §5.6(i), §6.8, §7.8, §8.10.
Encodings:: BibTeX

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W. Gautschi (1979b) A computational procedure for incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 466–481.

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External Links:: ISSN 0098-3500, Document, ZentralBlatt
Referenced by:: §8.25(iv), §8.25(v).
Encodings:: BibTeX

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A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:: Problem proposed by P. Smith.
External Links:: Document
Referenced by:: (9.10.14), (9.10.15), (9.10.16), §9.10(v).
Encodings:: BibTeX

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6: Bibliography S

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Z. Schulten, D. G. M. Anderson, and R. G. Gordon (1979) An algorithm for the evaluation of the complex Airy functions. J. Comput. Phys. 31 (1), pp. 60–75.

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External Links:: ISSN 0021-9991, Document, MathReview, ZentralBlatt
Referenced by:: (9.10.18), (9.10.19), §9.17(iii), Erratum (V1.0.10) for Equation (9.10.18), Erratum (V1.0.10) for Equation (9.10.19).
Encodings:: BibTeX

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N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

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M. E. Sherry (1959) The zeros and maxima of the Airy function and its first derivative to 25 significant figures. Report AFCRC-TR-59-135, ASTIA Document No. AD214568 Air Research and Development Command, U.S. Air Force, Bedford, MA.

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Notes:: Available from U.S. Department of Commerce, Office of Technical Services, Washington, D.C.
Referenced by:: 2nd item, §9.8(iv).
Encodings:: BibTeX

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A. D. Smirnov (1960) Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.

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Notes:: Translated from the Russian by D. G. Fry
External Links:: MathReview (W. F. Freiberger), ZentralBlatt
Referenced by:: (9.13.19), (9.13.20), (9.13.21), §9.13(i), 1st item.
Encodings:: BibTeX

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P. Spellucci and P. Pulay (1975) Effective calculation of the incomplete gamma function for parameter values

\alpha=(2n+1)/2

n=0,\ldots,5

. Angew. Informatik 17, pp. 30–32.

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Notes:: Includes Fortran code.
External Links:: ZentralBlatt
Referenced by:: §8.28(ii).
Encodings:: BibTeX

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7: Bibliography T

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N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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External Links:: ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

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N. M. Temme (1987) On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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External Links:: FullText, MathReview (G. Meinardus), ZentralBlatt
Referenced by:: §8.25(iii).
Encodings:: BibTeX

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N. M. Temme (1992a) Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §8.12, §8.12.
Encodings:: BibTeX

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N. M. Temme (1992b) Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.

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Notes:: Asymptotic methods in analysis and combinatorics
External Links:: ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt
Referenced by:: §8.18(ii).
Encodings:: BibTeX

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N. M. Temme (1995a) Asymptotics of zeros of incomplete gamma functions. Ann. Numer. Math. 2 (1-4), pp. 415–423.

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Notes:: Special functions (Torino, 1993)
External Links:: ISSN 1021-2655, FullText, MathReview (E. R. Love), ZentralBlatt
Referenced by:: §8.13(ii).
Encodings:: BibTeX

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8: Software Index

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	Open Source	With Book	Commercial
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9 Airy and Related Functions
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

9: Bibliography N

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National Bureau of Standards (1958) Integrals of Airy Functions. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: 2nd item, 3rd item.
Encodings:: BibTeX

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G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

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G. Nemes (2015c) The resurgence properties of the incomplete gamma function II. Stud. Appl. Math. 135 (1), pp. 86–116.

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External Links:: ISSN 0022-2526, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii).
Encodings:: BibTeX

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G. Nemes (2016) The resurgence properties of the incomplete gamma function, I. Anal. Appl. (Singap.) 14 (5), pp. 631–677.

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External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.11(iii), §8.11(v), §8.11(v), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

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E. Neuman (2013) Inequalities and bounds for the incomplete gamma function. Results Math. 63 (3-4), pp. 1209–1214.

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External Links:: ISSN 1422-6383, Document, FullText, MathReview (Pierpaolo Natalini), ZentralBlatt
Referenced by:: §8.10, §8.10.
Encodings:: BibTeX

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10: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(iii) Error Bounds for Real Variables

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§9.7(iv) Error Bounds for Complex Variables

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§9.7(v) Exponentially-Improved Expansions

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