Airy transform

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21: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
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Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
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… ►For example, the expression ${\mathrm{Ai}}^2+{\mathrm{Bi}}^2$ does not occur verbatim in DLMF, but ${\mathrm{Ai}}^2\left(z\right)+{\mathrm{Bi}}^2\left(z\right)$ and ${\mathrm{Ai}}^2\left(x\right)+{\mathrm{Bi}}^2\left(x\right)$ do. …

22: Bibliography B

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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External Links:

ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

BibTeX

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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:

ISSN 0893-9659, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

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External Links:

ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).

Encodings:

BibTeX

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Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

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23: Bibliography F

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Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Software 30 (4), pp. 471–490.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§9.17(ii), §9.17(v), §9.8(iv).

Encodings:

BibTeX

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

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

Referenced by:

3rd item, 2nd item, 1st item.

Encodings:

BibTeX

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On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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External Links:

ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

Encodings:

BibTeX

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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24: Bibliography T

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

Encodings:

BibTeX

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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:

ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

Encodings:

BibTeX

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

Encodings:

BibTeX

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

BibTeX

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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

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External Links:

ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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25: 1.17 Integral and Series Representations of the Dirac Delta

… ►Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9) (1.14.11), (1.14.10) (1.14.12), respectively. … ►

Airy Functions (§9.2)

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1.17.16 $$\delta \left(x-a\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{Ai}\left(t-x\right)\mathrm{Ai}\left(t-a\right)dt.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.17(ii)

Permalink:

http://dlmf.nist.gov/1.17.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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26: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

Encodings:

BibTeX

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A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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External Links:

ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)

Referenced by:

§18.16(iii).

Encodings:

BibTeX

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Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

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External Links:

Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(iii), §2.5(ii).

Encodings:

BibTeX

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:

ISSN 0096-3518, FullText, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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External Links:

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

Referenced by:

§9.13(i).

Encodings:

BibTeX

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27: Bibliography L

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Monotonicity properties of zeros of generalized Airy functions. Z. Angew. Math. Phys. 39 (2), pp. 267–271.

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External Links:

ISSN 0044-2275, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§9.13(i).

Encodings:

BibTeX

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Moment integrals of powers of Airy functions. Z. Angew. Math. Phys. 44 (5), pp. 891–908.

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External Links:

ISSN 0044-2275, Document, MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

(9.11.15), (9.11.18), §9.11(v), §9.11(v).

Encodings:

BibTeX

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The inhomogeneous Airy functions, $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ . J. Chem. Phys. 72 (1), pp. 332–336.

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External Links:

ISSN 0021-9606, Document, MathReview (V. L. Deshpande)

Referenced by:

(9.12.21), §9.12(vii), §9.17(iii).

Encodings:

BibTeX

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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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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:

ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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28: Bibliography O

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Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:

§1.14(viii).

Encodings:

BibTeX

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

Encodings:

BibTeX

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Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

External Links:

ISBN 0-387-05997-0, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).

Encodings:

BibTeX

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Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.

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External Links:

ISBN 0-387-06942-9, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §20.10(i), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii).

Encodings:

BibTeX

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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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