Airy transform

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1: 9.10 Integrals

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§9.10(v) Laplace Transforms

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§9.10(vi) Mellin Transform

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§9.10(vii) Stieltjes Transforms

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§9.10(ix) Compendia

►For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).

2: Bibliography W

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The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt

Referenced by:

(9.10.13), §9.10(ix), §9.10(v).

Encodings:

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3: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. …

4: Bibliography G

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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:

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5: Bibliography S

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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:

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6: 2.8 Differential Equations with a Parameter

… ►First we apply the Liouville transformation (§1.13(iv)) to (2.8.1). … ►The transformed equation has the form … ►The transformed differential equation is … ►For $\mathrm{Ai}$ and $\mathrm{Bi}$ see §9.2. … ►of smallest absolute value, and define the envelopes of $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ by …

7: 9.12 Scorer Functions

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9.12.24 $$\mathrm{Hi}\left(z\right)=\frac{3^{-2/3}}{2{\pi }^2\mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)\mathrm{\Gamma }\left(-t\right){(3^{1/3}{\mathrm{e}}^{\pi \mathrm{i}}z)}^{t}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Keywords:

Mellin transform

Source:

Exton (1983, (2.1), p. 100)

Permalink:

http://dlmf.nist.gov/9.12.E24

Encodings:

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See also:

Annotations for §9.12(vii), §9.12(vii), §9.12 and Ch.9

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8: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

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Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.

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External Links:

ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

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Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-285-8, MathReview (S. Hitotumatu), ZentralBlatt

Referenced by:

§3.11(v).

Encodings:

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

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Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§15.8(v).

Encodings:

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9: Bibliography P

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:

ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.37.

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

Referenced by:

§1.14(vii).

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Inequalities for the zeros of the Airy functions. SIAM J. Math. Anal. 22 (1), pp. 260–267.

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External Links:

ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§9.9(iv).

Encodings:

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Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.

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External Links:

ISSN 0098-3500, Document, GAMS (module 498; package TOMS), ZentralBlatt

Referenced by:

1st item, 4th item.

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10: Bibliography J

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Numerical calculation of Bessel, Hankel and Airy functions. Computer Physics Communications 183 (3), pp. 506–519.

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External Links:

Document, FullText, ZentralBlatt

Referenced by:

§10.19(iii), §10.19(iii).

Encodings:

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A note on sampling expansion for a transform with parabolic cylinder kernel. Inform. Sci. 26 (2), pp. 155–158.

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External Links:

ISSN 0020-0255, Document, MathReview, ZentralBlatt

Referenced by:

§12.16.

Encodings:

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Fast Hankel transforms. Geophysical Prospecting 27 (4), pp. 876–901.

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External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

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