Airy equation

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21: 9.8 Modulus and Phase

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§9.8(i) Definitions

… ►Graphs of $M\left(x\right)$ and $N\left(x\right)$ are included in §9.3(i). … ►

§9.8(ii) Identities

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§9.8(iii) Monotonicity

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22: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

23: 32.5 Integral Equations

§32.5 Integral Equations

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32.5.1 $$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

►where $k$ is a real constant, and $\mathrm{Ai}\left(z\right)$ is defined in §9.2. … ►

32.5.3 $$w(z)\sim k\mathrm{Ai}\left(z\right),$$ $z\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $z$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

24: 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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25: Bibliography C

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A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives. Molecular Phys. 48 (6), pp. 1305–1330.

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External Links:

ISSN 0026-8976, Document, MathReview (Peter Giblin)

Referenced by:

§36.10(i), §36.10(ii), §36.15(v), §36.8.

Encodings:

BibTeX

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26: Bibliography M

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The Airy Integral, Giving Tables of Solutions of the Differential Equation $y^{\prime \prime }=xy$ . British Association for the Advancement of Science, Mathematical Tables Part-Vol. B, Cambridge University Press, Cambridge.

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External Links:

MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

1st item, 1st item, §9.2(i), §9.2(ii), §9.2(v), §9.4, (9.6.2), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.8(i), §9.8(ii), §9.8(iv), §9.9(ii), Ch.9.

Encodings:

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27: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

Encodings:

BibTeX

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Tables of Airy Functions and Their Derivatives. Izdat. Nauka, Moscow (Russian).

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Referenced by:

4th item.

Encodings:

BibTeX

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28: 13.6 Relations to Other Functions

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§13.6(iii) Modified Bessel Functions

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13.6.11 $$U(\frac{5}{6},\frac{5}{3},\frac{4}{3}{z}^{3/2})=\sqrt{\pi }\frac{3^{5/6}\exp \left(\frac{2}{3}{z}^{3/2}\right)}{2^{2/3}z}\mathrm{Ai}\left(z\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function and $z$: complex variable

A&S Ref:

13.6.25

Referenced by:

(9.10.18)

Permalink:

http://dlmf.nist.gov/13.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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13.6.11_1 $$M(\nu +\frac{1}{2},2\nu +1+n,2z)=\mathrm{\Gamma }\left(\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }\sum _{k=0}^n\frac{{\left(-n\right)}_k{\left(2\nu \right)}_k(\nu +k)}{{\left(2\nu +1+n\right)}_{k}k!}{I}_{\nu +k}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.6))

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.6.E11_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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13.6.11_2 $$M(\nu +\frac{1}{2},2\nu +1-n,2z)=\mathrm{\Gamma }\left(\nu -n\right){\mathrm{e}}^z{\left(z/2\right)}^{n-\nu }\sum _{k=0}^n{(-1)}^k\frac{{\left(-n\right)}_k{\left(2\nu -2n\right)}_k(\nu -n+k)}{{\left(2\nu +1-n\right)}_{k}k!}{I}_{\nu +k-n}\left(z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.7))

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.6.E11_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

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29: Guide to Searching the DLMF

… ►From there you can also access an advanced search page where you can control certain settings, narrowing the search to certain chapters, or restricting the results to equations, graphs, tables, or bibliographic items. … ►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. Therefore, if your query is Ai^2+Bi^2, the system modifies the query so it will find the equations containing the latter expressions. … ►To find more effectively the information you need, especially equations, you may at times wish to specify what you want with descriptive words that characterize the contents but do not occur literally. For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function. …

30: Bibliography

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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:

English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47

External Links:

ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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On the intensity of light in the neighbourhood of a caustic. Trans. Camb. Phil. Soc. 6, pp. 379–402.

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Referenced by:

§9.16, §9.16.

Encodings:

BibTeX

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Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

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Referenced by:

§9.16.

Encodings:

BibTeX

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Integrals involving Airy functions. J. Phys. A 19 (13), pp. 2663–2665.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

(9.11.11), (9.11.12), (9.11.13), (9.11.14), §9.11(iv).

Encodings:

BibTeX

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Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

(9.11.10), (9.11.5), (9.11.6), (9.11.7), (9.11.8), (9.11.9), §9.11(iv), §9.11(iv).

Encodings:

BibTeX

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