Airy function

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31: Bibliography Y

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

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Referenced by:

4th item.

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32: Bibliography R

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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Referenced by:

3rd item.

Encodings:

BibTeX

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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External Links:

ISSN 0098-3500, Document

Referenced by:

1st item.

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BibTeX

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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33: 36.13 Kelvin’s Ship-Wave Pattern

§36.13 Kelvin’s Ship-Wave Pattern

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36.13.8 $$z(\rho ,\varphi )=2\pi \left({\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))+{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\right),$$ $\rho \to \mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z(\varphi ,\rho )$: wave height approximant, $u(\varphi )$: function and $v(\varphi )$: function

Referenced by:

Figure 36.13.1, Figure 36.13.1

Permalink:

http://dlmf.nist.gov/36.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.13 and Ch.36

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34: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

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13.21.22 $$M_{\kappa ,\mu }\left(x\right)=\frac{1}{2\pi }\mathrm{\Gamma }\left(2\mu +1\right)\mathrm{\Gamma }\left(\kappa -\mu +\frac{1}{2}\right)\widehat{c}(\kappa ,\mu )\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\left(\sin \left(\kappa \pi -\mu \pi \right)\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)+\cos \left(\kappa \pi -\mu \pi \right)\mathrm{Bi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)+\mathrm{envBi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$ , $\sin z$: sine function, $x$: real variable, $\widehat{c}(\kappa ,\mu )$: function and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(iii), §13.21 and Ch.13

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13.21.24 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)={\mathrm{e}}^{(\kappa -\frac{1}{6})\pi \mathrm{i}}\widehat{c}(\kappa ,\mu )\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\left(\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }{\mathrm{e}}^{-\frac{2}{3}\pi \mathrm{i}}\right)+\mathrm{envBi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\widehat{c}(\kappa ,\mu )$: function and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(iii), §13.21 and Ch.13

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13.21.25 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)={\mathrm{e}}^{-(\kappa -\frac{1}{6})\pi \mathrm{i}}\widehat{c}(\kappa ,\mu )\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)\left(\mathrm{Ai}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }{\mathrm{e}}^{\frac{2}{3}\pi \mathrm{i}}\right)+\mathrm{envBi}\left({\kappa }^{\frac{2}{3}}\widehat{\zeta }\right)O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{envBi}\left(x\right)$: envelope of Airy function $\mathrm{Bi}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\widehat{c}(\kappa ,\mu )$: function and $\widehat{\mathrm{\Psi }}(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(iii), §13.21 and Ch.13

… ►For the functions $\mathrm{Ai}$ and $\mathrm{Bi}$ see §9.2(i), and for the $\mathrm{env}$ functions associated with $\mathrm{Ai}$ and $\mathrm{Bi}$ see §2.8(iii). … ►For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa ,\mu }\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu |$ bounded, and $x\in [\delta ,\mathrm{\infty })$ see Olver (1997b, Chapter 11, Ex. 7.3). …

35: 36.2 Catastrophes and Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): ►

36.2.13 $${\mathrm{\Psi }}_1\left(x\right)=\frac{2\pi }{3^{1/3}}\mathrm{Ai}\left(\frac{x}{3^{1/3}}\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real parameter

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E13

Encodings:

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See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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36.2.20 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)=2{\pi }^2{(\frac{2}{3})}^{2/3}\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{x+\mathrm{i}y}{{12}^{1/3}}\right)\mathrm{Bi}\left(\frac{x-\mathrm{i}y}{{12}^{1/3}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter and $x$: real parameter

Referenced by:

§36.2(ii), §36.7(iii), §36.8

Permalink:

http://dlmf.nist.gov/36.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

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36.2.21 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(x,y,0\right)=\frac{4{\pi }^2}{3^{2/3}}\mathrm{Ai}\left(\frac{x}{3^{1/3}}\right)\mathrm{Ai}\left(\frac{y}{3^{1/3}}\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $y$: real parameter and $x$: real parameter

Referenced by:

§36.2(ii)

Permalink:

http://dlmf.nist.gov/36.2.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(ii), §36.2 and Ch.36

►Addendum: For further special cases see §36.2(iv) …

36: 33.20 Expansions for Small $|ϵ|$

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§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

37: 34.8 Approximations for Large Parameters

… ►Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). …

38: Bibliography N

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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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Proofs of two conjectures on the real zeros of the cylinder and Airy functions. SIAM J. Math. Anal. 53 (4), pp. 4328–4349.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(10.18.17), (10.18.18), §10.18(iii), (9.8.22), (9.8.23), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv), Erratum (V1.1.8) for Section 10.18(iii).

Encodings:

BibTeX

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Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations $ϵ{(p{y}^{\prime })}^{\prime }+(q+ϵr)y=f$ . Pergamon Press, Oxford.

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Notes:

D. E. Brown, translator.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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39: 10.19 Asymptotic Expansions for Large Order

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10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

… ►Here $\mathrm{Ai}$ and $\mathrm{Bi}$ are the Airy functions (§9.2), and … ►

10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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40: 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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