generalized Airy functions

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11: 10.16 Relations to Other Functions

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Elementary Functions

… ►For these and general results when $\nu$ is half an odd integer see §§10.47(ii) and 10.49(i). ►

Airy Functions

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Parabolic Cylinder Functions

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Generalized Hypergeometric Functions

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12: 10.39 Relations to Other Functions

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Elementary Functions

… ►For these and general results when $\nu$ is half an odd integer see §§10.47(ii) and 10.49(ii). ►

Airy Functions

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Parabolic Cylinder Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

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13: 18.15 Asymptotic Approximations

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18.15.22 $$L_n^{(\alpha )}\left(\nu x\right)={(-1)}^n\frac{{\mathrm{e}}^{\frac{1}{2}\nu x}}{2^{\alpha -\frac{1}{2}}{x}^{\frac{1}{2}\alpha +\frac{1}{4}}}{\left(\frac{\zeta }{x-1}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{m=0}^{M-1}\frac{E_m(\zeta )}{{\nu }^{2m}}+\frac{{\mathrm{Ai}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{m=0}^{M-1}\frac{F_m(\zeta )}{{\nu }^{2m}}+\mathrm{envAi}\left({\nu }^{\frac{2}{3}}\zeta \right)O\left(\frac{1}{{\nu }^{2M-\frac{2}{3}}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathrm{envAi}\left(x\right)$: envelope of Airy function $\mathrm{Ai}\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$ , $\zeta$ , $E_m$: coefficient, $F_m$: coefficient and $x$: real variable

Referenced by:

Erratum (V1.0.11) for Equation (18.15.22)

Permalink:

http://dlmf.nist.gov/18.15.E22

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

See also:

Annotations for §18.15(iv), §18.15(iv), §18.15 and Ch.18

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14: 9.16 Physical Applications

§9.16 Physical Applications

►Airy functions are applied in many branches of both classical and quantum physics. … ►In fluid dynamics, Airy functions enter several topics. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). ►Airy functions play a prominent role in problems defined by nonlinear wave equations. …

15: 9.19 Approximations

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Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

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16: 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.

Encodings:

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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17: 18.32 OP’s with Respect to Freud Weights

… ►A Freud weight is a weight function of the form …For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). ►Generalized Freud weights have the form … ►For (generalized) Freud weights on a subinterval of $[0,\mathrm{\infty })$ see also Levin and Lubinsky (2005).

18: Bibliography M

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Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Two-point quasi-fractional approximations to the Airy function $\mathrm{Ai}(x)$ . J. Comput. Phys. 99 (2), pp. 337–340.

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Notes:

There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$.

External Links:

ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford Science Publications, The Clarendon Press Oxford University Press, New York.

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External Links:

ISBN 0-19-853595-3, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§16.18, §16.19, §16.20.

Encodings:

BibTeX

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The Second Painlevé Transcendent: A Nonlinear Airy Function. In Mechanics Today, Vol. 5, pp. 297–313.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

Encodings:

BibTeX

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The generalized integro-exponential function. Math. Comp. 44 (170), pp. 443–458.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt

Referenced by:

§8.19(v), §8.19(xi).

Encodings:

BibTeX

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19: 16.18 Special Cases

§16.18 Special Cases

►The ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions introduced in Chapters 13 and 15, as well as the more general ${}_p{}^{}F_q^{}$ functions introduced in the present chapter, are all special cases of the Meijer $G$-function. …As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).

20: 36.5 Stokes Sets

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$K=1$. Airy Function

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9). …For $\left|Y\right|>Y_1$ the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation … ►the intersection lines with the bifurcation set are generated by $|X|=X_2=0.45148$, $Y=Y_2=0.59693$. …