relation to Airy function

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31: 18.35 Pollaczek Polynomials

… ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) …the recurrence relation of form (18.2.11_5) becomes … ►As in the coefficients of the above recurrence relations $n$ and $c$ only occur in the form $n+c$, the type 3 Pollaczek polynomials may also be called the associated type 2 Pollaczek polynomials by using the terminology of §18.30. … ►we have the explicit representations … ►This expansion is in terms of the Airy function $\mathrm{Ai}\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\mathrm{\infty })$. …

32: 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).

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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).

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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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ISSN 0048-6604, Document, MathReview (T. Erber)

Referenced by:

§9.14.

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Recurrence relations for hypergeometric functions of unit argument. Math. Comp. 45 (172), pp. 521–535.

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

Referenced by:

§16.4(iii).

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Monotonic sequences related to zeros of Bessel functions. Numer. Algorithms 49 (1-4), pp. 221–233.

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ISSN 1017-1398, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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33: Bibliography W

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Elliptic Functions According to Eisenstein and Kronecker. Classics in Mathematics, Springer-Verlag, Berlin.

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Notes:

Reprint of the 1976 original

External Links:

ISBN 3-540-65036-9, MathReview, ZentralBlatt

Referenced by:

§22.12, §23.8(ii).

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

External Links:

ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

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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).

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

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Four-figure tables of the Airy function in the complex plane. Philos. Mag. (7) 37, pp. 236–261.

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External Links:

Document, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

1st item.

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34: 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.

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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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ISSN 0098-3500, Document

Referenced by:

1st item.

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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).

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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).

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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).

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35: Bibliography O

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Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.

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ISSN 0377-0427, Document, MathReview (Boris Petrovich Osilenker), ZentralBlatt

Referenced by:

§25.8.

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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).

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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.

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External Links:

Document, MathReview (T. E. Hull), ZentralBlatt

Referenced by:

§13.19, §13.7(ii).

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

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Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94007-3; 0-387-95000-1, MathReview, ZentralBlatt

Referenced by:

§32.13(i).

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36: Software Index

… ►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

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A cross index of mathematical software in use at NIST.

37: 32.11 Asymptotic Approximations for Real Variables

… ►where … ►Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to $k\mathrm{Ai}\left(x\right)$, for some nonzero real $k$, where $\mathrm{Ai}$ denotes the Airy function (§9.2). Conversely, for any nonzero real $k$, there is a unique solution $w_k(x)$ of (32.11.4) that is asymptotic to $k\mathrm{Ai}\left(x\right)$ as $x\to +\mathrm{\infty }$. … ►The connection formulas relating (32.11.25) and (32.11.26) are … ►Now suppose $x\to -\mathrm{\infty }$. …

38: 2.8 Differential Equations with a Parameter

… ►These are elementary functions in Case I, and Airy functions (§9.2) in Case II. … ►Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that are $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and as $u\to \mathrm{\infty }$ … ►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 … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. …

39: 36.5 Stokes Sets

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$K=1$. Airy Function

… ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$-axis by $2\pi /3$. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

40: 36.12 Uniform Approximation of Integrals

… ►The function $g$ has a smooth amplitude. … ►Define a mapping $u(t;𝐲)$ by relating $f(u;𝐲)$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;𝐱)$ in the following way: … ►For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … ►For $\mathrm{Ai}$ and ${\mathrm{Ai}}^{\prime }$ see §9.2. …The coefficients of $\mathrm{Ai}$ and ${\mathrm{Ai}}^{\prime }$ are real if $y$ is real and $g$ is real analytic. …