generalized Airy functions

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21: 1.17 Integral and Series Representations of the Dirac Delta

… ►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that … ►Hence comparison with (1.17.5) shows that (1.17.9) can be interpreted as a generalized integral (1.17.3) with … ►For a generalization of (1.17.14) see Maximon (1991). … ►

Airy Functions (§9.2)

… ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. …

22: 36.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$l,m,n$	integers.
…
$\mathrm{Ai}$, $\mathrm{Bi}$	Airy functions (§9.2).

►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;𝐱)$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$; canonical integrals ${\mathrm{\Psi }}_K\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$ generated by the catastrophes. (There is no standard nomenclature for these functions.)

23: Bibliography Y

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

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4th item.

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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External Links:

MathReview (B. M. Agrawal), ZentralBlatt

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§35.10.

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The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.

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Document, ZentralBlatt

Referenced by:

§25.18(i).

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Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

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24: 36.12 Uniform Approximation of Integrals

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§36.12(i) General Theory for Cuspoids

… ►The function $g$ has a smooth amplitude. … ►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. …

25: 10.72 Mathematical Applications

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§10.72(i) Differential Equations with Turning Points

►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). … ►If $f(z)$ has a double zero $z_0$, or more generally $z_0$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots }$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. … ►

26: 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 $x\to +\mathrm{\infty }$. … ►where … ►In the generic case …

27: Bibliography W

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

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§4.12.

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On a generalization of the functions $\mathrm{ber}$ x, $\mathrm{bei}$ x, $\ker$ x, $\mathrm{kei}$ x. Quart. J. Pure Appl. Math. 42, pp. 316–342.

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External Links:

ZentralBlatt

Referenced by:

§10.61(iii), §10.61(iv), §10.65(i), §10.65(ii), §10.65(iii), §10.68(iii).

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Smoothing of Stokes’s discontinuity for the generalized Bessel function. II. Proc. Roy. Soc. London Ser. A 455, pp. 3065–3084.

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ISSN 1364-5021, Document, MathReview, ZentralBlatt

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§10.46.

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

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1st item.

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The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.46.

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28: Bibliography T

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Bernoulli polynomials old and new: Generalizations and asymptotics. CWI Quarterly 8 (1), pp. 47–66.

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ISSN 0922-5366, FullText, MathReview (Michael Hauss), ZentralBlatt

Referenced by:

§24.11, §24.16(i).

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

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FullText, ZentralBlatt

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§12.21(ii).

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Special functions in phase space: Mathieu functions. J. Phys. A 31 (31), pp. 6725–6739.

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ISSN 0305-4470, Document, MathReview (Bing Hong Wang), ZentralBlatt

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8th item.

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

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29: 9.12 Scorer Functions

§9.12 Scorer Functions

… ►The general solution is given by …where $A$ and $B$ are arbitrary constants, $w_1(z)$ and $w_2(z)$ are any two linearly independent solutions of Airy’s equation (9.2.1), and $p(z)$ is any particular solution of (9.12.1). … ► $-\mathrm{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ on the interval . $\mathrm{Hi}\left(x\right)$ is a numerically satisfactory companion to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ on the interval . …

30: Philip J. Davis

… ►After being asked by Milton Abramowitz to work on the project, he chose to write the Chapter “Gamma Function and Related Functions. … ►Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. … ►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined. … ►DLMF users can rotate, rescale, zoom and otherwise explore mathematical function surfaces. The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …