generalized Airy functions

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1: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

… ►Swanson and Headley (1967) define independent solutions

A_{n}\left(z\right)

and

B_{n}\left(z\right)

of (9.13.1) by … ► … ►Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … ►

2: 9.1 Special Notation

… ► ►►

$k$	nonnegative integer, except in §9.9(iii).
…

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3: 9.18 Tables

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Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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§9.18(vii) Generalized Airy Functions

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Smirnov (1960) tabulates $U_{1}(x,\alpha)$ , $U_{2}(x,\alpha)$ , defined by (9.13.20), (9.13.21), and also $\ifrac{\partial U_{1}(x,\alpha)}{\partial x}$ , $\ifrac{\partial U_{2}(x,\alpha)}{\partial x}$ , for $\alpha=1$ , $x=-6(.01)10$ to 5D or 5S, and also for $\alpha=\pm\tfrac{1}{4}$ , $\pm\tfrac{1}{3}$ , $\pm\tfrac{1}{2}$ , $\pm\tfrac{2}{3}$ , $\pm\tfrac{3}{4}$ , $\tfrac{5}{4}$ , $\tfrac{4}{3}$ , $\tfrac{3}{2}$ , $\tfrac{5}{3}$ , $\tfrac{7}{4}$ , 2, $x=0(.01)6$ ; 4D.

4: Bibliography N

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L. N. Nosova and S. A. Tumarkin (1965) Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations

\epsilon(py^{\prime})^{\prime}+(q+\epsilon r)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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5: Bibliography H

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V. B. Headley and V. K. Barwell (1975) On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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External Links:: FullText, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §9.13(i).
Encodings:: BibTeX

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6: Bibliography L

… ►

A. Laforgia and M. E. Muldoon (1988) 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).
Encodings:: BibTeX

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7: Bibliography B

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P. Baldwin (1985) Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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External Links:: ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §9.13(ii).
Encodings:: BibTeX

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8: Bibliography C

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R. C. Y. Chin and G. W. Hedstrom (1978) A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.

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External Links:: ISSN 0025-5718, FullText, MathReview (Philip Brenner), ZentralBlatt
Referenced by:: §9.13(ii).
Encodings:: BibTeX

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9: 36.13 Kelvin’s Ship-Wave Pattern

§36.13 Kelvin’s Ship-Wave Pattern

►A ship moving with constant speed

V

on deep water generates a surface gravity wave. … ►The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency

\omega

as a function of wavevector

\mathbf{k}

: … ►Then with the definitions (36.12.12), and the real functions … ►

36.13.8

z(\rho,\phi)=2\pi\left(\rho^{-1/3}u(\phi)\cos\left(\rho\widetilde{f}(\phi)% \right)\operatorname{Ai}\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho% \right))+\rho^{-2/3}v(\phi)\sin\left(\rho\widetilde{f}(\phi)\right)% \operatorname{Ai}'\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho\right% ))\right),

\rho\to\infty

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z(\NVar{\phi},\NVar{\rho})$ : wave height approximant, $u(\phi)$ : function and $v(\phi)$ : 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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10: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing

\operatorname{Ai}\left(z\right)

in the complex plane, once values of this function can be generated on the positive real axis. …