# generalized Airy functions

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##### 1: 9.13 Generalized Airy Functions
###### §9.13 GeneralizedAiryFunctions
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). …
##### 3: 9.18 Tables
• 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.

• ###### §9.18(vii) GeneralizedAiryFunctions
• 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
• 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.
• ##### 5: Bibliography H
• 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.
• ##### 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.
• ##### 7: Bibliography B
• P. Baldwin (1985) Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.
• ##### 8: Bibliography C
• 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.
• ##### 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 $z(\rho,\phi)=2\pi\left(\rho^{-1/3}u(\phi)\cos\left(\rho\widetilde{f}(\phi)% \right)\operatorname{Ai}\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho% \right))+\rho^{-2/3}v(\phi)\sin\left(\rho\widetilde{f}(\phi)\right)% \operatorname{Ai}'\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho\right% ))\right),$ $\rho\to\infty$.
##### 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. …