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1: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
Swanson and Headley (1967) define independent solutions A n ( z ) and B n ( z ) 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 Ai ( x ) , Ai ( x ) for x = 20 ( .01 ) 2 ; log 10 Ai ( x ) , Ai ( x ) / Ai ( x ) for x = 0 ( .1 ) 25 ( 1 ) 75 ; Bi ( x ) , Bi ( x ) for x = 10 ( .1 ) 2.5 ; log 10 Bi ( x ) , Bi ( x ) / Bi ( x ) for x = 0 ( .1 ) 10 ; M ( x ) , N ( x ) , θ ( x ) , ϕ ( x ) (respectively F ( x ) , G ( x ) , χ ( x ) , ψ ( 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) Generalized Airy Functions
  • Smirnov (1960) tabulates U 1 ( x , α ) , U 2 ( x , α ) , defined by (9.13.20), (9.13.21), and also U 1 ( x , α ) / x , U 2 ( x , α ) / x , for α = 1 , x = 6 ( .01 ) 10 to 5D or 5S, and also for α = ± 1 4 , ± 1 3 , ± 1 2 , ± 2 3 , ± 3 4 , 5 4 , 4 3 , 3 2 , 5 3 , 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 ϵ ( p y ) + ( q + ϵ 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 ω as a function of wavevector 𝐤 : … Then with the definitions (36.12.12), and the real functions
    36.13.8 z ( ρ , ϕ ) = 2 π ( ρ 1 / 3 u ( ϕ ) cos ( ρ f ~ ( ϕ ) ) Ai ( ρ 2 / 3 Δ ( ϕ ) ) ( 1 + O ( 1 / ρ ) ) + ρ 2 / 3 v ( ϕ ) sin ( ρ f ~ ( ϕ ) ) Ai ( ρ 2 / 3 Δ ( ϕ ) ) ( 1 + O ( 1 / ρ ) ) ) , ρ .
    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 Ai ( z ) in the complex plane, once values of this function can be generated on the positive real axis. …