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1: 9.9 Zeros
9.9.7 Ai ( a k ) = ( 1 ) k 1 V ( 3 8 π ( 4 k 1 ) ) ,
9.9.9 Ai ( a k ) = ( 1 ) k 1 W ( 3 8 π ( 4 k 3 ) ) .
2: 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.

  • 3: Bibliography R
  • M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.
  • M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.
  • W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.
  • W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.
  • W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.
  • 4: Bibliography G
  • W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • A. Gil, J. Segura, and N. M. Temme (2001) On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comp. 70 (235), pp. 1183–1194.
  • A. Gil, J. Segura, and N. M. Temme (2002c) Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • 5: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • National Bureau of Standards (1958) Integrals of Airy Functions. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • G. Nemes (2021) Proofs of two conjectures on the real zeros of the cylinder and Airy functions. SIAM J. Math. Anal. 53 (4), pp. 4328–4349.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 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.
  • 6: 30.9 Asymptotic Approximations and Expansions
    §30.9(i) Prolate Spheroidal Wave Functions
    For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). …
    §30.9(ii) Oblate Spheroidal Wave Functions
    For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …
    §30.9(iii) Other Approximations and Expansions
    7: 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.
  • B. J. Laurenzi (1993) Moment integrals of powers of Airy functions. Z. Angew. Math. Phys. 44 (5), pp. 891–908.
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • Soo-Y. Lee (1980) The inhomogeneous Airy functions, Gi ( z )  and Hi ( z ) . J. Chem. Phys. 72 (1), pp. 332–336.
  • L. Levey and L. B. Felsen (1969) On incomplete Airy functions and their application to diffraction problems. Radio Sci. 4 (10), pp. 959–969.
  • 8: 9.7 Asymptotic Expansions
    §9.7 Asymptotic Expansions
    §9.7(iii) Error Bounds for Real Variables
    §9.7(iv) Error Bounds for Complex Variables
    §9.7(v) Exponentially-Improved Expansions
    9: 12.11 Zeros
    §12.11(i) Distribution of Real Zeros
    For large negative values of a the real zeros of U ( a , x ) , U ( a , x ) , V ( a , x ) , and V ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …Here α = μ 4 3 a s , a s denoting the s th negative zero of the function Ai (see §9.9(i)). … where β = μ 4 3 a s , a s denoting the s th negative zero of the function Ai and … For further information, including associated functions, see Olver (1959).
    10: Bibliography M
  • A. J. MacLeod (1994) Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.
  • P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function Ai ( x ) . J. Comput. Phys. 99 (2), pp. 337–340.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • J. W. Miles (1980) The Second Painlevé Transcendent: A Nonlinear Airy Function. In Mechanics Today, Vol. 5, pp. 297–313.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.