Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the distribution of the zeros of generalized Airy functions


Authors: V. B. Headley and V. K. Barwell
Journal: Math. Comp. 29 (1975), 863-877
MSC: Primary 65D20; Secondary 33A70
DOI: https://doi.org/10.1090/S0025-5718-1975-0378360-3
MathSciNet review: 0378360
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give tables of zeros and values of the generalized Airy functions introduced by Swanson and Headley [SIAM J. Appl. Math., v. 15, 1967, pp. 1400-1412]. The tables enable us to sharpen substantially results on the distribution of the zeros. We show that the nonreal zeros are asymptotically close to the boundary rays of the sectors obtained in the paper cited. We conjecture from the numerical evidence that the zeros monotonically approach the rays.


References [Enhancements On Off] (What's this?)

  • [1] H. BREMMER, Terrestrial Radio Waves. Theory of Propagation, Elsevier, New York, 1949. MR 11, 295. MR 0032462 (11:295a)
  • [2] DONALD LUDWIG, "Uniform asymptotic expansions for wave propagation and diffraction problems," SIAM Rev., v. 12, 1970, pp. 325-331. MR 42 #1407. MR 0266502 (42:1407)
  • [3] J. C. P. MILLER, The Airy Integral, Giving Tables of Solutions of the Differential Equation $ y' = xy$, British Association for the Advancement of Science, Mathematical Tables, Part-volume B., Cambridge Univ. Press, Cambridge, 1946. MR 8, 353. MR 0018971 (8:353c)
  • [4] F. W. J. OLVER, "A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations," Proc. Cambridge Philos. Soc., v. 46, 1950, pp. 570-580. MR 12, 288. MR 0037609 (12:288b)
  • [5] F. W. J. OLVER, "A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order," Proc. Cambridge Philos. Soc., v. 47, 1951, pp. 699-712. MR 13, 283. MR 0043551 (13:283c)
  • [6] F. W. J. OLVER, "The asymptotic expansion of Bessel functions of large order," Philos. Trans. Roy. Soc. London Ser. A, v. 247, 1954, pp. 328-368. MR 16, 696. MR 0067250 (16:696a)
  • [7] F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, New York and London, 1974. MR 0435697 (55:8655)
  • [8] C. A. SWANSON, Properties of Airy Functions, Technical Report, California Institute of Technology, Pasadena, Calif., 1956.
  • [9] C. A. SWANSON & V. B. HEADLEY, "An extension of Airy's equation," SIAM J. Appl. Math., v. 15, 1967, pp. 1400-1412. MR 37 #482. MR 0224883 (37:482)
  • [10] WOLFGANG WASOW, Asymptotic Expansions for Ordinary Differential Equations, Wiley, New York and London, 1965. MR 34 #3041. MR 0203188 (34:3041)
  • [11] G. N. WATSON, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. MR 1349110 (96i:33010)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D20, 33A70

Retrieve articles in all journals with MSC: 65D20, 33A70


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0378360-3
Keywords: Generalized Airy functions, tables of zeros, asymptotic distribution of zeros, turning points
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society