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

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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.

**[1]**H. Bremmer,*Terrestrial Radio Waves. Theory of Propagation*, Elsevier Publishing Company, Inc., New York, N. Y., Amsterdam, London, Brussels, 1949. MR**0032462****[2]**Donald Ludwig,*Uniform asymptotic expansions for wave propagation and diffracton problems*, SIAM Rev.**12**(1970), 325–331. MR**0266502**, https://doi.org/10.1137/1012077**[3]**J. C. P. Miller,*The Airy Integral, Giving Tables of Solutions of the Differential Equation 𝑦”=𝑥𝑦*, Cambridge, at the University Press; New York, The Macmillan Company, 1946. MR**0018971****[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.**46**(1950), 570–580. MR**0037609****[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.**47**(1951), 699–712. MR**0043551****[6]**F. W. J. Olver,*The asymptotic expansion of Bessel functions of large order*, Philos. Trans. Roy. Soc. London. Ser. A.**247**(1954), 328–368. MR**0067250**, https://doi.org/10.1098/rsta.1954.0021**[7]**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697****[8]**C. A. SWANSON,*Properties of Airy Functions*, Technical Report, California Institute of Technology, Pasadena, Calif., 1956.**[9]**C. A. Swanson and V. B. Headley,*An extension of Airy’s equation*, SIAM J. Appl. Math.**15**(1967), 1400–1412. MR**0224883**, https://doi.org/10.1137/0115123**[10]**Wolfgang Wasow,*Asymptotic expansions for ordinary differential equations*, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0203188****[11]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110**

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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