Airy Function Zeros

 and  have zeros on the negative real axis only.  and  have zeros on the negative real axis and in the sector . The th (real) roots on the negative real axis of , , , and  are denoted , , , and , respectively (Abramowitz and Stegun 1972, p. 450; Fabijonas 2004; Fabijonas et al. 2004). Similarly, the th complex root of  and  in the sector  and of  and  in the sector  are denoted , , , and , respectively (Abramowitz and Stegun 1972, p. 450; Fabijonas 2004; Fabijonas et al. 2004).

The first few real roots of  are approximately  (OEIS A096714), , , , , , .... Similarly, the first few real roots of  are approximately  (OEIS A096715),, , , , , , ....

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Airy Functions." §10.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 446-452, 1972.

Fabijonas, B. R. "Algorithm 838: Airy Functions." ACM Trans. Math. Software 30, 491-501, 2004.

Fabijonas, B. R.; Lozier, D. W.; and Olver, F. W. J. "Computation of Complex Airy Functions and Their Zeros Using Asymptotics and the Differential Equation." ACM Trans. Math. Software 30, 471-490, 2004.

Sherry, M. "The Zeros and Maxima of the Airy Function and Its First Derivative to 25 Significant Figures." Air Force Cambridge Research Center. Tech. Rep. AFCRC-TR-59-135, ASTIA AD214568. April, 1959.