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Airy Function Zeros


Ai(z) and Ai^'(z) have zeros on the negative real axis only. Bi(z) and Bi^'(z) have zeros on the negative real axis and in the sector pi/3<|argz|<pi/2. The nth (real) roots on the negative real axis of Ai(z), Ai^'(z), Bi(z), and Bi^'(z) are denoted a_n, a_n^', b_n, and b_n^', respectively (Abramowitz and Stegun 1972, p. 450; Fabijonas 2004; Fabijonas et al. 2004). Similarly, the nth complex root of Bi(z) and Bi^'(z) in the sector pi/3<argz<pi/2 and of Bi(z) and Bi^'(z) in the sector -pi/2<argz<pi/3 are denoted beta_n, beta_n^', beta^__n, and beta^__n, respectively (Abramowitz and Stegun 1972, p. 450; Fabijonas 2004; Fabijonas et al. 2004).

AiryRoots

The first few real roots of Ai(x) are approximately -2.33811 (OEIS A096714), -4.08795, -5.52056, -6.7867144, -7.94413, -9.02265, .... Similarly, the first few real roots of Bi(x) are approximately -1.17371 (OEIS A096715), -3.27109, -4.83074, -6.16985, -7.37676, -8.49195, -9.53819, ....


See also

Airy Functions, Root, Zero

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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.

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Airy Function Zeros

Cite this as:

Weisstein, Eric W. "Airy Function Zeros." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AiryFunctionZeros.html

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