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Let $a_i$ be the zeros of the Airy function, which is the solution top the ODE $y''-xy=0$, such that Ai(a_i)=0. According to WolframMathWorld e.g., $a_{1..4}= -2.33811, -4.08795, -5.52056, -6.7867144$.

Is there a way to obtain an asymptotic formula (as opposed to the numerical approach) for the real roots of the Airy function, for large $i$?

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Yes, and this is called WKB method by physicists and Green-Liouville method by mathematicians. The idea is to approximate Airy's function itself, and then locate its zeros by an elementary use of Rouche's theorem. To approximate Airy's function, you rewrite the differential equation as an integral equation, and solve it by successive approximation.

For details, see, for example Olver, Asymptotics and special functions.

Remarks. It helps to know in advance that all these zeros are real. This is because they are simply related to eigenvalues of a self-adjoint boundary value problem for the Airy equation with boundary conditions $y(0)=y(+\infty)=0$.

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If you're only interested in the result rather than the method for obtaining it, you can find it online in the DLMF at http://dlmf.nist.gov/9.9

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