# How to obtain an asymptotic formula for the zeros of the Airy function ($a_i$ for large $i$)?

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 $i$?

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 $y(0)=y(+\infty)=0$.