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The question is quite formal. I recall the definition of Airy function $$Ai(\tau^{2/3}\zeta)=\frac{\tau^{1/3}}{2\pi}\int e^{i(\sigma^3/3+\sigma\zeta)}d\sigma,\quad Ai'(\tau^{2/3}\zeta)=\frac{i\tau^{1/3}}{2\pi}\int e^{i(\sigma^3/3+\sigma\zeta)}\tau^{1/3}\sigma d\sigma.$$ I recall moreover that $Ai(x)$ is the solution of the ODE $y''+xy=0$. My question is the following. Assume that $g(\tau,\sigma)$ is a smooth function decaying in $\sigma$ uniformly in $\tau$. Is it true that there are functions $g_1(\tau)$ and $g_2(\tau)$ such that $$\tau^{1/3}\int e^{i\tau(\sigma^3/3+\sigma\zeta)}g(\tau,\sigma)d\sigma=g_1(\tau)Ai(\tau^{2/3}\zeta)+g_2(\tau)\tau^{-1/3}Ai'(\tau^{2/3}\zeta)?$$

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