POTW Estimate of a Principal Value Integral

[Euge]

For $x\in \mathbb{R}$, let $$A(x) = \frac{1}{2\pi}\, P.V. \int_{-\infty}^\infty e^{i(xy + \frac{y^3}{3})}\, dy$$ Show that the integral defining $A(x)$ exists and $|A(x)| \le M(1 + |x|)^{-1/4}$ for some numerical constant $M$.

 

[pasmith]

I assume by A(x) you mean \operatorname{Ai}(x) or vice-versa.

 

[Euge]

I meant $A(x)$, sorry.