I want Fourier transformation of $\exp({-8\pi^3 x^3 i})$.
Could you give me some hint?
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Mathematica is not amused : $$ \frac{(\left|t \right|+t)\text{Ai}\left(- \frac{\left|t \right|}{2\sqrt[3]{3}\pi } \right)+(\left|t \right|-t)\text{Ai}\left( \frac{\left|t \right|}{2\sqrt[3]{3} \pi} \right)}{2\sqrt[3]{3}\sqrt{2\pi}\left|t \right|} $$ – Zubzub Jun 06 '17 at 14:42
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Off hand I wouldn't expect the Fourier transform to be expressible as a function, since your function is not $L^1$. (It blows up for large negative $x$) The result will likely be a distribution as with the Fourier transform for $e^x$, but it would be hard to show that using the integral definition of the transform. – Joel Jun 06 '17 at 14:44