I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{xix^2},$$ or to know where I can find some techniques to have some asymptotical expansion of such Fourier transform.

2$\begingroup$ it's a Fresnel integral $\endgroup$– Carlo BeenakkerJun 5, 2015 at 11:12

$\begingroup$ Thanks. I noticed it, but are there some explicit expression for these integrals? $\endgroup$– Felice IandoliJun 5, 2015 at 11:50

2$\begingroup$ well, it's called a Fresnel integral because it cannot be written in terms of elementary functions (very much like the error function) $\endgroup$– Carlo BeenakkerJun 5, 2015 at 11:53

$\begingroup$ Doesn't repeated integration by parts of the Fourier transform integral give an asymptotic expansion? $\endgroup$– paul garrettJun 5, 2015 at 14:36