I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to have some asymptotical expansion of such Fourier transform.
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2$\begingroup$ it's a Fresnel integral $\endgroup$– Carlo BeenakkerCommented Jun 5, 2015 at 11:12
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$\begingroup$ Thanks. I noticed it, but are there some explicit expression for these integrals? $\endgroup$– Felice IandoliCommented Jun 5, 2015 at 11:50
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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 BeenakkerCommented Jun 5, 2015 at 11:53
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$\begingroup$ Doesn't repeated integration by parts of the Fourier transform integral give an asymptotic expansion? $\endgroup$– paul garrettCommented Jun 5, 2015 at 14:36
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