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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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    $\begingroup$ it's a Fresnel integral $\endgroup$ Jun 5, 2015 at 11:12
  • $\begingroup$ Thanks. I noticed it, but are there some explicit expression for these integrals? $\endgroup$ Jun 5, 2015 at 11:50
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    $\begingroup$ well, it's called a Fresnel integral because it cannot be written in terms of elementary functions (very much like the error function) $\endgroup$ Jun 5, 2015 at 11:53
  • $\begingroup$ Doesn't repeated integration by parts of the Fourier transform integral give an asymptotic expansion? $\endgroup$ Jun 5, 2015 at 14:36

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