For $\omega\ne0$, insert the Gaussian damping factor $e^{-\varepsilon x^2/2}$ and use the <Gaussian integral>:
$$
\int_{\mathbb R}
e^{-(\varepsilon-i\omega)x^2/2-i\lambda x}\,dx
=\sqrt{\frac{2\pi}{\varepsilon-i\omega}}
\exp\left[-\frac{\lambda^2}{2(\varepsilon-i\omega)}\right].
$$
Taking $\varepsilon\downarrow0$ in $\mathcal S'$ with the continuous square-root branch gives the <Fresnel integral>
$$
\boxed{
\mathcal F\left[e^{i\omega x^2/2}\right](\lambda)
=\sqrt{\frac{2\pi}{|\omega|}}
e^{i\pi\operatorname{sgn}(\omega)/4}
e^{-i\lambda^2/(2\omega)}}.
$$
For $\omega=0$, the function is constant and its Fourier transform is $\boxed{2\pi\delta_0}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 327 / 3 / a / Solution

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The Fresnel integral is an oscillatory Gaussian integral. In the angular-frequency convention, its distributional form is
$$
\int_{\mathbb R}e^{i\omega x^2/2-i\lambda x}\,dx
=\sqrt{\frac{2\pi}{|\omega|}}
e^{i\pi\operatorname{sgn}(\omega)/4}
e^{-i\lambda^2/(2\omega)}
$$
for nonzero real $\omega$.


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