Fresnel integral
= Fresnel integral
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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 $\omega$.