Oscillatory Gaussian integral with negative phase
= Oscillatory Gaussian integral with negative phase
For $t>0$, contour rotation of the <Gaussian integral> gives
$$
\int_{-\infty}^{\infty}e^{-itu^2}du
=e^{-i\pi/4}\sqrt{\frac\pi t}.
$$
= Oscillatory Gaussian integral with negative phase
For $t>0$, contour rotation of the <Gaussian integral> gives
$$
\int_{-\infty}^{\infty}e^{-itu^2}du
=e^{-i\pi/4}\sqrt{\frac\pi t}.
$$