Oscillatory Gaussian integral with negative phase

For

t > 0

, contour rotation of the Gaussian integral gives

\int_{- \infty}^{\infty} e^{- i t u^{2}} d u = e^{- iπ /4} \frac{π}{t} .

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 31B / Solution

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