 Cubic oscillatory Gamma integral

ID: cubic-oscillatory-gamma-integral

For

t > 0

, contour rotation through angle

- π /6

gives

\int_{0}^{\infty} e^{- i t u^{3}} d u = e^{- iπ /6} t^{- 1/3} Γ (\frac{4}{3}) .

Taking real parts yields

\int_{0}^{\infty} cos (u^{3}) d u = Γ (\frac{4}{3}) cos \frac{π}{6} .

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