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Gamma function

Gamma integral

For $t>0$, contour rotation through angle $-\pi/6$ gives
$$
\int_0^\infty e^{-itu^3}du
=e^{-i\pi/6}t^{-1/3}\Gamma\left(\frac43\right).
$$
Taking real parts yields
$$
\int_0^\infty\cos(u^3)du
=\Gamma\left(\frac43\right)\cos\frac\pi6.
$$


Cubic oscillatory Gamma integral

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

Cubic oscillatory Gamma integral

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Cubic oscillatory Gamma integral

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