Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 30C / b / Solution

The endpoint $t=0$ is treated by rotating onto its method of steepest descent ray. It contributesAt $t=1$, put $s=1-t$ and expandTermwise endpoint integration, using ${\int }_0^{\infty }{s}^n{e}^{-ixs}ds=n!(ix{)}^{-n-1}$ on the rotated ray, giveswhereThe two displayed contributions come from the two endpoints; the remaining deformed contour is exponentially or algebraically smaller after any fixed truncation.