Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 79 / 2 / a / Solution

The factorial tail provides the connection with the supplied Borel summation formula. Near the first switching line choose $\lambda =-z^3$ with $\arg \lambda =3(\arg z-\pi /3)$, so $\lambda$ is near the positive real axis and ${\lambda }^{-1/3}=e^{i\pi /3}/z$. The endpoint contribution through $N-1$ terms isWriting $r=N+p$ in its formal remainder, with $\gamma =N+2/3$, givesThus the Borel remainder for a cubic exponential integral, using the contour just above the pole specified in the paper, isThe continuation is chosen from the sector below the switching line. The term $C/z$ fixes that continuation; an alternative lateral pole contour would shift it by the corresponding residue. This is why the contour prescription must accompany the divergent factorial series.