= Borel remainder for a cubic exponential integral
{c}
{title2=$R_N=\frac C z+\frac{e^{i\pi/3}}{3\Gamma(2/3)z}I(-z^3,N+2/3)$}
Near $\arg z=\pi/3$, set $\lambda=-z^3$ on the branch with argument near zero and truncate the endpoint expansion at $N$ terms. Its remaining factorial series, indexed by $r=N+p$, is $\lambda^{-1/3}/[3\Gamma(2/3)]$ times the <Borel summation> of $\sum_{p\geq0}\Gamma(N+2/3+p)e^\lambda/\lambda^{N+2/3+p}$. The upper-pole contour fixes the stated remainder and its continuation from the sector below the switching line. With $N+2/3\sim|z|^3$, the <Gaussian pole transition for a large gamma parameter> smooths the algebraic coefficient from $C$ to $Ce^{2\pi i/3}$.
Back to article page
