Endpoint contribution in steepest descent

If a deformed contour starts at an endpoint $z_0$ with ${\varphi }^{\prime }(z_0)\ne 0$ and initially enters a decaying sector, local integration gives the leading contribution $-a(z_0)e^{\lambda \varphi (z_0)}/[\lambda {\varphi }^{\prime }(z_0)]$, with the sign adjusted to the contour orientation.