Laplace endpoint estimate
= Laplace endpoint estimate
{c}
{title2=$\frac{a(b)e^{\lambda\phi(b)}}{\lambda\phi\prime(b)}$}
If a real smooth phase has its strict maximum at the upper endpoint $b$ and $\phi\prime(b)>0$, then $\int_a^b a(t)e^{\lambda\phi(t)}dt\sim a(b)e^{\lambda\phi(b)}/(\lambda\phi\prime(b))$. The endpoint coordinate $b-t$ has scale $1/\lambda$. At a nondegenerate stationary endpoint the scale is instead $\lambda^{-1/2}$ and the leading contribution is half a <Gaussian integral>.