Laplace's method at a nondegenerate stationary endpoint

If the maximum $t_0$ of the smooth real phase $\varphi$ is an endpoint, ${\varphi }^{\prime }(t_0)=0$, and ${\varphi }^{\prime \prime }(t_0)<0$, the local Gaussian is integrated over only one side of its peak. Consequentlywith the analogous formula for a right endpoint.