 Two-term monotone-endpoint Laplace expansion

For a smooth real phase decreasing away from its endpoint maximum, put $t=\varphi (a)-\varphi (x)$ and write the transformed amplitude as $G(t)=-g(x(t))/{\varphi }^{\prime }(x(t))$. Its first two Taylor coefficients are $G(0)=-g(a)/{\varphi }^{\prime }(a)$ and $G^{\prime }(0)=g^{\prime }(a)/{\varphi }^{\prime }(a{)}^2-g(a){\varphi }^{\prime \prime }(a)/{\varphi }^{\prime }(a{)}^3$. Applying Watson lemma gives the displayed two-term Laplace integral expansion. Localization and a Taylor remainder estimate justify termwise integration.