= Two-term monotone-endpoint Laplace expansion
{title2=$e^{\lambda\phi(a)}[-g(a)/(\lambda\phi'(a))+(g'(a)/\phi'(a)^2-g(a)\phi''(a)/\phi'(a)^3)/\lambda^2]$}
For a smooth real phase decreasing away from its endpoint maximum, put $t=\phi(a)-\phi(x)$ and write the transformed amplitude as $G(t)=-g(x(t))/\phi'(x(t))$. Its first two Taylor coefficients are $G(0)=-g(a)/\phi'(a)$ and $G'(0)=g'(a)/\phi'(a)^2-g(a)\phi''(a)/\phi'(a)^3$. Applying <Watson lemma> gives the displayed two-term <Laplace integral> expansion. Localization and a Taylor remainder estimate justify termwise integration.
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