= Degenerate endpoint in Laplace's method
Suppose a real phase has its unique minimum at the left endpoint and $\phi(t)-\phi(0)\sim ct^m$, with $c>0$ and integer $m\geq1$. For a continuous amplitude $a$ with $a(0)\ne0$, localization followed by $s=(\nu c)^{1/m}t$ gives
$$
\int_0^\infty a(t)e^{-\nu\phi(t)}\,dt\sim a(0)e^{-\nu\phi(0)}\frac{\Gamma(1/m)}{m(\nu c)^{1/m}}.
$$
This uses the <Gamma integral>. It assumes sufficient decay away from the minimum to make the remaining integral negligible. A cubic endpoint has width $\nu^{-1/3}$; an ordinary quadratic endpoint has width $\nu^{-1/2}$.
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