Laplace's method
= Laplace's method
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{wiki}
= Laplace method
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If a smooth real phase $\phi$ has a unique nondegenerate interior maximum at $t_0$, then
$$
\int a(t)e^{x\phi(t)}dt
\sim a(t_0)e^{x\phi(t_0)}
\sqrt{\frac{2\pi}{x|\phi''(t_0)|}}.
$$
For a maximum at an endpoint $b$ with $\phi'(b)>0$, the leading factor is $a(b)e^{x\phi(b)}/(x\phi'(b))$.