 Laplace's method

If a smooth real phase $\varphi$ has a unique nondegenerate interior maximum at $t_0$, thenFor a maximum at an endpoint $b$ with ${\varphi }^{\prime }(b)>0$, the leading factor is $a(b)e^{x\varphi (b)}/(x{\varphi }^{\prime }(b))$.

Laplace's method is a technique used in asymptotic analysis to approximate integrals of the form \[ I_n = \int_{a}^{b} e^{n f(x)} g(x) \, dx \] as \( n \) becomes large, where \( f(x) \) is a smooth function and \( g(x) \) is another function that is reasonably well-behaved.